Related papers: The PBW Filtration, Demazure Modules and Toroidal …
In this paper we study the PBW filtration on irreducible integrable highest weight representations of affine Kac-Moody algebra $\gh$. The $n$-th space of this filtration is spanned with the vectors $x_1... x_s v$, where $x_i\in\gh$, $s\le…
We study the PBW-filtration on the highest weight representations $V(\la)$ of the Lie algebras of type ${\tt A}_{n}$ and ${\tt C}_{n}$. This filtration is induced by the standard degree filtration on $\U(\fn^-)$. In previous papers, the…
We study the PBW filtration on the highest weight representations $V(\la)$ of $\msp_{2n}$. This filtration is induced by the standard degree filtration on $U(\n^-)$. We give a description of the associated graded $S(\n^-)$-module $gr…
We study the PBW filtration on the highest weight representations $V(\la)$ of $\msl_{n+1}$. This filtration is induced by the standard degree filtration on $U(\n^-)$. We give a description of the associated graded $S(\n^-)$-module $gr…
We study the PBW filtration on the irreducible highest weight representations of simple complex finite-dimensional Lie algebras. This filtration is induced by the standard degree filtration on the universal enveloping algebra. For certain…
We study the category of finite--dimensional bi--graded representations of toroidal current algebras associated to finite--dimensional complex simple Lie algebras. Using the theory of graded representations for current algebras, we…
We study the PBW filtration on irreducible finite--dimensional representations for the Lie algebra of type $\tt B_n$. We prove in several cases, including all multiples of the adjoint representation and all irreducible finite--dimensional…
In this paper, we are interested in the decomposition of the tensor product of two representations of a symmetrizable Kac-Moody Lie algebra $\mathfrak g$. Let $P\_+$ be the set of dominant integral weights. For $\lambda\in P\_+$ ,…
We study the Brylinski filtration induced by a principal Heisenberg subalgebra of an affine Kac-Moody algebra $\mathfrak{g}$, a notion first introduced by Slofstra. The associated graded space of this filtration on dominant weight spaces of…
For positive integer p=k+2, we construct a logarithmic extension of the ^sl(2)_k conformal field theory of integrable representations by taking the kernel of two fermionic screening operators in a three-boson realization of ^sl(2)_k. The…
We will introduce an $\mathbb{N}$-filtration on the negative part of a quantum group of type $A_n$, such that the associated graded algebra is a q-commutative polynomial algebra. This filtration is given in terms of the representation…
The survey is devoted to associative $\Z_{\ge0}$-graded algebras presented by n generators and n(n-1)/2 quadratic relations and satisfying the so-called Poincare-Birkhoff-Witt condition (PBW-algebras). We consider examples of such algebras…
We study the structure of the finite-dimensional representations of $\mathfrak{sl}_2[t]$, the current Lie algebra type of $A_1$, which are obtained by taking tensor products of special Demazure modules. We show that these representations…
Our main goal is to show that the Gelfand--Tsetlin toric degeneration of the type A flag variety can be obtained within a degenerate representation-theoretic framework similar to the theory of PBW degenerations. In fact, we provide such…
This paper is the detailed version of math.QA/0403477 (T. Arakawa, Quantized Reductions and Irreducible Representations of W-Algebras) with extended results; We study the representation theory of the W-algebra $W_k(g)$ associated with a…
Let h \subset g be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the h-module U(g)/U(g)h whose associated graded h-module is isomorphic to S(n). We give a necessary and sufficient condition…
The affinization morphism for the stack $\mathfrak{M}(\Pi_Q)$ of representations of a preprojective algebra $\Pi_Q$ is a local model for the morphism from the stack of objects in a general 2-Calabi-Yau category to the good moduli space. We…
We develop an elementary method for proving the PBW theorem for associative algebras with an ascending filtration. The idea is roughly the following. At first, we deduce a proof of the PBW property for the {\it ascending} filtration (with…
We consider graded deformations and PBW deformations of algebras defined over noncommutative algebras. We explain how fibers of graded deformations correspond to filtered algebras admitting a PBW property, with focus on smash product…
In this survey, we review some of the recent connections between the representation theory of (untwisted) quantum affine algebras and the representation theory of current algebras. We mainly focus on the finite-dimensional representations…