Transformation Groups 

About: Transformation Groups is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Lie algebra & Algebraic group. It has an ISSN identifier of 1083-4362. Over the lifetime, 941 publications have been published receiving 16424 citations.

Module categories, weak Hopf algebras and modular invariants

Abstract: We develop a theory of module categories over monoidal categories (this is a straightforward categorization of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and we classify module categories over the fusion category of sl(2) at a positive integer level where we meet once again the ADE classification pattern.

Cones, crystals, and patterns

Abstract: There are two well known combinatorial tools in the representation theory ofSLn, the semi-standard Young tableaux and the Gelfand-Tsetlin patterns. Using the path model and the theory of crystals, we generalize the concept of patterns to arbitrary complex semi-simple algebraic groups.

Equivariant Chow groups for torus actions

Abstract: We study Edidin and Graham's equivariant Chow groups in the case of torus actions. Our main results are: (i) a presentation of equivariant Chow groups in terms of invariant cycles, which shows how to recover usual Chow groups from equivariant ones; (ii) a precise form of the localization theorem for torus actions on projective, nonsingular varieties; (iii) a construction of equivariant multiplicities, as functionals on equivariant Chow groups; (iv) a construction of the action of operators of divided differences on theT-equivariant Chow group of any scheme with an action of a reductive group with maximal torusT. We apply these results to intersection theory on varieties with group actions, especially to Schubert calculus and its generalizations. In particular, we obtain a presentation of the Chow ring of any smooth, projective spherical variety.

Quasi-Hopf twistors for elliptic quantum groups

Abstract: The Yang-Baxter equation admits two classes of elliptic solutions, the vertex type and the face type. On the basis of these solutions, two types of elliptic quantum groups have been introduced (Foda et al. [FIJKMY1], Felder [Fe]). Fronsdal [Fr1, Fr2] made a penetrating observation that both of them are quasi-Hopf algebras, obtained by twisting the standard quantum affine algebraU q(g). In this paper we present an explicit formula for the twistors in the form of an infinite product of the universalR matrix ofU q(g). We also prove the shifted cocycle condition for the twistors, thereby completing Fronsdal's findings. This construction entails that, for generic values of the deformation parameters, the representation theory forU q(g) carries over to the elliptic algebras, including such objects as evaluation modules, highest weight modules and vertex operators. In particular, we confirm the conjectures of Foda et al. concerning the elliptic algebraA q,p ( $$\widehat{\mathfrak{s}\mathfrak{l}}_2 $$ ).

Hyperbolic localization of intersection cohomology

Abstract: For a normal variety X defined over an algebraically closed field with an action of the multiplicative group T = Gm, we consider the "hyperbolic localization" functor Db(X) → Db(XT), which localizes using closed supports in the directions flowing into the fixed points, and compact supports in the directions flowing out. We show that the hyperbolic localization of the intersection cohomology sheaf is a direct sum of intersection cohomology sheaves.