Celebratio Mathematica

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

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

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

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.

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

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

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.

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

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

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 \( s \).

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

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

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

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

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

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

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

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

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 \( p - 2 \).

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

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

year	title	people

			clear

Georgia Benkart

Complete Bibliography

Works connected to J. Marshall Osborn

Filter the Bibliography List