Celebratio Mathematica — Benkart

[1] M. Benkart, Georgia: Inner ideals and the structure of Lie algebras. Ph.D. thesis, Yale University, 1974. Advised by N. Jacobson. MR 2625003 phdthesis

[2] G. Benkart: “The Lie inner ideal structure of associative rings,” J. Algebra 43 : 2 (December 1976), pp. 561–584. MR 435149 Zbl 0342.16009 article

[3] G. M. Benkart and I. M. Isaacs: “On the existence of ad-nilpotent elements,” Proc. Am. Math. Soc. 63 : 1 (1977), pp. 39–40. MR 432721 Zbl 0359.17008 article

[4] G. Benkart: “On inner ideals and ad-nilpotent elements of Lie algebras,” Trans. Am. Math. Soc. 232 (1977), pp. 61–81. MR 466242 Zbl 0373.17003 article

An inner ideal of a Lie algebra \( L \) over a commutative ring \( k \) is a \( k \)-submodule \( B \) of \( L \) such that \( [B[BL]] \subseteq B \). This paper investigates properties of inner ideals and obtains results relating ad-nilpotent elements and inner ideals. For example, let \( L \) be a simple Lie algebra in which \( D_y^2 = 0 \) implies \( y = 0 \), where \( D_y \) denotes the adjoint mapping determined by \( y \). If \( L \) satisfies the descending chain condition on inner ideals and has proper inner ideals, then \( L \) contains a subalgebra \( S = \langle e,f,h\rangle \), isomorphic to the split 3-dimensional simple Lie algebra, such that \( D_e^3 = D_f^3 = 0 \). Lie algebras having such 3-dimensional subalgebras decompose into the direct sum of two copies of a Jordan algebra, two copies of a special Jordan module, and a Lie subalgebra of transformations of the Jordan algebra and module. The main feature of this decomposition is the correspondence between the Lie and the Jordan structures. In the special case when \( L \) is a finite dimensional, simple Lie algebra over an algebraically closed field of characteristic \( p > 5 \) this decomposition yields:

\( L \) is classical if and only if there is an \( x \neq 0 \) in \( L \) such that \( D_x^{p-1} = 0 \) and if \( D_y^2 = 0 \) implies \( y = 0 \).

The proof involves actually constructing a Cartan subalgebra which has 1-dimensional root spaces for nonzero roots and then using the Block axioms.

[5] G. M. Benkart, I. M. Isaacs, and J. M. Osborn: “Lie algebras with self-centralizing ad-nilpotent elements,” J. Algebra 57 : 2 (April 1979), pp. 279–309. MR 533800 Zbl 0402.17013 article

Irving Martin Isaacs	Related
J. Marshall Osborn	Related

[6] G. M. Benkart, I. M. Isaacs, and J. M. Osborn: “Albert–Zassenhaus Lie algebras and isomorphisms,” J. Algebra 57 : 2 (April 1979), pp. 310–338. MR 533801 Zbl 0402.17014 article

Irving Martin Isaacs	Related
J. Marshall Osborn	Related

[7] G. M. Benkart and I. M. Isaacs: “Lie algebras with nilpotent centralizers,” Canad. J. Math. 31 : 5 (1979), pp. 929–941. MR 546949 Zbl 0373.17004 article

[8] G. Benkart: “Derivations and automorphisms of matrices symmetric relative to a canonical involution,” J. Algebra 62 : 2 (February 1980), pp. 418–429. To Nathan Jacobson on his 70th birthday. MR 563238 Zbl 0424.17007 article

[9] G. M. Benkart, J. M. Osborn, and D. J. Britten: “Flexible Lie-admissible algebras with the solvable radical of \( A^- \) abelian and Lie algebras with nondegenerate forms,” Hadronic J. 4 : 2 (1980–1981), pp. 274–326. MR 613337 Zbl 0456.17002 article

If \( A \) is a flexible Lie-admissible algebra, then \( A \) under the product \( (xy) = xy - yx \) is a Lie algebra, denoted by \( A^- \). This paper investigates finite-dimensional, simple, flexible, Lie-admissible algebra \( A \) over an algebraically closed field of characteristic zero, for which the solvable radical \( R \) of \( A^- \) is Abelian. The technique employed is to view \( A \) as a module for a semisimple Lie algebra of derivations, and then to use representation theory to gain information about products in \( A \). In the final section of the paper we construct examples of simple flexible Lie-admissible algebras from Lie algebras with nondegenerate associative symmetric bilinear forms. These examples illustrate the great diversity of algebras which can occur when the assumption that \( R \) is Abelian is dropped.

Daniel John Britten	Related
J. Marshall Osborn	Related

[10] G. Benkart and J. M. Osborn: “Real division algebras and other algebras motivated by physics,” pp. 392–443 in Proceedings of the third workshop on Lie-admissible formulations (4–9 August 1980, Boston), published as Hadronic J. 4 : 2. Hadronic Press (Nonantum, MA), 1980–1981. MR 613340 Zbl 0456.17005 incollection

In this survey we discuss several general techniques which have been productive in the study of real division algebras, flexible Lie-admissible algebras, and other nonassociative algebras, and we summarize results obtained using these methods. The principal method involved in this work is to view an algebra \( A \) as a module for a semisimple Lie algebra of derivations of \( A \) and to use representation theory to study products in \( A \). In the case of real division algebras, we also discuss the use of isotopy and the use of a generalized Peirce decomposition. Most of the work summarized here has appeared in more detail in various other papers. The exceptions are results on a class of algebras of dimension 15, motivated by physics, which admit the Lie algebra \( sl(3) \) as an algebra of derivations.

[11] G. Benkart, J. M. Osborn, and D. Britten: “On applications of isotopy to real division algebras,” pp. 497–529 in Proceedings of the third workshop on Lie-admissible formulations (4–9 August 1980, Boston), published as Hadronic J. 4 : 2. Hadronic Press (Nonantum, MA), 1980–1981. MR 613342 Zbl 0451.17002 incollection

In this paper we illustrate how the notion of isotopy can be used to solve various problems concerning finite-dimensional division algebras over the real numbers. In particular, we show that the 8-dimensional division algebras which have the same derivation algebra as the octonions, and hence which most resemble the octonions, are not in general isotopes of the octonions. Secondly, using a result of Hopf, we argue that every commutative division algebra is the reals or is isomorphic to a special kind of isotope of the complex numbers. Finally, by considering a certain class of algebras, we show how isotopy is a useful tool for determining necessary and sufficient conditions on the multiplication constants in order to have a division algebra.

Daniel John Britten	Related
J. Marshall Osborn	Related

[12] G. M. Benkart and J. M. Osborn: “Derivations and automorphisms of nonassociative matrix algebras,” Trans. Am. Math. Soc. 263 : 2 (1981), pp. 411–430. MR 594417 Zbl 0453.16020 article

[13] G. M. Benkart and J. M. Osborn: “Flexible Lie-admissible algebras,” J. Algebra 71 : 1 (July 1981), pp. 11–31. MR 627422 Zbl 0467.17001 article

This paper investigates finite-dimensional flexible Lie-admissible algebras \( A \) over fields of characteristic 0. Under these hypotheses the vector space \( A \) with the Lie product \( [x,y] = xy - yx \) is a Lie algebra, denoted by \( A^- \). The main result of this work gives a characterization of those flexible Lie-admissible algebras for which the solvable radical of \( A^- \) is a direct summand of \( A^- \). Included in this class of algebras are all flexible Lie-admissible \( A \) for which \( A^- \) is a reductive Lie algebra. Our technique is to view \( A \) as a module for a certain semisimple Lie algebra of derivations of \( A \) and to see what restrictions the module structure imposes on the multiplication of \( A \). \( A \) subsequent investigation will show that this module approach can also be used to determine the flexible Lie-admissible algebras \( A \) for which the radical of \( A^- \) is abelian.

[14] G. M. Benkart and J. M. Osborn: “The derivation algebra of a real division algebra,” Am. J. Math. 103 : 6 (December 1981), pp. 1135–1150. MR 636955 Zbl 0474.17002 article

[15] G. M. Benkart and J. M. Osborn: “An investigation of real division algebras using derivations,” Pacific J. Math. 96 : 2 (1981), pp. 265–300. MR 637973 Zbl 0474.17003 article

[16] G. M. Benkart: “The construction of examples of Lie-admissible algebras,” pp. 461–493 in Proceedings of the first international conference on nonpotential interactions and their Lie-admissible treatment (5–9 January 1982, Orléans, France), published as Hadronic J. 5 : 5. Hadronic Press (Nonantum, MA), 1981–1982. MR 659292 Zbl 0481.17009 incollection

In this discussion we examine general methods of constructing Lie-admissible algebras. Many of the techniques which we survey have been known since the inception of Lie-admissible studies, many have developed as the subject has evolved especially in the last five years, and many are appearing for the first time in this article. In presenting these examples we have taken two different approaches. First we have looked for common themes to unite the seemingly disparate array of algebras in the literature. In this regard we discuss six general classes of Lie-admissible algebras: (1) algebras arising by adjoining a symmetric multiplication to a Lie algebra product; (2) deformations and cohomology extensions of Lie, associative and Lie-admissible algebras; (3) mutation algebras; (4) nodal algebras; (5) Lie superalgebras; and (6) algebras resulting from structure theorems. Our second approach is to single out particular algebras within our general classes to illustrate with special cases how some concrete calculations might proceed. The special cases studied have been chosen because of their potential physical relevance.

[17] G. M. Benkart and J. M. Osborn: “Power-associative products on matrices,” pp. 1859–1892 in Proceedings of the first international conference on nonpotential interactions and their Lie-admissible treatment (5–9 January 1982, Orléans, France), published as Hadronic J. 5 : 5. Hadronic Press (Nonantum, MA), 1981–1982. MR 683312 Zbl 0507.17008 incollection

The main purpose of this work is to classify all power-associative products \( * \) that can be defined on the algebra \( A \) of \( n{\times}n \) matrices over a field \( \mathbf{F} \) of characteristic not 2 or 3, satisfying the condition that \[ x*y - y*x = xy - yx \] for all \( x,y\in A \), where \( xy \) denotes the usual associative prouct in \( A \). Such products \( * \) are automatically Lie-admissible, and it is shown that they are Jordan-admissible also. The motivation for this investigation comes from recent studies of the spin of a proton or neutron using such products defined on the associative envelope of the spin \( 1/2 \) matrices. Included in the present paper are results showing how the classification of the products \( * \) on \( n{\times}n \) matrices can be carried over to the associative envelope of spin \( s \) matrices for arbitrary \( s \).

[18] G. M. Benkart, D. J. Britten, and J. M. Osborn: “Real flexible division algebras,” Canad. J. Math. 34 : 3 (1982), pp. 550–588. MR 663304 Zbl 0469.17001 article

In this paper we classify finite-dimensional flexible division algebras over the real numbers. We show that every such algebra is either (i) commutative and of dimension one or two, (ii) a slight variant of a noncommutative Jordan algebra of degree two, or (iii) an algebra defined by putting a certain product on the \( 3{\times} 3 \) complex skew-Hermitian matrices of trace zero. A precise statement of this result is given at the end of this section after we have developed the necessary background and terminology. In Section 3 we show that, if one also assumes that the algebra is Lie-admissible, then the structure follows rapidly from results in [Benkart and Osborn 1981a, 1981b].

