Title:

qSchur algebras and quantized enveloping algebras

The main aim of this thesis is to investigate the relationship between the quantized enveloping algebra U(gln) (corresponding to the Lie algebra gln) and the qSchur algebra, Sq(n, r). It was shown in [BLM] that there is a surjective algebra homomorphism θr : (gln)→Z[v, v 1] ⓍSq(n,r), where q = v2. §1 is devoted to background material. In §2, we show explicitly how to embed the qSchur algebra into the rth tensor power of a suitable n x n matrix ring. This gives a product rule for the qSchur algebra with similar properties to Schur's product rule for the unquantized Schur algebra. A corollary of this is that we can describe, in §2.3, a certain family of subalgebras of the qSchur algebra. In §3, we use the product rule of §2 to prove a qanalogue of Woodcock's straightening formula for codeterminants. This gives a basis of "standard quantized codeterminants" for Sq(n, r) which is heavily used in chapters 4, 5 and 6. In §4, we use the theory of quantized codeterminants developed in §3 to describe preimages under the homomorphism Or and the kernel of Or. In §5, we use the results of §3 and §4 to link the representation theories of U(gln} and Sq(n, r). We also obtain a simplified proof of Dipper and James' "semistandard basis theorem" for q Weyl modules of qSchur algebras. In §6, we show how to make the set of qSchur algebras Sq(n, r) (for a fixed n) into an inverse system. We prove that the resulting inverse limit, Sv(n), is a cellular algebra which is closely related to the quantized enveloping algebra U(sln) and Lusztig's algebra U.
