Presenting affine infinitesimal Schur algebras.

Let U ( gl ˆ n) be the universal enveloping algebra of gl ˆ n. A certain Z -form U Z ( gl ˆ n) of U ( gl ˆ n) was introduced in [15] such that the affine Schur algebra S ▵ (n , r) Z is a homomorphic image of U Z ( gl ˆ n). Let k be a field and let U k ( gl ˆ n) = U Z ( gl ˆ n) ⊗ k. Let S ▵ (n , r) k = S ▵ (n , r) Z ⊗ k. By tensoring with k , we have a surjective algebra homomorphism ξ r , k : = ξ r ⊗ i d : U k ( gl ˆ n) ↠ S ▵ (n , r) k. For h ⩾ 1 and r ∈ N let s ▵ (n , r) h = ξ r , k (s ▵ (n) h) , where s ▵ (n) h is a certain subalgebra of U k ( gl ˆ n). The algebra s ▵ (n , r) h is the affine analogue of infinitesimal Schur algebras introduced in [7]. In this paper we will give a presentation of s ▵ (n , r) 1 by generators and defining relations. [ABSTRACT FROM AUTHOR]