17 research outputs found
Brundan-Kazhdan-Lusztig conjecture for general linear Lie superalgebras
In the framework of canonical and dual canonical bases of Fock spaces,
Brundan in 2003 formulated a Kazhdan-Lusztig type conjecture for the characters
of the irreducible and tilting modules in the BGG category for the general
linear Lie superalgebra for the first time. In this paper, we prove Brundan's
conjecture and its variants associated to all Borel subalgebras in full
generality.Comment: 64 pages, Notes in the Introduction and Remark 3.14 adde
Character Formula for Infinite Dimensional Unitarizable Modules of the General Linear Superalgebra
The Fock space of bosonic and fermionic quantum oscillators forms
a unitarizable module of the general linear superalgebra . Its
tensor powers decompose into direct sums of infinite dimensional irreducible
highest weight -modules. We obtain an explicit decomposition of
any tensor power of this Fock space into irreducibles, and develop a character
formula for the irreducible -modules arising in this way.Comment: 25 Pages, LaTeX forma
Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras
Let be a tensor product of unitarizable irreducible highest weight
modules over the Lie (super)algebra , where is
, or . We
show, using super duality, that the singular eigenvectors of the (super) Gaudin
Hamiltonians for on can be obtained from the singular
eigenvectors of the Gaudin Hamiltonians for the corresponding Lie algebras on
some tensor products of finite-dimensional irreducible modules. As a
consequence, the (super) Gaudin Hamiltonians for are
diagonalizable on the space spanned by singular vectors of and hence on
. In particular, we establish the diagonalization of the Gaudin
Hamiltonians, associated to any of the orthogonal Lie algebra
and the symplectic Lie algebra , on the
tensor product of infinite-dimensional unitarizable irreducible highest weight
modules
Quadratic and cubic Gaudin Hamiltonians and super Knizhnik-Zamolodchikov equations for general linear Lie superalgebras
We show that under a generic condition, the quadratic Gaudin Hamiltonians
associated to are diagonalizable on any singular
weight space in any tensor product of unitarizable highest weight
-modules. Moreover, every joint eigenbasis of the
Hamiltonians can be obtained from some joint eigenbasis of the quadratic Gaudin
Hamiltonians for the general linear Lie algebra on the
corresponding singular weight space in the tensor product of some
finite-dimensional irreducible -modules for and
sufficiently large. After specializing to , we show that similar results
hold as well for the cubic Gaudin Hamiltonians associated to
.
We also relate the set of singular solutions of the (super)
Knizhnik-Zamolodchikov equations for to the set of
singular solutions of the Knizhnik-Zamolodchikov equations for
for and sufficiently large
Finite conformal modules over the N=2,3,4 superconformal algebras
In this paper we continue the study of representation theory of formal
distribution Lie superalgebras initiated in q-alg/9706030. We study finite
Verma-type conformal modules over the N=2, N=3 and the two N=4 superconformal
algebras and also find explicitly all singular vectors in these modules. From
our analysis of these modules we obtain a complete list of finite irreducible
conformal modules over the N=2, N=3 and the two N=4 superconformal algebras.Comment: 36 pages, no figures, LaTeX forma