Daniel John Britten	Related
J. Marshall Osborn	Related

[19] G. M. Benkart and J. M. Osborn: “Representations of rank one Lie algebras of characteristic \( p \),” pp. 1–37 in Lie algebras and related topics (29–31 May 1981, New Brunswick, NJ). Edited by D. Winter. Lecture Notes in Mathematics 933. Springer (Berlin), 1982. MR 675104 Zbl 0491.17003 incollection

David J. Winter	Related
J. Marshall Osborn	Related

[20] G. M. Benkart and J. M. Osborn: “On the determination of rank one Lie algebras of prime characteristic,” pp. 263–265 in Algebraists’ homage: Papers in ring theory and related topics. Edited by S. A. Amitsur, D. J. Saltman, and G. B. Seligman. Contemporary Mathematics 13. American Mathematical Society (Providence, RI), 1982. Zbl 0504.17003 incollection

David J. Saltman	Related
Shimshon Avraham Amitsur	Related
George Benham Seligman	Related
J. Marshall Osborn	Related

[21] G. M. Benkart: “Bimodules for flexible Lie-admissible algebras,” Algebras Groups Geom. 1 : 1 (1984), pp. 109–126. MR 744732 Zbl 0535.17013 article

[22] G. M. Benkart and J. M. Osborn: “Rank one Lie algebras,” Ann. of Math. (2) 119 : 3 (May 1984), pp. 437–463. MR 744860 Zbl 0563.17011 article

[23] G. M. Benkart: “Power-associative Lie-admissible algebras,” J. Algebra 90 : 1 (September 1984), pp. 37–58. MR 757079 Zbl 0542.17012 article

Associated with any algebra \( A \) over a field of characteristic not 2 are two algebras denoted by \( A^- \) and \( A^+ \). These algebras have the same underlying vector space as \( A \), but are given the products \[ [x, y] = x*y - y*x \quad\text{and}\quad x \circ y = \tfrac{1}{2}(x*y + y*x) ,\] respectively, where \( * \) is the multiplication in \( A \). The algebra \( A \) is said to be Lie-admissible if \( A^- \) is a Lie algebra. If each element in \( A \) generates an associative subalgebra of \( A \) under the \( * \) product, then \( A \) is called power-associative. In this investigation we determine all finite-dimensional power-associative Lie-admissible algebras \( A \) over a field of characteristic zero such that \( A^- \) is a semisimple Lie algebra. In fact, we determine these algebras under an assumption weaker than power associativity, namely that the third power identity \[ x * (x * x) = (x * x) * x \] holds for all \( x \) in \( A \).

The motivation to study this class of algebras comes from the Lie-admissible formulations [Santilli 1992] proposed in the treatment of forces not derivable from a potential. In these physical contexts power-associativity is generally required to have a well-defined notion of exponential.

[24] G. Benkart: “A Kac–Moody bibliography and some related references,” pp. 111–135 in Lie algebras and related topics (26 June–6 July 1984, Windsor, ON). Edited by D. J. Britten, F. W. Lemire, and R. V. Moody. CMS Conference Proceedings 5. American Mathematical Society (Providence, RI), 1986. MR 832196 Zbl 0578.17013 incollection

Frank W. Lemire	Related
Robert V. Moody	Related
Daniel John Britten	Related

[25] G. Benkart: “Cartan subalgebras in Lie algebras of Cartan type,” pp. 157–187 in Lie algebras and related topics (26 June–6 July 1984, Windsor, ON). Edited by D. J. Britten, F. W. Lemire, and R. V. Moody. CMS Conference Proceedings 5. American Mathematical Society (Providence, RI), 1986. MR 832198 Zbl 0581.17006 incollection

The known finite dimensional simple Lie algebras over an algebraically closed field of characteristic \( p\geq 7 \) are either classical (analogues of complex Lie algebras) or of Cartan type. Each Lie algebra \( L \) of Cartan type possesses a certain maximal subalgebra \( L_0 \) which is invariant under the automorphisms of \( L \), and every Cartan subalgebra \( L_0 \) is a Cartan subalgebra of \( L \). The main result of this work is to show that the Cartan subalgebras of \( L_0 \) are conjugate under the group generated by the generalized exponential mappings defined by \[ E^z = 1 + \sum_{j=1}^{p-1} \left(\prod_{\nu = 1}^j (\nu \vert -\psi)^{-1}\right)(adz)^j \] where \( z\in L_0 \) has \( (adz)^{p^k} = 0 \), and where \[ \psi = (adz)^p + \cdots + (adz)^{p^{k-1}} .\] As a consequence, all the Cartan subalgebras of \( L \) in \( L_0 \) have the same dimension, and this dimension is computed for each type of algebra.

Frank W. Lemire	Related
Robert V. Moody	Related
Daniel John Britten	Related

[26] G. M. Benkart and R. V. Moody: “Derivations, central extensions, and affine Lie algebras,” Algebras Groups Geom. 3 : 4 (1986), pp. 456–492. MR 901810 Zbl 0619.17014 article

[27] G. Benkart and J. M. Osborn: “Toral rank one Lie algebras,” J. Algebra 115 : 1 (May 1988), pp. 238–250. MR 937612 Zbl 0644.17009 article

[28] G. M. Benkart, T. B. Gregory, J. M. Osborn, H. Strade, and R. L. Wilson: “Isomorphism classes of Hamiltonian Lie algebras,” pp. 42–57 in Lie algebras (23–28 August 1987, Madison, WI). Edited by G. Benkart and J. M. Osborn. Lecture Notes in Mathematics 1373. Springer (Berlin), 1989. MR 1007323 Zbl 0677.17012 incollection

Helmut Strade	Related
Thomas Bradford Gregory	Related
Robert Lee Wilson	Related
J. Marshall Osborn	Related

[29] G. Benkart and T. Gregory: “Graded Lie algebras with classical reductive null component,” Math. Ann. 285 : 1 (1989), pp. 85–98. MR 1010192 Zbl 0648.17005 article

[30] Lie algebras (23–28 August 1987, Madison, WI). Edited by G. Benkart and J. M. Osborn. Lecture Notes in Mathematics 1373. Springer (Berlin), 1989. Zbl 0661.00006 book

[31] G. M. Benkart, D. J. Britten, and F. W. Lemire: Stability in modules for classical Lie algebras — a constructive approach. Memoirs of the American Mathematical Society 430. American Mathematical Society (Providence, RI), 1990. MR 1010997 Zbl 0706.17003 book

Frank W. Lemire	Related
Daniel John Britten	Related

[32] G. Benkart: “Simple modular Lie algebras with 1-sections that are classical or solvable,” Comm. Algebra 18 : 11 (1990), pp. 3633–3638. MR 1068610 Zbl 0723.17019 article

[33] Lie algebras and related topics (22 May–1 June 1988, Madison, WI). Edited by G. Benkart and J. M. Osborn. Contemporary Mathematics 110. American Mathematical Society (Providence, RI), 1990. MR 1079096 Zbl 0704.00015 book

[34] G. Benkart: “Partitions, tableaux, and stability in the representation theory of classical Lie algebras,” pp. 47–76 in Lie theory, differential equations and representation theory (1–11 August 1989, Montreal). Edited by V. Hussin. Publications CRM (Montreal), 1990. MR 1121952 Zbl 0735.17007 incollection

[35] G. Benkart and J. M. Osborn: “Simple Lie algebras of characteristic \( p \) with dependent roots,” Trans. Am. Math. Soc. 318 : 2 (April 1990), pp. 783–807. MR 955488 Zbl 0703.17009 article

We investigate finite dimensional simple Lie algebras over an algebraically closed \( \mathbf{F} \) of characteristic \( p\geq 7 \) having a Cartan subalgebra \( H \) whose roots are dependent over \( \mathbf{F} \). We show that \( H \) must be one-dimensional or for some root \( \alpha \) relative to \( H \) there is a 1-section \( L^{(\alpha)} \) such that the core of \( L^{(\alpha)} \) is a simple Lie algebra of Cartan type \[ H(2:\underline{m}:\Phi)^{(2)} \quad\text{or}\quad W(1:\underline{n}) \] for some \( n > 1 \). The results we obtain have applications to studying the local behavior of simple Lie algebras and to classifying simple Lie algebras which have a Cartan subalgebra of dimension less than \( p - 2 \).

[36] G. Benkart and J. Stroomer: “Tableaux and insertion schemes for spinor representations of the orthogonal Lie algebra \( so(2r+1,\mathbb{C}) \),” J. Combin. Theory Ser. A 57 : 2 (July 1991), pp. 211–237. MR 1111558 Zbl 0747.17006 article

[37] G. Benkart, D. Britten, and F. Lemire: “Projection maps for tensor products of \( \mathfrak{gl}(r,\mathbb{C}) \)-representations,” Publ. Res. Inst. Math. Sci. 28 : 6 (1992), pp. 983–1010. MR 1203757 Zbl 0830.17004 article

We investigate the tensor product \[ \mathcal{T} = V(\lambda^1)\otimes \cdots \otimes V(\lambda^m) \] of the finite dimensional irreducible \( \mathcal{G} = \mathfrak{gl}(r,\mathbb{C}) \) modules labelled by partitions \( \lambda^1,\dots \), \( \lambda^m \) of \( m \) not necessarily distinct numbers \( n_1,\dots \), \( n_m \) respectively. We determine the centralizer algebra \( \operatorname{End}_{\mathcal{G}}(\mathcal{T}) \) and the projection maps of \( \mathcal{T} \) onto its irreducible \( \mathcal{G} \)-summands and give an explicit construction of the corresponding maximal vectors. In the special case that \( n_i = 1 \) for \( i = 1,\dots \), \( m \), the results reduce to the well-known results of Schur and Weyl.

Frank W. Lemire	Related
Daniel John Britten	Related

[38] G. Benkart and J. Stroomer: “A combinatorial model for tensor products of the spin representation,” pp. 37–51 in Hadronic mechanics and nonpotential interactions (13–17 August 1990, Cedar Falls, IA), part 1: Mathematics. Edited by H. C. Myung. Nova Science Publishers, 1992. MR 1269550 Zbl 0813.17006 incollection

Jeffrey D. Stroomer	Related
Hyo Chul Myung	Related

[39] G. Benkart, S.-J. Kang, and K. C. Misra: “Graded Lie algebras of Kac–Moody type,” Adv. Math. 97 : 2 (February 1993), pp. 154–190. MR 1201842 Zbl 0854.17026 article

Kailash C. Misra	Related
Seok-Jin Kang	Related

[40] G. Benkart and C. Lee: “Stability in modules for general linear Lie superalgebras,” Nova J. Algebra Geom. 2 : 4 (1993), pp. 383–409. MR 1285098 Zbl 0873.17003 article

[41] G. Benkart, J. M. Osborn, and H. Strade: “Contributions to the classification of simple modular Lie algebras,” Trans. Am. Math. Soc. 341 : 1 (1994), pp. 227–252. MR 1129435 Zbl 0792.17016 article

We develop results directed towards the problem of classifying the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic \( p > 7 \). A 1-section of such a Lie algebra relative to a torus \( T \) of maximal absolute toral rank possesses a unique subalgebra maximal with respect to having a composition series with factors which are abelian or classical simple. In this paper we show that the sum \( Q \) of those compositionally classical subalgebras is a subalgebra. This extends to the general case a crucial step in the classification by Block and Wilson of the restricted simple Lie algebras. We derive properties of the filtration which can be constructed using \( Q \) and obtain structural information about the 1-sections and 2-sections of \( Q \) relative to \( T \). We further classify all those algebras in which \( Q \) is solvable.

