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, 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, 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, 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},
}
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
Abstract
People
BibTeX
We investigate weight modules for finite and infinite Weyl algebras, reducing the classification of such simple modules to the determination of maximal ideals in certain polynomial algebras and maximal left ideals in certain skew-polynomial algebras. We also study the representation type of the blocks of locally-finite weight module categories and describe indecomposable modules in tame blocks.
@incollection {key2056678m,
AUTHOR = {Bekkert, Viktor and Benkart, Georgia
and Futorny, Vyacheslav},
TITLE = {Weight modules for {W}eyl algebras},
BOOKTITLE = {Kac--{M}oody {L}ie algebras and related
topics},
EDITOR = {Sthanumoorthy, N. and Misra, Kailash
C.},
SERIES = {Contemporary Mathematics},
NUMBER = {343},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2004},
PAGES = {17--42},
NOTE = {(28--31 January 2002, Chennai, India).
ArXiv:math/0202222. MR:2056678. Zbl:1057.16021.},
ISSN = {0271-4132},
ISBN = {9780821833377},
}
B. Allison and G. Benkart :
“Unitary Lie algebras and Lie tori of type \( \mathrm{BC}_r \) , \( r\geq 3 \) 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
People
BibTeX
@incollection {key2642560m,
AUTHOR = {Allison, Bruce and Benkart, Georgia},
TITLE = {Unitary {L}ie algebras and {L}ie tori
of type \$\mathrm{BC}_r\$, \$r\geq 3\$},
BOOKTITLE = {Quantum affine algebras, extended affine
{L}ie algebras, and their applications},
EDITOR = {Gao, Yun and Jing, Naihuan and Lau,
Michael and Misra, Kailash C.},
SERIES = {Contemporary Mathematics},
NUMBER = {506},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2010},
PAGES = {1--47},
DOI = {10.1090/conm/506/09934},
NOTE = {(2--7 March 2008, Banff, AB). ArXiv:0811.3263.
MR:2642560. Zbl:1262.17011.},
ISSN = {0271-4132},
ISBN = {9780821845073},
}
G. Benkart, S. A. Lopes, and M. Ondrus :
“A parametric family of subalgebras of the Weyl algebra, II: Irreducible modules 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)2014 preprint .
MR
3203899 Zbl
1308.16022 ArXiv
1212.1404 incollection
Abstract
People
BibTeX
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.
@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},
}
