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
People
BibTeX
@article {key1201842m,
AUTHOR = {Benkart, Georgia and Kang, Seok-Jin
and Misra, Kailash C.},
TITLE = {Graded {L}ie algebras of {K}ac--{M}oody
type},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {97},
NUMBER = {2},
MONTH = {February},
YEAR = {1993},
PAGES = {154--190},
DOI = {10.1006/aima.1993.1005},
NOTE = {MR:1201842. Zbl:0854.17026.},
ISSN = {0001-8708},
}
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
Abstract
People
BibTeX
@article {key1275194m,
AUTHOR = {Benkart, Georgia and Kang, Seok-Jin
and Misra, Kailash C.},
TITLE = {Indefinite {K}ac--{M}oody algebras of
classical type},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {105},
NUMBER = {1},
MONTH = {April},
YEAR = {1994},
PAGES = {76--110},
DOI = {10.1006/aima.1994.1040},
NOTE = {MR:1275194. Zbl:0824.17025.},
ISSN = {0001-8708},
}
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
Abstract
People
BibTeX
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.
@article {key1363869m,
AUTHOR = {Benkart, Georgia and Kang, Seok-Jin
and Misra, Kailash C.},
TITLE = {Indefinite {K}ac--{M}oody algebras of
special linear type},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {170},
NUMBER = {2},
MONTH = {October},
YEAR = {1995},
PAGES = {379--404},
DOI = {10.2140/pjm.1995.170.379},
URL = {http://projecteuclid.org/euclid.pjm/1102370875},
NOTE = {MR:1363869. Zbl:0857.17020.},
ISSN = {0030-8730},
}
G. Benkart :
“Commuting actions — a tale of two groups 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
People
BibTeX
@incollection {key1395593m,
AUTHOR = {Benkart, Georgia},
TITLE = {Commuting actions---a tale of two groups},
BOOKTITLE = {Lie algebras and their representations},
EDITOR = {Kang, Seok-Jin and Kim, Myung-Hwan and
Lee, Insok},
SERIES = {Contemporary Mathematics},
NUMBER = {194},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1996},
PAGES = {1--46},
DOI = {10.1090/conm/194/02387},
NOTE = {(23--27 January 1995, Seoul). MR:1395593.
Zbl:0874.17004.},
ISSN = {0271-4132},
ISBN = {9780821805121},
}
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
Abstract
People
BibTeX
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.
@article {key1421398m,
AUTHOR = {Benkart, Georgia and Kang, Seok-Jin
and Misra, Kailash C.},
TITLE = {Weight multiplicity polynomials for
affine {K}ac--{M}oody algebras of type
\$A^{(1)}_r\$},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {104},
NUMBER = {2},
YEAR = {1996},
PAGES = {153--187},
URL = {http://www.numdam.org/item?id=CM_1996__104_2_153_0},
NOTE = {MR:1421398. Zbl:0862.17016.},
ISSN = {0010-437X},
}
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
Abstract
People
BibTeX
@article {key1451594m,
AUTHOR = {Benkart, Georgia and Kang, Seok-Jin
and Melville, Duncan},
TITLE = {Quantized enveloping algebras for {B}orcherds
superalgebras},
JOURNAL = {Trans. Am. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical
Society},
VOLUME = {350},
NUMBER = {8},
YEAR = {1998},
PAGES = {3297--3319},
DOI = {10.1090/S0002-9947-98-02058-3},
NOTE = {MR:1451594. Zbl:0913.17008.},
ISSN = {0002-9947},
}
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 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
People
BibTeX
@incollection {key1745252m,
AUTHOR = {Benkart, Georgia and Kang, Seok-Jin
and Lee, Hyeonmi and Shin, Dong-Uy},
TITLE = {The polynomial behavior of weight multiplicities
for classical simple {L}ie algebras
and classical affine {K}ac--{M}oody
algebras},
BOOKTITLE = {Recent developments in quantum affine
algebras and related topics},
EDITOR = {Jing, Naihuan and Misra, Kailash C.},
SERIES = {Contemporary Mathematics},
NUMBER = {248},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1999},
PAGES = {1--29},
DOI = {10.1090/conm/248/03815},
NOTE = {(21--24 May 1998, Raleigh, NC). MR:1745252.
Zbl:0948.17011.},
ISSN = {0271-4132},
ISBN = {9780821811993},
}
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
Abstract
People
BibTeX
A crystal base theory is introduced for the quantized enveloping algebra of the general linear Lie superalgebra \( \mathfrak{gl}(m,n) \) , and an explicit realization of the crystal base is given in terms of semistandard tableaux.