Helmut Strade	Related
J. Marshall Osborn	Related

[42] G. Benkart, S.-J. Kang, and K. C. Misra: “Indefinite Kac–Moody algebras of classical type,” Adv. Math. 105 : 1 (April 1994), pp. 76–110. MR 1275194 Zbl 0824.17025 article

Kailash C. Misra	Related
Seok-Jin Kang	Related

[43] G. Benkart, M. Chakrabarti, T. Halverson, R. Leduc, C. Lee, and J. Stroomer: “Tensor product representations of general linear groups and their connections with Brauer algebras,” J. Algebra 166 : 3 (June 1994), pp. 529–567. For J. Marshall Osborn and Louis Solomon on their 60th birthdays. MR 1280591 Zbl 0815.20028 article

For the complex general linear group \( G = GL(r,\mathbb{C}) \) we investigate the tensor product module \[\textstyle T = (\bigotimes^p V)\otimes(\bigotimes^q V^*) \] of \( p \) copies of its natural representation \( V = \mathbb{C}^r \) and \( q \) copies of the dual space \( V^* \) of \( V \). We describe the maximal vectors of \( T \) and from that obtain an explicit decomposition of \( T \) into its irreducible \( G \)-summands. Knowledge of the maximal vectors allows us to determine the centralizer algebra \( \mathscr{C} \) of all transformations on \( T \) commuting with the action of \( G \), to construct the irreducible \( \mathscr{C} \)-representations, and to identify \( \mathscr{C} \) with a certain subalgebra \( \mathscr{B}^{(r)}_{p,q} \) of the Brauer algebra \( \mathscr{B}(r)_{p+q} \).

Jeffrey D. Stroomer	Related
Thomas Michael Halverson	Related
Louis Solomon	Related
Robert Edgar Leduc	Related
Chanyoung Lee Shader	Related
Manish Chakrabarti	Related
J. Marshall Osborn	Related

[44] G. Benkart and S. Kass: “Weight multiplicities for affine Kac–Moody algebras,” pp. 1–12 in Modern trends in Lie algebra representation theory (20–22 May 1993, Kingston, ON). Edited by V. Futorny and D. Pollack. Queen’s Papers in Pure and Applied Mathematics 94. Queen’s University (Kingston, ON), 1994. Conference held on the occasion of Albert John Coleman’s 75th birthday. MR 1281175 Zbl 0818.17027 incollection

Albert John Coleman	Related
Vyacheslav M. Futorny	Related
Steven Neil Kass	Related

[45] G. Benkart and E. Zelmanov: “Lie algebras graded by root systems,” pp. 31–38 in Non-associative algebra and its applications (12–17 July 1993, Oviedo, Spain). Edited by S. González. Mathematics and its Applications 303. Kluwer Academic (Dordrecht), 1994. MR 1338154 Zbl 0826.17030 incollection

Let \( \Delta \) be a finite irreducible reduced root system. Assume \( \Gamma \) is the integer lattice generated by \( \Delta \). In [1992] S. Berman and R. Moody initiated the study of Lie algebras graded by the root system \( \Delta \). Following them we say that a Lie algebra is graded by \( \Delta \) or is \( \Delta \)-graded if

\( L \) has a \( \Gamma \)-gradation \( L = \sum_{\gamma\in\Gamma}L_{\gamma} \) in which \( L_{\gamma}\neq (0) \) if and only if \( \gamma\in \Delta\cup\{0\} \);
the split simple Lie algebra \( S \) whose root system is \( \Delta \) is a subalgebra of \( L \), and relative to some split Cartan subalgebra \( \mathcal{H} \) of \( \mathcal{G} \) we have \( L_{\gamma} \supseteq \mathcal{G}_{\gamma} \) for all \( \gamma \in \Delta\cup\{0\} \);
for all \( h\in\mathcal{H} \) the operator \( ad(h) \) acts diagonally on \( L_{\gamma} \) with eigenvalue \( \gamma(h) \); and
\( L \) is generated by its nonzero root spaces \( L_{\gamma} \) where \( \gamma\in\Delta \).

[46] G. Benkart, A. I. Kostrikin, and M. I. Kuznetsov: “Finite-dimensional simple Lie algebras with a nonsingular derivation,” J. Algebra 171 : 3 (February 1995), pp. 894–916. MR 1315926 Zbl 0815.17018 article

Michael Ivanovich Kuznetsov	Related
Aleksei Ivanovich Kostrikin	Related

[47] G. Benkart, S.-J. Kang, and K. C. Misra: “Indefinite Kac–Moody algebras of special linear type,” Pacific J. Math. 170 : 2 (October 1995), pp. 379–404. MR 1363869 Zbl 0857.17020 article

From the special linear Lie algebra \( A_n = \mathfrak{sl}(n + 1,\mathbb{C}) \) we construct certain indefinite Kac–Moody Lie algebras \( IA_n(a,b) \) and then use the representation theory of \( A_n \) to determine explicit closed form root multiplicity formulas for the roots \( \alpha \) of \( IA_n(a,b) \) whose degree satisfies \( |\deg(\alpha)| \leq 2a + 1 \). These expressions involve the well-known Littlewood–Richardson coefficients and Kostka numbers. Using the Euler–Poincaré Principle and Kostant’s formula, we derive two expressions, one of which is recursive and the other closed form, for the multiplicity of an arbitrary root \( \alpha \) of \( IA_n(a,b) \) as a polynomial in Kostka numbers.

Kailash C. Misra	Related
Seok-Jin Kang	Related

[48] G. Benkart, A. I. Kostrikin, and M. I. Kuznetsov: “The simple graded Lie algebras of characteristic three with classical reductive component \( L_0 \),” Comm. Algebra 24 : 1 (1996), pp. 223–234. MR 1370532 Zbl 0846.17022 article

Michael Ivanovich Kuznetsov	Related
Aleksei Ivanovich Kostrikin	Related

[49] G. Benkart: “Commuting actions — a tale of two groups,” pp. 1–46 in Lie algebras and their representations (23–27 January 1995, Seoul). Edited by S.-J. Kang, M.-H. Kim, and I. Lee. Contemporary Mathematics 194. American Mathematical Society (Providence, RI), 1996. MR 1395593 Zbl 0874.17004 incollection

Myung-Hwan Kim	Related
Seok-Jin Kang	Related

[50] G. Benkart, F. Sottile, and J. Stroomer: “Tableau switching: Algorithms and applications,” J. Combin. Theory Ser. A 76 : 1 (October 1996), pp. 11–43. MR 1405988 Zbl 0858.05099 article

We define and characterize switching, an operation that takes two tableaux sharing a common border and “moves them through each other” giving another such pair. Several authors, including James and Kerber, Remmel, Haiman, and Shimozono, have defined switching operations; however, each of their operations is somewhat different from the rest and each imposes a particular order on the switches that can occur. Our goal is to study switching in a general context, thereby showing that the previously defined operations are actually special instances of a single algorithm. The key observation is that switches can be performed in virtually any order without affecting the final outcome. Many known proofs concerning the jeu de taquin, Schur functions, tableaux, characters of representations, branching rules, and the Littlewood–Richardson rule use essentially the same mechanism. Switching provides a common framework for interpreting these proofs. We relate Schützenberger’s evacuation procedure to switching and in the process obtain further results concerning evacuation. We define reversal, an operation which extends evacuation to tableaux of arbitrary skew shape, and apply reversal and related mappings to give combinatorial proofs of various symmetries of Littlewood–Richardson coefficients.

Jeffrey D. Stroomer	Related
Frank Sottile	Related

[51] G. Benkart and E. Zelmanov: “Lie algebras graded by finite root systems and intersection matrix algebras,” Invent. Math. 126 : 1 (1996), pp. 1–45. MR 1408554 Zbl 0871.17024 article

[52] G. Benkart, S.-J. Kang, and K. C. Misra: “Weight multiplicity polynomials for affine Kac–Moody algebras of type \( A^{(1)}_r \),” Compositio Math. 104 : 2 (1996), pp. 153–187. MR 1421398 Zbl 0862.17016 article

For the affine Kac–Moody algebras \( X_r^{(1)} \) it has been conjectured by Benkart and Kass that for fixed dominant weights \( \lambda \), \( \mu \), the multiplicity of the weight \( \mu \) in the irreducible \( X_r^{(1)} \)-module \( L(\lambda) \) of highest weight \( \lambda \) is a polynomial in \( r \) which depends on the type \( X \) of the algebra. In this paper we provide a precise conjecture for the degree of that polynomial for the algebras \( A_r^{(1)} \). To offer evidence for this conjecture we prove it for all dominant weights \( \lambda \) and all weights \( \mu \) of depth \( \leq 2 \) by explicitly exhibiting the polynomials as expressions involving Kostka numbers.

Kailash C. Misra	Related
Seok-Jin Kang	Related

[53] G. Benkart, D. Britten, and F. Lemire: “Modules with bounded weight multiplicities for simple Lie algebras,” Math. Z. 225 : 2 (June 1997), pp. 333–353. MR 1464935 Zbl 0884.17004 article

Frank W. Lemire	Related
Daniel John Britten	Related

[54] Y. Bahturin and G. Benkart: “Highest weight modules for locally finite Lie algebras,” pp. 1–31 in Modular interfaces (18–20 Feburary 1995, Riverside, CA). Edited by V. Chari and I. B. Penkov. AMS/IP Studies in Advanced Mathematics 4. American Mathematical Society (Providence, RI), 1997. MR 1483900 Zbl 0920.17011 incollection

Vyjayanthi Chari	Related
Yuri Alexandrovich Bahturin	Related

[55] G. Benkart, S.-J. Kang, and D. Melville: “Quantized enveloping algebras for Borcherds superalgebras,” Trans. Am. Math. Soc. 350 : 8 (1998), pp. 3297–3319. MR 1451594 Zbl 0913.17008 article

Duncan James Galfred Melville	Related
Seok-Jin Kang	Related

[56] G. Benkart: “Derivations and invariant forms of Lie algebras graded by finite root systems,” Canad. J. Math. 50 : 2 (1998), pp. 225–241. MR 1618175 Zbl 0913.17015 article

[57] G. Benkart, C. L. Shader, and A. Ram: “Tensor product representations for orthosymplectic Lie superalgebras,” J. Pure Appl. Algebra 130 : 1 (August 1998), pp. 1–48. MR 1632811 Zbl 0932.17008 ArXiv math/9607232 article

We derive a general result about commuting actions on certain objects in braided rigid monoidal categories. This enables us to define an action of the Brauer algebra on the tensor space \( V^{\otimes k} \) which commutes with the action of the orthosymplectic Lie superalgebra \( \operatorname{spo}(V) \) and the orthosymplectic Lie color algebra \( \operatorname{spo}(V,\beta) \). We use the Brauer algebra action to compute maximal vectors in \( V^{\otimes k} \) and to decompose \( V^{\otimes k} \) into a direct sum of submodules \( T^{\lambda} \). We compute the characters of the modules \( T^{\lambda} \), give a combinatorial description of these characters in terms of tableaux, and model the decomposition of \( V^{\otimes k} \) into the submodules \( T^{\lambda} \) with a Robinson–Schensted–Knuth-type insertion scheme.

