We describe the structure of a Verma module with a generic highest weight at the critical level over a symmetrizable affine Lie superalgebra ĝ 6= A(2k, 2l). We obtain the character formula for a simple module with a generic highest weight at the critical level conjectured by V. G. Kac and D. A. Kazhdan.

We define a special sort of weighted oriented graphs, signed quivers. Each of these yields a symmetric quiver, i.e., a quiver endowed with an involutive anti-automorphism and the inherited signs. We develop a representation theory of symmetric quivers, in particular we describe the indecom-posable symmetric representations. Their dimensions constitute root systems corresponding to certain symme...

A Bernstein-Gelfand-Gelfand resolution for arbitrary Kac-Moody algebras and arbitrary subsets of the set of simple roots is proven. Moreover, quantum group analogs of the Bernstein-Gelfand-Gelfand resolution for symmetrizable Kac-Moody algebras are established. For quantized enveloping algebras with fixed deformation parameter q ∈ C \ {0} exactness is proven for all q which are not a root of un...

In this paper we study the cocenter of cyclotomic quiver Hecke algebra $${\mathscr {R}}^\Lambda _\alpha $$ associated to an arbitrary symmetrizable Cartan matrix $$A=(a_{ij})_{{i,j}\in I}$$ , $$\Lambda \in P^+$$ and $$\alpha Q_n^+$$ . We introduce a notion called “piecewise dominant sequence” use it construct some explicit homogeneous elements which span Our first main result shows that minimal...

We determine the Lie subalgebra gnil of a Borcherds symmetrizable generalized Kac-Moody algebra g generated by ad-locally nilpotent elements and show that it is ‘essentially’ same as Levi with its simple roots precisely real g.