Abstract: We will introduce the Clifford algebra associated to a vector space equipped a quadratic form. As an important case, then we give a description of the Clifford algebra for $\mathbb{R}^n$ and $\mathbb{C}^n$, equipped with the standard quadratic form, as a subalgebra of matrix algebras and prove the periodicity theorem for these algebras. Moreover, representation theory of the Clifford algebras will be discussed.

Abstract: Nakajima constructed highest weight representations of A-type affine Kac-Moody algebras by using the (equivariant) cohomology of cyclic quiver varieties. In the present talk I will describe a geometric realization of level one highest weight representations of A-type affine Kac-Moody algebras by using moduli spaces of sheaves on a 2-dimensional root toric stack over the minimal resolution of the Kleinian singularity $\mathbb{C}^2/\mathbb{Z}_k$. If time permits, I will explain the conjectural relation between these two geometric constructions.