Arun Ram	Related
Chanyoung Lee Shader	Related

[58] G. Benkart, T. Gregory, and M. I. Kuznetsov: “On graded Lie algebras of characteristic three with classical reductive null component,” pp. 149–164 in The Monster and Lie algebras (May 1996, Columbus, OH). Edited by J. Ferrar and K. Harada. Ohio State University Mathematical Research Institute Publications 7. de Gruyter (Berlin), 1998. MR 1650657 Zbl 0926.17016 incollection

Thomas Bradford Gregory	Related
Michael Ivanovich Kuznetsov	Related

[59] G. Benkart and T. Roby: “Down-up algebras,” J. Algebra 209 : 1 (November 1998), pp. 305–344. An addendum to this article was published in J. Algebra 213:1 (1999). MR 1652138 Zbl 0922.17006 ArXiv math/9803159 article

The algebra generated by the down and up operators on a differential or uniform partially ordered set (poset) encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinite-dimensional associative algebras called down-algebras. We show that down-up algebras exhibit many of the important features of the universal enveloping algebra \( U(\mathfrak{sl}_2) \) of the Lie algebra \( \mathfrak{sl}_2 \) including a Poincaré–Birkhoff–Witt type basis and a well-behaved representation theory. We investigate the structure and representations of down-up algebras and focus especially on Verma modules, highest weight representations, and category \( \mathscr{O} \) modules for them. We calculate the exact expressions for all the weights, since that information has proven to be particularly useful in determining structural results about posets.

[60] G. Benkart: “Down-up algebras and Witten’s deformations of the universal enveloping algebra of \( \mathfrak{sl}_2 \),” pp. 29–45 in Recent progress in algebra (11–15 August 1997, Taejon, South Korea). Edited by S. G. Hahn, H. C. Myung, and E. Zelmanov. Contemporary Mathematics 224. American Mathematical Society (Providence, RI), 1999. MR 1653061 Zbl 0922.17007 incollection

Efim Isaakovich Zelmanov	Related
Hyo Chul Myung	Related

[61] G. Benkart and T. Roby: “Addendum: ‘Down-up algebras’,” J. Algebra 213 : 1 (1999), pp. 378. Addendum to an article published in J. Algebra 209:1 (1998). MR 1674692 article

[62] G. Benkart, S.-J. Kang, H. Lee, and D.-U. Shin: “The polynomial behavior of weight multiplicities for classical simple Lie algebras and classical affine Kac–Moody algebras,” pp. 1–29 in Recent developments in quantum affine algebras and related topics (21–24 May 1998, Raleigh, NC). Edited by N. Jing and K. C. Misra. Contemporary Mathematics 248. American Mathematical Society (Providence, RI), 1999. MR 1745252 Zbl 0948.17011 incollection

Kailash C. Misra	Related
Seok-Jin Kang	Related
Dong-Uy Shin	Related
Hyeonmi Lee	Related

[63] G. Benkart: “Lie algebras graded by finite root systems from AD to BC,” pp. 39–45 in Proceedings of the international conference on Jordan structures (June 1997, Málaga, Spain). Edited by A. Castellón Serrano, J. A. Cuenca Mira, A. Fernández López, and C. Martín González. Universidad de Málaga, 1999. MR 1746560 Zbl 1006.17501 incollection

[64] G. Benkart and J. M. Pérez-Izquierdo: “A quantum octonion algebra,” Trans. Am. Math. Soc. 352 : 2 (2000), pp. 935–968. To the memory of Alberto Izquierdo. MR 1637137 Zbl 0931.17011 ArXiv math/9801141 article

Using the natural irreducible 8-dimensional representation and the two spin representations of the quantum group \( U_q(\mathrm{D}_4) \) of \( \mathrm{D}_4 \), we construct a quantum analogue of the split octonions and study its properties. We prove that the quantum octonion algebra satisfies the \( \mathrm{q} \)-Principle of Local Triality and has a nondegenerate bilinear form which satisfies a \( \mathrm{q} \)-version of the composition property. By its construction, the quantum octonion algebra is a nonassociative algebra with a Yang–Baxter operator action coming from the \( \mathrm{R} \)-matrix of \( U_q(\mathrm{D}_4) \). The product in the quantum octonions is a \( U_q(\mathrm{D}_4) \)-module homomorphism. Using that, we prove identities for the quantum octonions, and as a consequence, obtain at \( q = 1 \) new “representation theory” proofs for very well-known identities satisfied by the octonions. In the process of constructing the quantum octonions we introduce an algebra which is a \( \mathrm{q} \)-analogue of the 8-dimensional para-Hurwitz algebra.

[65] G. Benkart, S.-J. Kang, and M. Kashiwara: “Crystal bases for the quantum superalgebra \( U_q(\mathfrak{gl}(m,n) \),” J. Am. Math. Soc. 13 : 2 (2000), pp. 295–331. MR 1694051 Zbl 0963.17010 ArXiv math/9810092 article

Seok-Jin Kang	Related
Masaki Kashiwara	Related

[66] B. Allison, G. Benkart, and Y. Gao: “Central extensions of Lie algebras graded by finite root systems,” Math. Ann. 316 : 3 (2000), pp. 499–527. MR 1752782 Zbl 0989.17004 article

Bruce N. Allison	Related
Yun Gao	Related

[67]G. Benkart, I. Kaplansky, K. McCrimmon, D. J. Saltman, and G. B. Seligman: “Nathan Jacobson (1910–1999),” Notices Amer. Math. Soc. 47 : 9 (2000), pp. 1061–1071. MR 1777887 Zbl 1028.01010

David J. Saltman	Related
Kevin McCrimmon	Related
Irving Kaplansky	Related	Volume
George Benham Seligman	Related
Nathan Jacobson	Related

[68] G. Benkart and S.-J. Kang: “Crystal bases for quantum superalgebras,” pp. 21–54 in Combinatorial methods in representation theory (21–31 July and 26 October–6 November 1998, Kyoto). Edited by K. Koike, M. Kashiwara, S. Okada, I. Terada, and H.-F. Yamada. Advanced Studies in Pure Mathematics 28. Kinokuniya (Tokyo), 2000. MR 1855589 Zbl 1027.17009 incollection

Seok-Jin Kang	Related
Masaki Kashiwara	Related

[69] G. Benkart, S.-J. Kang, H. Lee, K. C. Misra, and D.-U. Shin: “The polynomial behavior of weight multiplicities for the affine Kac–Moody algebras \( A_r^{(1)} \),” Compositio Math. 126 : 1 (2001), pp. 91–111. MR 1827864 Zbl 0997.17013 ArXiv math/9809026 article

Kailash C. Misra	Related
Seok-Jin Kang	Related
Dong-Uy Shin	Related
Hyeonmi Lee	Related

[70] G. Benkart and S. Witherspoon: “A Hopf structure for down-up algebras,” Math. Z. 238 : 3 (2001), pp. 523–553. MR 1869697 Zbl 1006.16028 article

[71] B. Allison, G. Benkart, and Y. Gao: Lie algebras graded by the root systems \( \textrm{BC}_r \), \( r\geq 2 \). Memoirs of the American Mathematical Society 751. American Mathematical Society (Provicence, RI), 2002. MR 1902499 Zbl 0998.17031 book

Bruce N. Allison	Related
Yun Gao	Related

[72] G. Benkart and A. Elduque: “A new construction of the Kac Jordan superalgebra,” Proc. Am. Math. Soc. 130 : 11 (2002), pp. 3209–3217. To Irving Kaplansky. MR 1912998 Zbl 1083.17014 article

Alberto Elduque	Related
Irving Kaplansky	Related	Volume

[73] G. Benkart and D. Moon: “Tensor product representations of Temperley–Lieb algebras and their centralizer algebras,” pp. 151–166 in Topics in Young diagrams and representation theory (6–9 November 2001, Kyoto). Edited by M. Kosuda. Sūrikaisekikenkyūsho Kōkyūroku 1262. 2002. Also published in Combinatorial and geometric representation theory (2003). MR 1929395 incollection

Masashi Kosuda	Related
Dong Ho Moon	Related

[74] G. Benkart and A. Elduque: “Lie superalgebras graded by the root systems \( C(n) \), \( D(m,n) \), \( D(2,1;\alpha) \), \( F(4) \), \( G(3) \),” Canad. Math. Bull. 45 : 4 (2002), pp. 509–524. To Professor Robert Moody with our best wishes on his sixtieth birthday. MR 1941225 Zbl 1040.17026 article

Robert V. Moody	Related
Alberto Elduque	Related

[75] G. Benkart and S. Doty: “Derangements and tensor powers of adjoint modules for \( \mathfrak{sl}_n \),” J. Algebraic Combin. 16 : 1 (2002), pp. 31–42. MR 1941983 Zbl 1018.17003 ArXiv math/0108106 article

[76] D. Spellman, G. M. Benkart, A. M. Gaglione, W. D. Joyner, M. E. Kidwell, M. D. Meyerson, and W. P. Wardlaw: “Principal ideals and associate rings,” JP J. Algebra Number Theory Appl. 2 : 2 (2002), pp. 181–193. MR 1942384 Zbl 1046.13004 article

William Patterson Wardlaw	Related
Mark Elliot Kidwell	Related
Anthony Michael Gaglione	Related
Mark Daniel Meyerson	Related
William David Joyner	Related
Dennis Spellman	Related

[77] G. Benkart and O. Smirnov: “Lie algebras graded by the root system \( \mathrm{BC}_1 \),” J. Lie Theory 13 : 1 (2003), pp. 91–132. MR 1958577 Zbl 1015.17028 article

[78] G. Benkart and D. Moon: “Tensor product representations of Temperley–Lieb algebras and their centralizer algebras,” pp. 31–49 in Combinatorial and geometric representation theory (22–26 October 2001, Seoul). Edited by S.-J. Kang and K.-H. Lee. Contemporary Mathematics 325. American Mathematical Society (Providence, RI), 2003. Also published in Topics in Young diagrams and representation theory (2002). MR 1988984 Zbl 1031.17003 incollection

Dong Ho Moon	Related
Seok-Jin Kang	Related
Kyu-Hwan Lee	Related

[79] G. Benkart and A. Elduque: “The Tits construction and the exceptional simple classical Lie superalgebras,” Q. J. Math. 54 : 2 (June 2003), pp. 123–137. MR 1989868 Zbl 1045.17002 article

[80] G. Benkart and A. Elduque: “Lie superalgebras graded by the root system \( A(m,n) \),” J. Lie Theory 13 : 2 (2003), pp. 387–400. MR 2003150 Zbl 1030.17029 ArXiv math/0202284 article

[81] G. Benkart and A. Elduque: “Lie superalgebras graded by the root system \( \mathrm{B}(m,n) \),” Selecta Math. (N.S.) 9 : 3 (2003), pp. 313–360. MR 2006571 Zbl 1040.17027 article

[82] Y. Bahturin and G. Benkart: “Some constructions in the theory of locally finite simple Lie algebras,” J. Lie Theory 14 : 1 (2004), pp. 243–270. MR 2040179 Zbl 1138.17311 article

[83] G. Benkart and S. Witherspoon: “Representations of two-parameter quantum groups and Schur–Weyl duality,” pp. 65–92 in Hopf algebras (1–3 February 2002, Chicago). Edited by J. Bergen, S. Catoiu, and W. Chin. Lecture Notes in Pure and Applied Mathematics 237. Dekker (New York), 2004. MR 2051731 Zbl 1048.16021 ArXiv math/0108038 incollection