@article {key1694051m,
AUTHOR = {Benkart, Georgia and Kang, Seok-Jin
and Kashiwara, Masaki},
TITLE = {Crystal bases for the quantum superalgebra
\$U_q(\mathfrak{gl}(m,n)\$},
JOURNAL = {J. Am. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical
Society},
VOLUME = {13},
NUMBER = {2},
YEAR = {2000},
PAGES = {295--331},
DOI = {10.1090/S0894-0347-00-00321-0},
NOTE = {ArXiv:math/9810092. MR:1694051. Zbl:0963.17010.},
ISSN = {0894-0347},
}
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
People
BibTeX
@incollection {key1855589m,
AUTHOR = {Benkart, Georgia and Kang, Seok-Jin},
TITLE = {Crystal bases for quantum superalgebras},
BOOKTITLE = {Combinatorial methods in representation
theory},
EDITOR = {Koike, Kazuhiko and Kashiwara, Masaki
and Okada, Soichi and Terada, Itaru
and Yamada, Hiro-Fumi},
SERIES = {Advanced Studies in Pure Mathematics},
NUMBER = {28},
PUBLISHER = {Kinokuniya},
ADDRESS = {Tokyo},
YEAR = {2000},
PAGES = {21--54},
NOTE = {(21--31 July and 26 October--6 November
1998, Kyoto). MR:1855589. Zbl:1027.17009.},
ISSN = {0920-1971},
ISBN = {9784314101417},
}
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
Abstract
People
BibTeX
@article {key1827864m,
AUTHOR = {Benkart, Georgia and Kang, Seok-Jin
and Lee, Hyeonmi and Misra, Kailash
C. and Shin, Dong-Uy},
TITLE = {The polynomial behavior of weight multiplicities
for the affine {K}ac--{M}oody algebras
\$A_r^{(1)}\$},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {126},
NUMBER = {1},
YEAR = {2001},
PAGES = {91--111},
DOI = {10.1023/A:1017584131106},
NOTE = {ArXiv:math/9809026. MR:1827864. Zbl:0997.17013.},
ISSN = {0010-437X},
}
G. Benkart and D. Moon :
“Tensor product representations of Temperley–Lieb algebras and their centralizer algebras 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
People
BibTeX
@incollection {key1988984m,
AUTHOR = {Benkart, Georgia and Moon, Dongho},
TITLE = {Tensor product representations of {T}emperley--{L}ieb
algebras and their centralizer algebras},
BOOKTITLE = {Combinatorial and geometric representation
theory},
EDITOR = {Kang, Seok-Jin and Lee, Kyu-Hwan},
SERIES = {Contemporary Mathematics},
NUMBER = {325},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2003},
PAGES = {31--49},
DOI = {10.1090/conm/325/05663},
NOTE = {(22--26 October 2001, Seoul). Also published
in \textit{Topics in Young diagrams
and representation theory} (2002). MR:1988984.
Zbl:1031.17003.},
ISSN = {0271-4132},
}
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
Abstract
People
BibTeX
We describe Poincaré–Birkhoff–Witt bases for the two-parameter quantum groups
\[ U = U_{r,s}(\mathfrak{sl}_n) \]
following Kharchenko and show that the positive part of \( U \) has the structure of an iterated skew polynomial ring. We define an ad-invariant bilinear form on \( U \) , which plays an important role in the construction of central elements. We introduce an analogue of the Harish-Chandra homomorphism and use it to determine the centre of \( U \) .
@article {key2227803m,
AUTHOR = {Benkart, Georgia and Kang, Seok-Jin
and Lee, Kyu-Hwan},
TITLE = {On the centre of two-parameter quantum
groups},
JOURNAL = {Proc. Roy. Soc. Edinburgh Sect. A},
FJOURNAL = {Proceedings of the Royal Society of
Edinburgh. Section A. Mathematics},
VOLUME = {136},
NUMBER = {3},
YEAR = {2006},
PAGES = {445--472},
DOI = {10.1017/S0308210500005011},
NOTE = {MR:2227803. Zbl:1106.17013.},
ISSN = {0308-2105},
}
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
Abstract
People
BibTeX
@article {key2272099m,
AUTHOR = {Benkart, Georgia and Frenkel, Igor and
Kang, Seok-Jin and Lee, Hyeonmi},
TITLE = {Level 1 perfect crystals and path realizations
of basic representations at \$q = 0\$},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
VOLUME = {2006},
YEAR = {2006},
DOI = {10.1155/IMRN/2006/10312},
NOTE = {Article ID 10312, 28 pp. ArXiv:math/0507114.
MR:2272099. Zbl:1149.17016.},
ISSN = {1073-7928},
}
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
Abstract
People
BibTeX
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. \( B(\lambda) \) ).
@article {key3178600m,
AUTHOR = {Benkart, Georgia and Kang, Seok-Jin
and Oh, Se-jin and Park, Euiyong},
TITLE = {Construction of irreducible representations
over {K}hovanov--{L}auda--{R}ouquier
algebras of finite classical type},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
VOLUME = {2014},
NUMBER = {5},
MONTH = {January},
YEAR = {2014},
PAGES = {1312--1366},
DOI = {10.1093/imrn/rns244},
NOTE = {In memory of Professor Hyo Chul Myung.
ArXiv:1108.1048. MR:3178600. Zbl:1355.17009.},
ISSN = {1073-7928},
}
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
Abstract
People
BibTeX
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.
@article {key3473434m,
AUTHOR = {Benkart, Georgia and Guay, Nicolas and
Jung, Ji Hye and Kang, Seok-Jin and
Wilcox, Stewart},
TITLE = {Quantum walled {B}rauer--{C}lifford
superalgebras},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {454},
MONTH = {May},
YEAR = {2016},
PAGES = {433--474},
DOI = {10.1016/j.jalgebra.2015.04.038},
NOTE = {ArXiv:1404.0443. MR:3473434. Zbl:1342.17002.},
ISSN = {0021-8693},
}
