Related papers: Tilting modules and highest weight theory for redu…

Affine Lie algebras admit non-classical highest-weight theories through alternative partitions of the root system. Although significant inroads have been made, much of the classical machinery is inapplicable in this broader context, and…

The notion of Weyl modules, both local and global, goes back to Chari and Pressley in the case of affine Lie algebras, and has been extensively studied for various Lie algebras graded by root systems. We extend that definition to a certain…

We propose a notion of algebra of {\it twisted} chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex…

Zhao and the second author (2013) constructed a functor from o(k)-Mod to o(k + 2)-Mod. In this paper, we use the functor successively to obtain an universal first-order differential operator realization for any highest-weight representation…

By introducing $N$-framed quivers, we define the localization of Lusztig's sheaves for $N$-framed quivers and functors $E^{(n)}_{i}, F^{(n)}_{i}, K^{\pm}_i$ for localizations. This gives a categorical realization of tensor products of…

We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced…

Fix any complex Kac-Moody Lie algebra $\mathfrak{g}$, and Cartan subalgebra $\mathfrak{h}\subset \mathfrak{g}$. We study arbitrary highest weight $\mathfrak{g}$-modules $V$ (with any highest weight $\lambda\in \mathfrak{h}^*$, and let…

We use the category of linear complexes of tilting modules for the BGG category O, associated with a semi-simple complex finite-dimensional Lie algebra g, to reprove in purely algebraic way several known results about O obtained earlier by…

In this paper we consider the structure and representation theory of truncated current algebras $\mathfrak{g}_m = \mathfrak{g}[t]/(t^{m+1})$ associated to the Lie algebra $\mathfrak{g}$ of a standard reductive group over a field of positive…

We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group $\overline{U}_q^H(\mfg)$ of a simple Lie algebra $\mfg$ at roots of unity, and study their categories…

We use the Kazhdan-Lusztig tensoring to define affine translation functors, describe annihilating ideals of highest weight modules over an affine Lie algebra in terms of the corresponding VOA, and to sketch a functorial approach to ``affine…

Let $\frak g$ be a finite dimensional complex semi-simple Lie algebra with Weyl group $W$ and simple reflections $S$. For $I\subseteq S$ let $\frak g_I$ be the corresponding semi-simple subalgebra of $\frak g$. Denote by $W_I$ the Weyl…

Suppose $g=g_0+g_1$ is a finite-dimensional restricted Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$. In this article, we propose a conjecture for maximal dimensions of irreducible modules over the…

The exchange graph of a cluster algebra encodes the combinatorics of mutations of clusters. Through the recent "categorifications" of cluster algebras using representation theory one obtains a whole variety of exchange graphs associated…

For any nullity $2$ extended affine Lie algebra $\mathcal{E}$ of maximal type and $\ell\in\mathbb{C}$, we prove that there exist a vertex algebra $V_{\mathcal{E}}(\ell)$ and an automorphism group $G$ of $V_{\mathcal{E}}(\ell)$ equipped with…

Motivated by the study of invariant rings of finite groups on the first Weyl algebras $A_{1}$ (\cite{AHV}) and finding interesting families of new noetherian rings, a class of algebras similar to $U(sl_{2})$ were introduced and studied by…

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra $U_q(q(n))$. The key ingredients are the triangular decomposition of $U_q(q(n))$ and the classification of finite dimensional…

For a field $F$ of characteristic zero and an additive subgroup $G$ of $F$, a Lie algebra $B(G)$ of lock type is defined with basis $\{L_{a,i},c|a \in G, i>-2\}$ and relations…

Let $Q$ be a finite quiver of Dynkin type and $\Lambda=\Lambda_Q$ be the preprojective algebra of $Q$ over an algebraically closed field $k$. Let $\mathcal {T}_\Lambda$ be the mutation graph of maximal rigid $\Lambda$ modules. Geiss,…

Let L >= 3. Using the moduli interpretation, we define certain elliptic modular forms of level Gamma(L) over any field k where 6L is invertible and k contains the Lth roots of unity. These forms generate a graded algebra R_L, which, over C,…