[84] G. Benkart and O. Eng: “Weighted Aztec diamond graphs and the Weyl character formula,” Electron. J. Combin. 11 : 1 (2004). Research Paper 28, 16 pp. MR 2056080 Zbl 1053.52025 article

[85] V. Bekkert, G. Benkart, and V. Futorny: “Weight modules for Weyl algebras,” pp. 17–42 in Kac–Moody Lie algebras and related topics (28–31 January 2002, Chennai, India). Edited by N. Sthanumoorthy and K. C. Misra. Contemporary Mathematics 343. American Mathematical Society (Providence, RI), 2004. MR 2056678 Zbl 1057.16021 ArXiv math/0202222 incollection

Vyacheslav M. Futorny	Related
Viktor Ivanovich Bekkert	Related
Kailash C. Misra	Related

[86] G. Benkart and S. Witherspoon: “Restricted two-parameter quantum groups,” pp. 293–318 in Representations of finite dimensional algebras and related topics in Lie theory and geometry (15 July–10 August 2002, Toronto). Edited by V. Dlab and C. M. Ringel. Fields Institute Communications 40. American Mathematical Society (Providence, RI), 2004. MR 2057401 Zbl 1048.16020 incollection

Vlastimil B. Dlab	Related
Sarah J. Witherspoon	Related

[87] G. Benkart and S. Witherspoon: “Two-parameter quantum groups and Drinfel’d doubles,” Algebr. Represent. Theory 7 : 3 (2004), pp. 261–286. MR 2070408 Zbl 1113.16041 ArXiv math/0011064 article

[88] G. Benkart, A. Elduque, and C. Martínez: “\( A(n,n) \)-graded Lie superalgebras,” J. Reine Angew. Math. 573 (2004), pp. 139–156. MR 2084585 Zbl 1059.17017 ArXiv math/0309395 article

Alberto Elduque	Related
Consuelo Martínez López	Related

[89] G. Benkart and P. Terwilliger: “Irreducible modules for the quantum affine algebra \( U_q(\widehat{\mathfrak{sl}}_2) \) and its Borel subalgebra,” J. Algebra 282 : 1 (2004), pp. 172–194. MR 2095578 Zbl 1106.17014 ArXiv math/0311152 article

Let \( U_q(\mathfrak{b}) \) denote the standard Borel subalgebra of the quantum affine algebra \( U_q(\widehat{\mathfrak{sl}}_2) \). We show that the following hold for any choice of scalars \( \varepsilon_0 \), \( \varepsilon_1 \) from the set \( \{1,-1\} \):

Let \( V \) be a finite-dimensional irreducible \( U_q(\mathfrak{b}) \)-module of type \( (\varepsilon_0,\varepsilon_1) \). Then the action of \( U_q(\mathfrak{b}) \) on \( V \) extends uniquely to an action of \( U_q(\widehat{\mathfrak{sl}}_2) \) on \( V \). The resulting \( U_q(\widehat{\mathfrak{sl}}_2) \)-module structure on \( V \) is irreducible and of type \( (\varepsilon_0,\varepsilon_1) \).
Let \( V \) be a finite-dimensional irreducible \( U_q(\widehat{\mathfrak{sl}}_2) \)-module of type \( (\varepsilon_0,\varepsilon_1) \). When the \( U_q(\widehat{\mathfrak{sl}}_2) \)-action is restricted to \( U_q(\mathfrak{b}) \), the resulting \( U_q(\mathfrak{b}) \)-module structure on \( V \) is irreducible and of type \( (\varepsilon_0,\varepsilon_1) \).

[90] G. Benkart and D. Moon: “Tensor product representations of Temperley–Lieb algebras and Chebyshev polynomials,” pp. 57–80 in Representations of algebras and related topics (15 July–10 August 2002, Toronto). Edited by R.-O. Buchweitz and H. Lenzing. Fields Institute Communications 45. American Mathematical Society (Providence, RI), 2005. To Professor Vlastimil Dlab with our best wishes on our seventieth birthday. MR 2146240 Zbl 1179.17009 incollection

The Brauer algebra \( B_k(n) \) determines the centralizer algebra of the action of the orthogonal group \( O_n \) on the \( k \)-fold tensor power \( V^{\otimes k} \) of its natural \( n \)-dimensional module \( V \). As a subalgebra of the Brauer algebra, the Temperley–Lieb alebra \( TL_k(n) \) inherits an action on \( V^{\otimes k} \). In this work, we investigate the centralizer algebra \( \mathcal{Z}_k(n) \) of the Temperley–Lieb algera on this tensor product space. We show that the dimesions of the irreducible \( \mathcal{Z}_k(n) \)-modules are related to vlaues of Chebyshev polynomials of the second kind. We determine the branching rule from \( B_k(n) \) to \( TL_k(n) \) using inverse Chebyshev relations.

Dong Ho Moon	Related
Vlastimil B. Dlab	Related

[91] S. Cho, K.-C. Ha, Y.-O. Kim, and D. Moon: “Key exchange protocol using matrix algebras and its analysis,” J. Korean Math. Soc. 42 : 6 (2005), pp. 1287–1309. MR 2176265 Zbl 1083.94007 article

Young-One Kim	Related
Kil-Chan Ha	Related
Dong Ho Moon	Related
Soojin Cho	Related

[92] G. Benkart and E. Neher: “The centroid of extended affine and root graded Lie algebras,” J. Pure Appl. Algebra 205 : 1 (April 2006), pp. 117–145. MR 2193194 Zbl 1163.17306 ArXiv math/0502561 article

[93] G. Benkart, X. Xu, and K. Zhao: “Classical Lie superalgebras over simple associative algebras,” Proc. London Math. Soc. (3) 92 : 3 (May 2006), pp. 581–600. MR 2223537 Zbl 1129.17009 article

Over arbitrary fields of characteristic not equal to 2, we construct three families of simple Lie algebras and six families of simple Lie superalgebras of matrices with entries chosen from different one-sided ideals of a simple associative algebra. These families correspond to the classical Lie algebras and superalgebras. Our constructions intermix the structure of the associative algebra and the structure of the matrix algebra in an essential, compatible way. Many examples of simple associative algebras without an identity element arise as a by-product. The study of conformal algebras and superalgebras often involves matrix algebras over associative algebras such as Weyl algebras, and for that reason, we illustrate our constructions by taking various one-sided ideals from a Weyl algebra or a quantum torus.

Kaiming Zhao	Related
Xiaoping Xu	Related

[94] G. Benkart, S.-J. Kang, and K.-H. Lee: “On the centre of two-parameter quantum groups,” Proc. Roy. Soc. Edinburgh Sect. A 136 : 3 (2006), pp. 445–472. MR 2227803 Zbl 1106.17013 article

Seok-Jin Kang	Related
Kyu-Hwan Lee	Related

[95] G. Benkart and A. Labra: “Representations of rank 3 algebras,” Comm. Algebra 34 : 8 (2006), pp. 2867–2877. MR 2250574 Zbl 1127.17001 article

[96] Representations of algebraic groups, quantum groups, and Lie algebras (11–15 July 2004, Snowbird, UT). Edited by G. Benkart, J. C. Jantzen, Z. Lin, D. K. Nakano, and B. J. Parshall. Contemporary Mathematics 413. American Mathematical Society (Providence, RI), 2006. MR 2259846 Zbl 1097.20500 book

Zong Zhu Lin	Related
Daniel Ken Nakano	Related
Jens Carsten Jantzen	Related
Brian J. Parshall	Related

[97] G. Benkart, I. Frenkel, S.-J. Kang, and H. Lee: “Level 1 perfect crystals and path realizations of basic representations at \( q = 0 \),” Int. Math. Res. Not. 2006 (2006). Article ID 10312, 28 pp. MR 2272099 Zbl 1149.17016 ArXiv math/0507114 article

Seok-Jin Kang	Related
Hyeonmi Lee	Related
Igor Borisovich Frenkel	Related

[98] G. Benkart and Y. Yoshii: “Lie \( G \)-tori of symplectic type,” Q. J. Math. 57 : 4 (December 2006), pp. 425–448. Dedicated to Professor George Seligman with admiration. MR 2277593 Zbl 1223.17022 ArXiv math/0509183 article

George Benham Seligman	Related
Yoji Yoshii	Related

[99] G. Benkart and S. Witherspoon: “Quantum group actions, twisting elements, and deformations of algebras,” J. Pure Appl. Algebra 208 : 1 (January 2007), pp. 371–389. MR 2270011 Zbl 1116.16035 ArXiv math/0503310 article

[100] G. Benkart and P. Terwilliger: “The universal central extension of the three-point \( \mathfrak{sl}_2 \) loop algebra,” Proc. Am. Math. Soc. 135 : 6 (2007), pp. 1659–1668. MR 2286073 Zbl 1153.17008 ArXiv math/0512422 article

[101] G. Benkart, T. Gregory, and A. Premet: The recognition theorem for graded Lie algebras in prime characteristic. Memoirs of the American Mathematical Society 920. American Mathematical Society (Providence, RI), 2009. MR 2488391 Zbl 1167.17004 ArXiv math/0508373 book

Thomas Bradford Gregory	Related
Alexander A. Premet	Related

[102] G. Benkart and M. Ondrus: “Whittaker modules for generalized Weyl algebras,” Represent. Theory 13 (2009), pp. 141–164. MR 2497458 Zbl 1251.16020 ArXiv 0803.3570 article

We investigate Whittaker modules for generalized Weyl algebras, a class of associative algebras which includes the quantum plane, Weyl algebras, the universal enveloping algebra of \( \mathfrak{sl}_2 \) and of Heisenberg Lie algebras, Smith’s generalizations of \( U(\mathfrak{sl}_2) \), various quantum analogues of these algebras, and many others. We show that the Whittaker modules \( V = Aw \) of the generalized Weyl algebra \( A = R(\phi,t) \) are in bijection with the \( \phi \)-stable left ideals of \( R \). We determine the annihilator \( \operatorname{Ann}_A(w) \) of the cyclic generator \( w \) of \( V \). We also describe the annihilator ideal \( \operatorname{Ann}_A(V) \) under certain assumptions that hold for most of the examples mentioned above. As one special case, we recover Kostant’s well-known results on Whittaker modules and their associated annihilators for \( U(\mathfrak{sl}_2) \).

