Search

We extend [13] to the nonsemisimple case. We define and study exact factorizations B = A • C of a finite tensor category B into a product of two tensor subcategories A , C , and relate exact factorizations of finite tensor categories to exact sequences of finite tensor categories with respect to exact module categories [8]. We apply our results to study exact factorizations of quasi-Hopf algebras, and extensions of a finite group scheme theoretical tensor category [14] by another one. We also provide several examples to illustrate our results. [ABSTRACT FROM AUTHOR]

We use [ 11 ] to study the algebra structure of twisted cotriangular Hopf algebras |${}_J\mathcal{O}(G)_{J}$|⁠ , where |$J$| is a Hopf |$2$| -cocycle for a connected nilpotent algebraic group |$G$| over |$\mathbb{C}$|⁠. In particular, we show that |${}_J\mathcal{O}(G)_{J}$| is an affine Noetherian domain with Gelfand–Kirillov dimension |$\dim (G)$|⁠ , and that if |$G$| is unipotent and |$J$| is supported on |$G$|⁠ , then |${}_J\mathcal{O}(G)_{J}\cong U({\mathfrak{g}})$| as algebras, where |${\mathfrak{g}}={\textrm{Lie}}(G)$|⁠. We also determine the finite dimensional irreducible representations of |${}_J\mathcal{O}(G)_{J}$|⁠ , by analyzing twisted function algebras on |$(H,H)$| -double cosets of the support |$H\subset G$| of |$J$|⁠. Finally, we work out several examples to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract: Let be a finite-dimensional complex semisimple Lie algebra, and let V be a finite-dimensional complex representation of . We give a closed formula for the mth Frobenius–Schur indicator, , of V in representation-theoretic terms. We deduce that the indicators take integer values, and that for a large enough m, the mth indicator of V equals the dimension of the zero weight space of V. For the classical complex Lie algebras , , and , this is the case for m greater or equal to , , and , respectively. [Copyright &y& Elsevier]

Abstract: We construct new finite dimensional basic quasi-Hopf algebras of dimension , , parametrized by primitive roots of unity of order , with radical of codimension , which generalize the construction of the basic quasi-Hopf algebras of dimension 8 given in [3]. These quasi-Hopf algebras are not twist equivalent to a Hopf algebra, and may be regarded as quasi-Hopf analogs of Taft Hopf algebras. By [4], our construction is equivalent to the construction of new finite tensor categories whose simple objects form a cyclic group of order , and which are not tensor equivalent to a representation category of a Hopf algebra. We also prove that if is a finite dimensional radically graded quasi-Hopf algebra with , where is prime and is a nontrivial associator, such that is a free left module over of rank 1 (it is always free), then is isomorphic to . [Copyright &y& Elsevier]

Let k be an algebraically closed field of characteristic 0 or p > 2. Let G be an affine supergroup scheme over k. We classify the indecomposable exact module categories over the tensor category sCoh f (G) of (coherent sheaves of) finite dimensional O (G) -supermodules in terms of (H , Ψ) -equivariant coherent sheaves on G. We deduce from it the classification of indecomposable geometrical module categories over sRep (G). When G is finite, this yields the classification of all indecomposable exact module categories over the finite tensor category sRep (G). In particular, we obtain a classification of twists for the supergroup algebra k G of a finite supergroup scheme G , and then combine it with [7, Corollary 4.1] to classify finite dimensional triangular Hopf algebras with the Chevalley property over k. [ABSTRACT FROM AUTHOR]

Abstract: Let k be any field. We consider the Hopf–Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of finite-dimensional Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative finite-dimensional Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a finite-dimensional Hopf algebra over k, revealing so that the Hopf–Schur group can be much larger than the Schur group of k. [Copyright &y& Elsevier]