arXiv subjects
Publications and source records attributed to Libor Krizka.
We provide an explicit classification of all simple $\mathfrak{sl}_2$-modules that are torsion free of rank $1$ over the Cartan subalgebra. We also establish a similar result for the first Weyl algebra and for the Lie superalgebra $\mathfrak{osp}(1|2)$.
The closure of a Diximier sheet is the image of a generalized Grothendieck's simultaneous resolution. We show that the associated variety of simple affine vertex algebras is contained in the closure of the Diximier sheet when a chiralization of generalized Grothendieck's simultaneous resolution exists. This generalizes in a conceptual manner the results obtained by the first named author and Anne Moreau and, in particular, allows to extend them to the types A_2, C_n, E_6, E_7.
We associate to an arbitrary positive root $α$ of a complex semisimple finite-dimensional Lie algebra $\mfrak{g}$ a twisting endofunctor $T_α$ of the category of $\mfrak{g}$-modules. We apply this functor to generalized Verma modules in the category $\mcal{O}(\mfrak{g})$ and construct a family of $α$-Gelfand--Tsetlin modules with finite $Γ_α$-multiplicities, where $Γ_α$ is a commutative $\C$-subalgebra of the universal enveloping algebra of $\mfrak{g}$ generated by a Cartan subalgebra of $\mfrak{g}$ and by the Casimir element of the $\mfrak{sl}(2)$-subalgebra corresponding to the root $α$. This covers classical results of Andersen and Stroppel when $α$ is a simple root and previous results of the authors in the case when $\mfrak{g}$ is a complex simple Lie algebra and $α$ is the maximal root of $\mfrak{g}$. The significance of constructed modules is that they are Gelfand--Tsetlin modules with respect to any commutative $\C$-subalgebra of the universal enveloping algebra of $\mfrak{g}$ containing $Γ_α$. Using the Beilinson--Bernstein correspondence we give a geometric realization of these modules together with their explicit description. We also identify a tensor subcategory of the category of $α$-Gelfand--Tsetlin modules which contains constructed modules as well as the category $\mcal{O}(\mfrak{g})$.
We construct two examples of q-deformed classical Howe dual pairs (sl(2,C), sl(2,C)) and (sl(2,C), sl(n,C)). Moreover, we obtain a noncommutative version of the first fundamental theorem of classical invariant theory. Our approach to these duality differs from the paper of Lehrer-Zhang-Zhang. Furthermore, we solve the tensor product decomposition problem for Verma modules over U_q(sl(2,C)) provided q is not a root of unity.
In the present paper we describe a new class of Gelfand--Tsetlin modules for an arbitrary complex simple finite-dimensional Lie algebra g and give their geometric realization as the space of delta-functions" on the flag manifold G/B supported at the 1-dimensional submanifold. When g=sl(n) (or gl(n)) these modules form a subclass of Gelfand-Tsetlin modules with infinite dimensional weight subspaces. We discuss their properties and describe the simplicity criterion for these modules in the case of the Lie algebra sl(3,C).
We construct realizations of quantum generalized Verma modules for U_q(sl_n(C)) by quan- tum differential operators. Taking the classical limit q ! 1 provides a realization of classical generalized Verma modules for sl_n(C) by differential operators.
We discuss some aspects of the composition structure of twisted Verma modules for the Lie algebra $\mathfrak{sl}(3, \mathbb{C})$, including the explicit structure of singular vectors for both $\mathfrak{sl}(3, \mathbb{C})$ and one of its Lie subalgebras $\mathfrak{sl}(2, \mathbb{C})$, and also of their generators. Our analysis is based on the use of partial Fourier tranform applied to the realization of twisted Verma modules as $\mathrm{D}$-modules on the Schubert cells in the full flag manifold for $\mathrm{SL}(3, \mathbb{C})$.