[103] G. Benkart and A. Fernández López: “The Lie inner ideal structure of associative rings revisited,” Comm. Algebra 37 : 11 (2009), pp. 3833–3850. MR 2573222 Zbl 1210.16038 article

[104] B. Allison and G. Benkart: “Unitary Lie algebras and Lie tori of type \( \mathrm{BC}_r \), \( r\geq 3 \),” pp. 1–47 in Quantum affine algebras, extended affine Lie algebras, and their applications (2–7 March 2008, Banff, AB). Edited by Y. Gao, N. Jing, M. Lau, and K. C. Misra. Contemporary Mathematics 506. American Mathematical Society (Providence, RI), 2010. MR 2642560 Zbl 1262.17011 ArXiv 0811.3263 incollection

Bruce N. Allison	Related
Yun Gao	Related
Kailash C. Misra	Related

[105] G. Benkart, B. Srinivasan, M. Gray, E. Maycock, and L. Rothschild: “Alice Turner Schafer (1915–2009): remembrances,” Notices Am. Math. Soc. 57 : 9 (October 2010), pp. 1116–1119. edited by Anne Leggett. MR 2730368 Zbl 1197.01050 article

Ellen J. Maycock	Related
Mary Lee Wheat Gray	Related
Alice Turner Schafer	Related
Anne M. Leggett	Related
Bhama Srinivasan	Related
Linda Preiss Rothschild	Related

[106] G. Benkart, M. Pereira, and S. Witherspoon: “Yetter–Drinfeld modules under cocycle twists,” J. Algebra 324 : 11 (December 2010), pp. 2990–3006. To Susan Montgomery in honor of her distinguished career. MR 2732983 Zbl 1223.16011 ArXiv 0908.1563 article

We give an explicit formula for the correspondence between simple Yetter–Drinfeld modules for certain finite-dimensional pointed Hopf algebras \( H \) and those for cocycle twists \( H^{\sigma} \) of \( H \). This implies an equivalence between modules for their Drinfeld doubles. To illustrate our results, we consider the restricted two-parameter quantum groups \( \mathfrak{u}_{r,s}(\mathfrak{sl}_n) \) under conditions on the parameters guaranteeing that \( \mathfrak{u}_{r,s}(\mathfrak{sl}_n) \) is a Drinfeld double of its Borel subalgebra. We determine explicit correspondences between \( \mathfrak{u}_{r,s}(\mathfrak{sl}_n) \)-modules for different values of \( r \) and \( s \) and provide examples where no such correspondence can exist. Our examples were obtained via the computer algebra system SINGULAR::PLURAL.

Susan Montgomery	Related
Mariana Pereira	Related
Sarah J. Witherspoon	Related

[107] G. Benkart and P. Terwilliger: “The equitable basis for \( \mathfrak{sl}_2 \),” Math. Z. 268 : 1–2 (June 2011), pp. 535–557. MR 2805446 Zbl 1277.17013 ArXiv 0810.2066 article

This article contains an investigation of the equitable basis for the Lie algebra \( \mathfrak{sl}_2 \). Denoting this basis by \( \{x,y,z\} \), we have \[ [x,y] = 2x + 2y, \qquad [y,z] = 2y + 2z, \qquad [z,x] = 2z + 2x. \] We determine the group of automorphisms \( G \) generated by \[ \operatorname{exp}(\operatorname{ad} x^*), \quad\operatorname{exp}(\operatorname{ad} y^*), \quad\operatorname{exp}(\operatorname{ad} z^*) ,\] where \( \{x^*, y^*, z^*\} \) is the basis for \( \mathfrak{sl}_2 \) dual to \( \{x, y, z\} \) with respect to the trace form \( (u,v) = \operatorname{tr}(uv) \) and study the relationship of \( G \) to the isometries of the lattices \[ L = \mathbb{Z}x \oplus \mathbb{Z}y \oplus \mathbb{Z}z \quad\text{and}\quad L^* = \mathbb{Z}x^* \oplus \mathbb{Z}y^* \oplus \mathbb{Z}z^* .\] The matrix of the trace form is a Cartan matrix of hyperbolic type, and we identify the equitable basis with a set of simple roots of the corresponding Kac–Moody Lie algebra \( \mathfrak{g} \), so that \( L \) is the root lattice and \( \frac{1}{2}L^* \) is the weight lattice of \( \mathfrak{g} \). The orbit \( G(x) \) of \( x \) coincides with the set of real roots of \( \mathfrak{g} \). We determine the isotropic roots of \( \mathfrak{g} \) and show that each isotropic root has multiplicity 1. We describe the finite-dimensional \( \mathfrak{sl}_2 \)-modules from the point of view of the equitable basis. In the final section, we establish a connection between the Weyl group orbit of the fundamental weights of \( \mathfrak{g} \) and Pythagorean triples.

[108] S. Cho, E.-K. Jung, and D. Moon: “A hive-model proof of the second reduction formula of Littlewood–Richardson coefficients,” Ann. Comb. 15 : 2 (2011), pp. 223–231. MR 2813512 Zbl 1233.05210 article

Eun-Kyoung Jung	Related
Dong Ho Moon	Related
Soojin Cho	Related

[109] G. Benkart and A. Elduque: “Lie algebras with prescribed \( \mathfrak{sl}_3 \) decomposition,” Proc. Am. Math. Soc. 140 : 8 (2012), pp. 2627–2638. MR 2910750 Zbl 1329.17021 ArXiv 1101.0489 article

[110] G. Benkart: Multiparameter Weyl algebras. Preprint, June 2013. ArXiv 1306.0485 techreport

[111] V. Bekkert, G. Benkart, V. Futorny, and I. Kashuba: “New irreducible modules for Heisenberg and affine Lie algebras,” J. Algebra 373 (January 2013), pp. 284–298. MR 2995027 Zbl 1306.17009 ArXiv 1107.0893 article

We study \( \mathbb{Z} \)-graded modules of nonzero level with arbitrary weight multiplicities over Heisenberg Lie algebras and the associated generalized loop modules over affine Kac–Moody Lie algebras. We construct new families of such irreducible modules over Heisenberg Lie algebras. Our main result establishes the irreducibility of the corresponding generalized loop modules providing an explicit construction of many new examples of irreducible modules for affine Lie algebras. In particular, to any function \( \phi: \mathbb{N}\to \{\pm\} \) we associate a \( \phi \)-highest weight module over the Heisenberg Lie algebra and a \( \phi \)-imaginary Verma module over the affine Lie algebra. We show that any \( \phi \)-imaginary Verma module of nonzero level is irreducible.

Vyacheslav M. Futorny	Related
Viktor Ivanovich Bekkert	Related
Iryna Kashuba	Related

[112] G. Benkart, S. Madariaga, and J. M. Pérez-Izquierdo: “Hopf algebras with triality,” Trans. Am. Math. Soc. 365 : 2 (2013), pp. 1001–1023. MR 2995381 Zbl 1278.16032 ArXiv 1106.4302 article

[113] G. Benkart and D. Moon: “Planar rook algebras and tensor representations of \( \mathfrak{gl}(1|1) \),” Comm. Algebra 41 : 7 (2013), pp. 2405–2416. MR 3169400 Zbl 1269.05115 ArXiv 1201.2482 article

We establish a connection between planar rook algebras and tensor representations \( \mathbf{V}^{\otimes k} \) of the natural two-dimensional representation \( \mathbf{V} \) of the general linear Lie superalgebra \( \mathfrak{gl}(1|1) \). In particular, we show that the centralizer algebra \( \operatorname{E}_{\mathfrak{gl}(1|1)}(\mathbf{V}^{\otimes k}) \) is the planar rook algebra \( \mathbb{C}P_{k-1} \) for all \( k\geq 1 \), and we exhibit an explicit decomposition of \( \mathbf{V}^{\otimes k} \) into irreducible \( \mathfrak{gl}(1|1) \)-modules. We obtain similar results for the quantum enveloping algebra \( \mathbf{U}_{\mathbf{q}}(\mathfrak{gl}(1|1)) \) and its natural two-dimensional module \( \mathbf{V}_{\mathbf{q}} \).

[114] G. Benkart, S. A. Lopes, and M. Ondrus: “A parametric family of subalgebras of the Weyl algebra, II: Irreducible modules,” pp. 73–98 in Recent developments in algebraic and combinatorial aspects of representation theory (12–16 August 2010, Bangalore, India and 18–20 May 2012, Riverside, CA). Edited by V. Chari, J. Greenstein, K. C. Misra, K. N. Raghavan, and S. Viswanath. Contemporary Mathematics 602. American Mathematical Society (Providence, RI), 2013. Part I was published in Trans. Am. Math. Soc. 367:3 (2015). Part III appeared as a 2014 preprint. MR 3203899 Zbl 1308.16022 ArXiv 1212.1404 incollection

An Ore extension over a polynomial algebra \( \mathbf{F}[x] \) is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra \( \mathbf{A}_h \) generated by elements \( x, y \), which satisfy \( yx - xy = h \), where \( h \in \mathbf{F}[x] \). When \( h \neq 0 \), these algebras are subalgebras of the Weyl algebra \( \mathbf{A}_1 \) and can be viewed as differential operators with polynomial coefficients. In previous work, we studied the structure of \( \mathbf{A}_h \) and determined its automorphism group \( \operatorname{Aut}_{\mathbb{F}}(\mathbf{A}_h) \) and the subalgebra of invariants under \( \operatorname{Aut}_{\mathbb{F}}(\mathbf{A}_h) \). Here we determine the irreducible \( \mathbf{A}_h \)-modules. In a sequel to this paper, we completely describe the derivations of \( \mathbf{A}_h \) over any field.

Samuel Antonio Lopes	Related
Kailash C. Misra	Related
Matthew Ondrus	Related
Vyjayanthi Chari	Related

@incollection {key3203899m,
	AUTHOR = {Benkart, Georgia and Lopes, Samuel A.
		 and Ondrus, Matthew},
	TITLE = {A parametric family of subalgebras of
		 the {W}eyl algebra, {II}: {I}rreducible
		 modules},
	BOOKTITLE = {Recent developments in algebraic and
		 combinatorial aspects of representation
		 theory},
	EDITOR = {Chari, Vyjayanthi and Greenstein, Jacob
		 and Misra, Kailash C. and Raghavan,
		 K. N. and Viswanath, Sankaran},
	SERIES = {Contemporary Mathematics},
	NUMBER = {602},
	PUBLISHER = {American Mathematical Society},
	ADDRESS = {Providence, RI},
	YEAR = {2013},
	PAGES = {73--98},
	DOI = {10.1090/conm/602/12027},
	NOTE = {(12--16 August 2010, Bangalore, India
		 and 18--20 May 2012, Riverside, CA).
		 Part I was published in \textit{Trans.
		 Am. Math. Soc.} \textbf{367}:3 (2015).
		 Part III appeared as a 2014 preprint.
		 ArXiv:1212.1404. MR:3203899. Zbl:1308.16022.},
	ISSN = {0271-4132},
	ISBN = {9780821890370},
}

[115] G. Benkart, T. Halverson, and N. Harman: A parametric family of subalgebras of the Weyl algebra, III: Derivations. Preprint, October 2014. Part I was published in Trans. Am. Math. Soc. 367:3 (2015). Part II was published in Recent developments in algebraic and combinatorial aspects of representation theory (2013). ArXiv 1406.1508 techreport

An Ore extension over a polynomial algebra \( \mathbb{F}[x] \) is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra \( \mathrm{A}_h \) generated by elements \( x,y \), which satisfy \( yx-xy = h \), where \( h \) is in \( \mathbb{F}[x] \). When \( h \) is nonzero, these algebras are subalgebras of the Weyl algebra \( \mathrm{A}_1 \) and can be viewed as differential operators with polynomial coefficients. This paper determines the derivations of \( \mathrm{A}_h \) and the Lie structure of the first Hochschild cohomology group \[ \mathrm{HH}^1(\mathrm{A}_h) = \operatorname{Der}_{\mathbb{F}}(\mathrm{A}_h)/\operatorname{Inder}_{\mathbb{F}}(\mathrm{A}_h) \] of outer derivations over an arbitrary field. In characteristic 0, we show that \( \mathrm{HH}^1(\mathrm{A}_h) \) has a unique maximal nilpotent ideal modulo which it is 0 or a direct sum of simple Lie algebras that are field extensions of the one-variable Witt algebra. In positive characteristic we obtain decomposition theorems for \( \operatorname{Der}_{\mathbb{F}}(\mathrm{A}_h) \) and \( \mathrm{HH}^1(\mathrm{A}_h) \) and describe the structure of \( \mathrm{HH}^1(\mathrm{A}_h) \) as a module over the center of \( \mathrm{A}_h \).

