Infinitesimal quantum and little q-Schur algebras

We first follow De Concini and Kac [in: A. Connes, M. Dulfo, A. Joseph, R. Rentshler (Eds.), Operator Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory, 1990, pp. 471–506] to give a presentation for the infinitesimal quantum , uk(n), and then reconstruct or realize uk(n) in two different ways following the Beilinson–Lusztig–MacPherson geometric setting approach [Duke Math. J. 61 (1990) 655–677]. Thus, we obtain three new bases for uk(n). In the second part of the paper, we use uk(n) to introduce the little q-Schur algebra uk(n,r) as a subalgebra of the q-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 uk(n) and q-Schur algebras. We also obtain a formula for the dimension of uk(n,r).