Infinitesimal Quantum Gl_n And Associated Little Q-Schur Algebras

In [14], Doty, Nakano and Peters define infinitesimal Schur algebras, closely related to the Frobenius kernel of an algebraic group over a field of positive characteristic. A theory for a quantum version of the infinitesimal Schur algebra is studied by Anton Cox in [4]. Motivated by above papers, from the view of quantized enveloping algebras, we introduce the little g-Schur algebra.In this paper, We first follow De Concini and Kac [3] to give a presentation for the infinitesimal quantum gIn,, u(n), and then reconstruct or realize u(n) in two different ways following the Beilinson-Lusztig-MacPherson geometric setting approach [1]. Thus, we obtain three new bases for u(n). In the second part of the paper, we use u(n) to introduce the little g-Schur algebra u(n, r) as a sub-algebra of the g-Schur algebra Uk(n,r). The symmetry structure of a little q-Schur algebra is then investigated through the construction of various bases of monomial, BLM and PBW types for u(n) and g-Schur algebras. We also obtain a formula for the dimension of uk(n,r) and we shall describe the relation between the infinitesimal g-Schur algebra defined in [4] and the little g-Schur algebra in this paper. Furthermore, we obtain a new monomial basis fot the g-Schur algebra. And we shall use the new monomial basis to prove the conjecture in [13, 2.2]. Furthermore, we study the relation between the little and the infinitesimal g-Schur algebra. And, we shall study the little Weyl module fot the little quantum group. We shall give the complete set of the irreducible module for the little g-Schur algebra. And we shall study the conjecture in 13, 2.3]. This conjecture is not true, and we shall prove the corresponding result by adding a condition.