Thomas Michael Halverson	Related
Nate Harman	Related

[116] G. Benkart and T. Halverson: “Motzkin algebras,” European J. Combin. 36 (February 2014), pp. 473–502. MR 3131911 Zbl 1284.05333 ArXiv 1106.5277 article

We introduce an associative algebra \( \mathbf{M}_k(x) \) whose dimension is the \( 2k \)-th Motzkin number. The algebra \( \mathbf{M}_k(x) \) has a basis of “Motzkin diagrams”, which are analogous to Brauer and Temperley–Lieb diagrams. We show for a particular value of \( x \) that the algebra \( \mathbf{M}_k(x) \) is the centralizer algebra of the quantum enveloping algebra \( \mathbf{U}_{\mathbf{q}}(\mathfrak{gl}_2) \) acting on the \( k \)-fold tensor power of the sum of the 1-dimensional and 2-dimensional irreducible \( \mathbf{U}_{\mathbf{q}}(\mathfrak{gl}_2) \)-modules. We prove that \( \mathbf{M}_k(x) \) is cellular in the sense of Graham and Lehrer and construct indecomposable \( \mathbf{M}_k(x) \)-modules which are the left cell modules. When \( \mathbf{M}_k(x) \) is a semisimple algebra, these modules provide a complete set of representatives of isomorphism classes of irreducible \( \mathbf{M}_k(x) \)-modules. We compute the determinant of the Gram matrix of a bilinear form on the cell modules and use these determinants to show that \( \mathbf{M}_k(x) \) is semisimple exactly when \( x \) is not the root of certain Chebyshev polynomials.

[117] G. Benkart, S.-J. Kang, S.-j. Oh, and E. Park: “Construction of irreducible representations over Khovanov–Lauda–Rouquier algebras of finite classical type,” Int. Math. Res. Not. 2014 : 5 (January 2014), pp. 1312–1366. In memory of Professor Hyo Chul Myung. MR 3178600 Zbl 1355.17009 ArXiv 1108.1048 article

We give an explicit construction of irreducible modules over Khovanov–Lauda–Rouquier algebras \( R \) and their cyclotomic quotients \( R^{\lambda} \) for finite classical types using a crystal basis theoretic approach. More precisely, for each element \( \nu \) of the crystal \( B(\infty) \) (resp. \( B(\lambda) \)), we first construct certain modules \( \Delta(\mathbf{a};k) \) labeled by the adapted string \( \mathbf{a} \) of \( \nu \). We then prove that the head of the induced module \[ \operatorname{Ind}(\Delta(\mathbf{a};1) \boxtimes \cdots \boxtimes \Delta(\mathbf{a};n)) \] is irreducible and that every irreducible \( R \)-module (resp. \( R^{\lambda} \)-module) can be realized as the irreducible head of one of the induced modules \[ \operatorname{Ind}(\Delta(\mathbf{a};1) \boxtimes \cdots \boxtimes \Delta(\mathbf{a};n)) .\] Moreover, we show that our construction is compatible with the crystal structure on \( B(\infty) \) (resp. \( B(\lambda) \)).

Hyo Chul Myung	Related
Se-jin Oh	Related
Euiyong Park	Related
Seok-Jin Kang	Related

[118] G. Benkart, S. Cho, and D. Moon: “The combinatorics of \( \mathbf{A}_2 \)-webs,” Electron. J. Combin. 21 : 2 (2014). Research Paper 2.25, 33 pages. MR 3210659 Zbl 1300.05316 ArXiv 1312.1023 article

Dong Ho Moon	Related
Soojin Cho	Related

[119] G. Benkart, S. A. Lopes, and M. Ondrus: “A parametric family of subalgebras of the Weyl algebra, I: Structure and automorphisms,” Trans. Am. Math. Soc. 367 : 3 (2015), pp. 1993–2021. Part II was published in Recent developments in algebraic and combinatorial aspects of representation theory (2013). Part III appeared as a 2014 preprint. MR 3286506 Zbl 1312.16020 ArXiv 1210.4631 article

An Ore extension over a polynomial algebra \( \mathbb{F}[x] \) is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra \( \mathbf{A}_h \) generated by elements \( x,y \), which satisfy \( yx - xy = h \), where \( h\in \mathbb{F}[x] \). We investigate the family of algebras \( \mathbf{A}_h \) as \( h \) ranges over all the polynomials in \( \mathbb{F}[x] \). When \( h \neq 0 \), the algebras \( \mathbf{A}_h \) are subalgebras of the Weyl algebra \( \mathbf{A}_1 \) and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of \( \mathbf{A}_h \) over arbitrary fields \( \mathbb{F} \) and describe the invariants in \( \mathbf{A}_h \) under the automorphisms. We determine the center, normal elements, and height one prime ideals of \( \mathbf{A}_h \), localizations and Ore sets for \( \mathbf{A}_h \), and the Lie ideal \( [\mathbf{A}_h,\mathbf{A}_h] \). We also show that \( \mathbf{A}_h \) cannot be realized as a generalized Weyl algebra over \( \mathbb{F}[x] \), except when \( h\in\mathbb{F} \). In two sequels to this work, we completely describe the irreducible modules and derivations of \( \mathbf{A}_h \) over any field.

Matthew Ondrus	Related
Samuel Antonio Lopes	Related

[120] G. Benkart, S. A. Lopes, and M. Ondrus: “Derivations of a parametric family of subalgebras of the Weyl algebra,” J. Algebra 424 (February 2015), pp. 46–97. MR 3293213 Zbl 1312.16019 article

An Ore extension over a polynomial algebra \( \mathbb{F}[x] \) is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra \( \mathbf{A}_h \) generated by elements \( x,y \), which satisfy \( yx-xy = h \), where \( h\in \mathbb{F}[x] \). When \( h\neq 0 \), the algebra \( \mathbf{A}_h \) is subalgebra of the Weyl algebra \( \mathbf{A}_1 \) and can be viewed as differential operators with polynomial coefficients. This paper determines the derivations of \( \mathbf{A}_h \) and the Lie structure of the first Hochschild cohomology group \[ \mathbf{HH}^1(\mathbf{A}_h) = \operatorname{Der}_{\mathbb{F}}(\mathbf{A}_h)/\operatorname{Inder}_{\mathbb{F}}(\mathbf{A}_h) \] of outer derivations over an arbitrary field. In characteristic 0, we show that \( \mathbf{HH}^1(\mathbf{A}_h) \) has a unique maximal nilpotent ideal modulo which \( \mathbf{HH}^1(\mathbf{A}_h) \) is 0 or a direct sum of simple Lie algebras that are field extensions of the one-variable Witt algebra. In positive characteristic, we obtain decomposition theorems for \( \operatorname{Der}_{\mathbb{F}}(\mathbf{A}_h) \) and \( \mathbf{HH}^1(\mathbf{A}_h) \) and describe the structure of \( \mathbf{HH}^1(\mathbf{A}_h) \) as a module over the center of \( \mathbf{A}_h \).

Matthew Ondrus	Related
Samuel Antonio Lopes	Related

[121] G. Benkart, M. Džamonja, J. Roitman, I. Juhász, W. Fleissner, F. Tall, P. Nyikos, K. Kunen, and A. Miller: “Memories of Mary Ellen Rudin,” Notices Am. Math. Soc. 62 : 6 (2015), pp. 617–629. Coordinating editors were Georgia Benkart, Mirna Džamonja and Judith Roitman. MR 3362445 Zbl 1338.01028 article

István Juhász	Related
Kenneth Kunen	Related
William George Fleissner	Related
Judith Roitman	Related
Mirna Džamonja	Related
Arnold William Miller	Related
Peter Joseph Nyikos	Related
Franklin D. Tall	Related
Mary Ellen Rudin	Related	Volume

[122] G. Benkart and J. Feldvoss: “Some problems in the representation theory of simple modular Lie algebras,” pp. 207–228 in Lie algebras and related topics: Workshop on Lie algebras, in honor of Helmut Strade’s 70th birthday (22–24 May 2013, Milan). Edited by M. Avitabile, J. Feldvoss, and T. Weigel. Contemporary Mathematics 652. American Mathematical Society (Providence, RI), 2015. MR 3453057 Zbl 06622485 ArXiv 1503.06762 incollection

The finite-dimensional restricted simple Lie algebras of characteristic \( p > 5 \) are classical or of Cartan type. The classical algebras are analogues of the simple complex Lie algebras and have a well-advanced representation theory with important connections to Kazhdan–Lusztig theory, quantum groups at roots of unity, and the representation theory of algebraic groups. We survey progress that has been made towards developing a representation theory for the restricted simple Cartan-type Lie algebras, discuss comparable results in the classical case, formulate a couple of conjectures, and pose a dozen open problems for further study.

Helmut Strade	Related
Jörg Feldvoss	Related

[123] G. Benkart and A. Elduque: Cross products, invariants, and centralizers. Preprint, June 2016. Dedicated to Efim Zelmanov on the occasion of his 60th birthday. ArXiv 1606.07588 techreport

An algebra \( \mathrm{V} \) with a cross product \( \times \) has dimension 3 or 7. In this work, we use 3-tangles to describe, and provide a basis for, the space of homomorphisms from \( \mathrm{V}^{\otimes n} \) to \( \mathrm{V}^{\otimes m} \) that are invariant under the action of the automorphism group \( \operatorname{Aut}(\mathrm{V},\times) \) of \( V \), which is a special orthogonal group when \( \dim \mathrm{V} = 3 \), and a simple algebraic group of type \( \mathrm{G}_2 \) when \( \dim \mathrm{V} = 7 \). When \( m = n \), this gives a graphical description of the centralizer algebra \[ \operatorname{End}_{\operatorname{Aut}(\mathrm{V},\times)}(\mathrm{V}^{\otimes n}) ,\] and therefore, also a graphical realization of the \( \operatorname{Aut}(\mathrm{V},\times) \)-invariants in \( \mathrm{V}^{\otimes n}) \) equivalent to the First Fundamental Theorem of Invariant Theory. We show how the 3-dimensional simple Kaplansky Jordan superalgebra can be interpreted as a cross product (super)algebra and use 3-tangles to obtain a graphical description of the centralizers and invariants of the Kaplansky superalgebra relative to the action of the special orthosymplectic group.

Efim Isaakovich Zelmanov	Related
Alberto Elduque	Related

[124] G. Benkart, T. Halverson, and N. Harman: Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups. Preprint, May 2016. ArXiv 1605.06543 techreport

The partition algebra \( \mathrm{P}_k(n) \) and the symmetric group \( \mathrm{S}_n \) are in Schur–Weyl duality on the \( k \)-fold tensor power \( \mathrm{M}_n^{\otimes k} \) of the permutation module \( \mathrm{M}_n \) of \( \mathrm{S}_n \), so there is a surjection \[ \mathrm{P}_k(n)\to\mathrm{Z}_k(n) := \operatorname{End}_{\mathrm{S}_n}(\mathrm{M}_n^{\otimes k}) ,\] which is an isomorphism when \( n\neq 2k \). We prove a dimension formula for the irreducible modules of the centralizer algebra \( \mathrm{Z}_k(n) \) in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible \( \mathrm{S}_n \)-modules in \( \mathrm{M}_n^{\otimes k} \). Our dimension expressions hold for any \( n\geq 1 \) and \( k\geq 0 \). Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on \( \mathrm{M}_n^{\otimes k} \) and the quasi-partition algebra corresponding to tensor powers of the reflection representation of \( \mathrm{S}_n \).

