Highest-weight vectors in tensor products of Verma modules for U_q(sl_2)

We obtain an explicit basis for the subspace spanned by highest-weight vectors in a tensor product of two highest-weight modules for the quantized universal enveloping algebra of sl_2. The structure constants provide a generalization of the Clebsh-Gordan coefficients. As a byproduct, we give an alternative proof for the decomposition of these tensor products as direct sums of indecomposable modules and supply generators for all highest weight summands.

Orthogonal basis for the Shapovalov form on $U_q(\mathfrak{sl}(n + 1))$

Let U be either the classical or quantized universal enveloping algebra of the Lie algebra [Formula: see text] extended over the field of fractions of the Cartan subalgebra. We suggest a PBW basis in U over the extended Cartan subalgebra diagonalizing the contravariant Shapovalov form on generic Verma module. The matrix coefficients of the form are calculated and the inverse form is explicitly constructed.

Canonical bases and higher representation theory

This paper develops a general theory of canonical bases and how they arise naturally in the context of categorification. As an application, we show that Lusztig’s canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is of finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest- and highest-weight integrable representations. This generalizes past work of the author’s in the highest-weight case.

Assembling Crystals of Type A

Regular An-crystals are certain edge-colored directed graphs, which are related to representations of the quantized universal enveloping algebra Uq(𝔰𝔩n+1). For such a crystal K with colors 1,2,…,n, we consider its maximal connected subcrystals with colors 1,…,n-1 and with colors 2,…,n and characterize the interlacing structure for all pairs of these subcrystals. This enables us to give a recursive description of the combinatorial structure of K via subcrystals and develop an efficient procedure of assembling K.

THE HEISENBERG FAMILY FOR THE PRINCIPAL GRADING OF $U_q(\hat{\cal G})$

We describe existence conditions and explicitly construct elements for a Heisenberg family in the principal grading of the quantized universal enveloping algebra [Formula: see text] of an affine Kac–Moody algebra [Formula: see text] in the Drinfeld formulation.