The quantum general linear supergroup, canonical bases and Kazhdan-Lusztig polynomials

Canonical bases of the tensor powers of the natural $ U_q (\mathfrak{g}\mathfrak{l}_{m|n} ) $ -module V are constructed by adapting the work of Frenkel, Khovanov and Kirrilov to the quantum supergroup setting. This result is generalized in several directions. We first construct the canonical bases of the ?2-graded symmetric algebra of V and tensor powers of this superalgebra; then construct canonical bases for the superalgebra O q (M m|n ) of a quantum (m,n) × (m,n)-supermatrix; and finally deduce from the latter result the canonical basis of every irreducible tensor module for $ U_q (\mathfrak{g}\mathfrak{l}_{m|n} ) $ by applying a quantum analogue of the Borel-Weil construction.

Canonical bases of the tensor powers of the natural Uq(glm|n)-module V are constructed by adapting the work of Frenkel, Khovanov and Kirrilov to the quantum supergroup setting. This result is generalized in several directions. We first construct the canonical bases of the Z2-graded symmetric algebra of V and tensor powers of this superalgebra; then construct canonical bases for the superalgebra Oq(Mm|n) of a quantum (m, n) × (m, n)-supermatrix; and finally deduce from the latter result the canonical basis of every irreducible tensor module for Uq(glm|n) by applying a quantum analogue of the Borel-Weil construction.