Thomas Michael Halverson	Related
Nate Harman	Related

[125] G. Benkart, T. Halverson, and N. Harman: Chip firing on Dynkin diagrams and McKay quivers. Preprint, February 2016. ArXiv 1601.06849 techreport

Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay–Cartan matrices for finite subgroups \( \mathrm{G} \) of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights. In the McKay–Cartan case for finite subgroups of the special linear group, the cokernel is related to the abelianization of the subgroup \( \mathrm{G} \). In the special case of the classical McKay correspondence, the critical group and the abelianization are shown to be isomorphic.

Thomas Michael Halverson	Related
Nate Harman	Related

[126] G. Benkart and T. Halverson: Walks on graphs and their connections with tensor invariants and centralizer algebras. Preprint, October 2016. ArXiv 1610.07837 techreport

The number of walks of \( k \) steps from the node \( \mathbf{0} \) to the node \( \lambda \) on the representation graph (McKay quiver) determined by a finite group \( \mathrm{G} \) and a \( \mathrm{G} \)-module \( \mathrm{V} \) is the multiplicity of the irreducible \( \mathrm{G} \)-module \( \mathrm{G}_{\lambda} \) in the tensor power \( \mathrm{V}^{\otimes k} \), and it is also the dimension of the irreducible module labeled by \( \lambda \) for the centralizer algebra \[ \mathrm{Z}_k(\mathrm{G}) = \operatorname{End}_{\mathrm{G}}(\mathrm{V}^{\otimes k}) .\] This paper explores ways to effectively calculate that number using the character theory of \( \mathrm{G} \). We determine the corresponding Poincaré series. The special case \( \lambda = 0 \) gives the Poincaré series for the tensor invariants \[ \mathrm{T}(\mathrm{V})^{\mathrm{G}} = \bigoplus_{k=0}^{\infty}(\mathrm{V}^{\otimes k})^{\mathrm{G}} .\] When \( \mathrm{G} \) is abelian, we show that the exponential generating function for the number of walks is a product of generalized hyperbolic functions. Many graphs (such as circulant graphs) can be viewed as representation graphs, and the methods presented here provide efficient ways to compute the number of walks on them.

[127] G. Benkart: “Poincaré series for tensor invariants and the McKay correspondence,” Adv. Math. 290 (February 2016), pp. 236–259. MR 3451923 Zbl 1342.14030 ArXiv 1407.3997 article

For a finite group \( \mathbf{G} \) and a finite-dimensional \( \mathbf{G} \)-module \( \mathbf{V} \), we prove a general result on the Poincar/’e series for the \( \mathbf{G} \)-invariants in the tensor algebra \[ \mathbf{T}(\mathbf{V}) = \bigoplus_{k\geq 0}\mathbf{V}^{\otimes k} .\] We apply this result to the finite subgroups \( \mathbf{G} \) of the \( 2{\times}2 \) special unitary matrices and their natural module \( \mathbf{G} \) of \( 2{\times}1 \) column vectors. Because these subgroups are in one-to-one correspondence with the simply laced affine Dynkin diagrams by the McKay correspondence, the Poincaré series obtained are the generating functions for the number of walks on the simply laced affine Dynkin diagrams.

[128] J. M. Barnes, G. Benkart, and T. Halverson: “McKay centralizer algebras,” Proc. Lond. Math. Soc. (3) 112 : 2 (2016), pp. 375–414. MR 3471253 Zbl 1332.05147 ArXiv 1312.5254 article

For a finite subgroup \( \mathbf{G} \) of the special unitary group \( \mathrm{SU}_2 \), we study the centralizer algebra \[ Z_k(\mathbf{G}) = \operatorname{End}_{\mathbf{G}}(V^{\otimes k}) \] of \( \mathbf{G} \) acting on the \( k \)-fold tensor product of its defining representation \( \mathbf{V} = \mathbb{C}^2 \). These subgroups are in bijection with the simply laced affine Dynkin diagrams. The McKay correspondence relates the representation theory of these groups to the associated Dynkin diagram, and we use this connection to show that the structure and representation theory of \( \mathbf{Z}_k(\mathbb{G}) \) as a semisimple algebra is controlled by the combinatorics of the corresponding Dynkin diagram.

Jeffrey M. Barnes	Related
Thomas Michael Halverson	Related

[129] G. Benkart, N. Guay, J. H. Jung, S.-J. Kang, and S. Wilcox: “Quantum walled Brauer–Clifford superalgebras,” J. Algebra 454 (May 2016), pp. 433–474. MR 3473434 Zbl 1342.17002 ArXiv 1404.0443 article

We introduce a new family of superalgebras, the quantum walled Brauer–Clifford superalgebras \( \mathbf{BC}_{r,s}(q) \). The superalgebra \( \mathbf{BC}_{r,s}(q) \) is a quantum deformation of the walled Brauer–Clifford superalgebra \( \mathbf{BC}_{r,s} \) and a super version of the quantum walled Brauer algebra. We prove that \( \mathbf{BC}_{r,s}(q) \) is the centralizer superalgebra of the action of \( \mathfrak{U}_q(\mathfrak{q}(n)) \) on the mixed tensor space \[ \mathbf{V}_q^{r,s} = \mathbf{V}_q^{\otimes r} \otimes (\mathbf{V}_q^*)^{\otimes s} \] when \( n \geq r+s \), where \( \mathbf{V}_q = \mathbb{C}(q)^{(n|n)} \) is the natural representation of the quantum enveloping superalgebra \( \mathfrak{U}_q(\mathfrak{q}(n)) \) and \( \mathbf{V}_q^* \) is its dual space. We also provide a diagrammatic realization of \( \mathbf{BC}_{r,s}(q) \) as the \( (r,s) \)-bead tangle algebra \( \mathbf{BT}_{r,s}(q) \). Finally, we define the notion of \( q \)-Schur superalgebras of type \( \mathbf{Q} \) and establish their basic properties.

Stewart Wilcox	Related
Nicolas Guay	Related
Ji Hye Jung	Related
Seok-Jin Kang	Related

[130] G. Benkart and D. Moon: “A Schur–Weyl duality approach to walking on cubes,” Ann. Comb. 20 : 3 (2016), pp. 397–417. MR 3537911 Zbl 1347.05246 ArXiv 1409.8154 article

Walks on the representation graph \( \mathcal{R}_{\mathbf{V}}(\mathbf{G}) \) determined by a group \( \mathbf{G} \) and a \( \mathbf{G} \)-module \( \mathbf{V} \) are related to the centralizer algebras of the action of \( \mathbf{G} \) on the tensor powers \( \mathbf{V}^{\otimes k} \) via Schur–Weyl duality. This paper explores that connection when the group is \( \mathbb{Z}^n_2 \) and the module \( \mathbf{V} \) is chosen so the representation graph is the \( n \)-cube. We describe a basis for the centralizer algebras in terms of labeled partition diagrams. We obtain an expression for the number of walks by counting certain partitions and determine the exponential generating functions for the number of walks.

[131] G. Benkart and J. Meinel: “The center of the affine nilTemperley–Lieb algebra,” Math. Z. 284 : 1–2 (2016), pp. 413–439. MR 3545499 Zbl 06642709 ArXiv 1505.02544 article

[132] G. Benkart, L. Colmenarejo, P. E. Harris, R. Orellana, G. Panova, A. Schilling, and M. Yip: A minimaj-preserving crystal on ordered multiset partitions. Preprint, July 2017. ArXiv 1707.08709 techreport

We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization \( R_{n,k} \) due to Haglund, Rhoades and Shimozono of the coinvariant algebra \( R_n \). The crystal structure also enables us to demonstrate the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions.

Laura Colmenarejo Hernando	Related
Martha Yip	Related
Pamela Estephania Harris	Related
Rosa C. Orellana	Related
Greta Cvetanova Panova	Related
Anne Schilling	Related

[133] G. Benkart and T. Halverson: Partition algebras \( \mathrm{P}_k(n) \) with \( 2k > n \) and the fundamental theorems of invariant theory for the symmetric group \( \mathrm{S}_n \). Preprint, July 2017. ArXiv 1707.01410 techreport

Assume \( \mathrm{M}_n \) is the \( n \)-dimensional permutation module for the symmetric group \( \mathrm{S}_n \), and let \( \mathrm{M}_n^{\otimes k} \) be its \( k \)-fold tensor power. The partition algebra \( \mathrm{P}_k(n) \) maps surjectively onto the centralizer algebra \( \operatorname{End}_{\mathrm{S}_n}(\mathrm{M}_n^{\otimes k}) \) for all \( k,n\in\mathbb{Z}_{\geq 1} \) and isomorphically when \( n\geq 2k \). We describe the image of the surjection \[ \Phi_{k,n}:\mathrm{P}_k(n)\to \operatorname{End}_{\mathrm{S}_n}(\mathrm{M}_n^{\otimes k}) \] explicitly in terms of the orbit basis of \( \mathrm{P}_k(n) \) and show that when \( 2k > n \) the kernel of \( \Phi_{k,n} \) is generated by a single essential idempotent \( e_{k,n} \), which is an orbit basis element. We obtain a presentation for \( \operatorname{End}_{\mathrm{S}_n}(\mathrm{M}_n^{\otimes k}) \) by imposing one additional relation, \( e_{k,n} = 0 \), to the standard presentation of the partition algebra \( \mathrm{P}_k(n) \) when \( 2k > n \). As a consequence, we obtain the fundamental theorems of invariant theory for the symmetric group \( \mathrm{S}_n \). We show under the natural embedding of the partition algebra \( \mathrm{P}_n(n) \) into \( \mathrm{P}_n(n) \) for \( k\geq n \) that the essential idempotent \( e_{n,n} \) generates the kernel of \( \Phi_{k,n} \). Therefore, the relation \( e_{n,n} = 0 \) can replace \( e_{k,n}=0 \) when \( k\geq n \).

[134] G. Benkart, P. Diaconis, M. W. Liebeck, and P. H. Tiep: “Tensor product Markov chains,” J. Algebra 561 (2020), pp. 17–83. MR 4135538 Zbl 1467.60054 article

Pham Huu Tiep	Related
Martin Walter Liebeck	Related
Persi Diaconis	Related

[135] G. Benkart, R. Biswal, E. Kirkman, V. C. Nguyen, and J. Zhu: “McKay matrices for finite-dimensional Hopf algebras,” Canad. J. Math. 74 : 3 (2022), pp. 686–731. MR 4430927 Zbl 1490.19001 article

Rekha Biswal	Related
Ellen Elizabeth Kirkman	Related
Van Cat Nguyen	Related
Jieru Zhu	Related

[136] G. Benkart, R. Biswal, E. Kirkman, V. C. Nguyen, and J. Zhu: “Tensor representations for the Drinfeld double of the Taft algebra,” J. Algebra 606 (2022), pp. 764–797. MR 4432247 Zbl 07541265 article

Rekha Biswal	Related
Ellen Elizabeth Kirkman	Related
Van Cat Nguyen	Related
Jieru Zhu	Related

Complete Bibliography