A counterpart of the Verlinde algebra for the small quantum group. (English) Zbl 1036.17014

Lachowska, Anna

Duke Math. J. 118, No. 1, 37-60 (2003).

Let \(\mathfrak g\) be a semisimple complex Lie algebra, \(q\) a root of unity of odd order, and \(U_q^{\text{ fin}}({\mathfrak g})\) the associated small quantum group. The author shows that the ideal spanned by the characters of projective modules in the Grothendieck ring of the category of finite-dimensional modules over \(U_q^{\text{ fin}}({\mathfrak g})\) has a description parallel to that of the Verlinde algebra of the fusion category, with the character of the Steinberg module on the role of the identity.

Reviewer: Sorin Dascalescu (Safat)

MSC:

17B37	Quantum groups (quantized enveloping algebras) and related deformations
16W30	Hopf algebras (associative rings and algebras) (MSC2000)
81R50	Quantum groups and related algebraic methods applied to problems in quantum theory

Keywords:

small quantum group; fusion category; Steinberg module; Verlinde algebra

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