We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, having abelian group of grouplike elements, under some mild restrictions on the group order. The proof uses the recent classification by Andruskiewitsch and Schneider of such Hopf algebras. Examples include all of Lusztig’s small quantum groups, whose cohomology was first computed explicitly by Gin...

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. In order to do that we define the Hochschild cohomology of an algebra in an abelian monoidal category. Then we characterize those algebras which have dimension less than or equal to 1 with respect to Hochschild cohomolo...

The notion of crossed product with a coquasi-Hopf algebra H is introduced and studied. The result of such a crossed product is an algebra in the monoidal category of right H-comodules. We give necessary and sufficient conditions for two crossed products to be equivalent. Then, two structure theorems for coquasi Hopf modules are given. First, these are relative Hopf modules over the crossed prod...

We find a self-dual noncommutative and noncocommutative Hopf algebra HF acting as a universal symmetry on the modules over inner Frobenius algebras of modular categories (as used in two dimensional boundary conformal field theory) similar to the GrothendieckTeichmüller groupGT as introduced by Drinfeld as a universal symmetry of quasitriangular quasi-Hopf algebras. We discuss the relationship t...

Hopf algebra structure on the differential algebra of the extended q-plane is defined. An algebra of forms which is obtained from the generators of the extended q-plane is introduced and its Hopf algebra structure is given. E-mail: [email protected] E-mail: [email protected]

Let (H,R) be a quasitriangular weak Hopf algebra, and A a quantum commutative weak H-module algebra. We establish the relationship of homological dimensions between weak smash product algebra A#H and A under some conditions. As an application, we consider the case of twisted weak Hopf algebra. Mathematics Subject Classification (2010): 16T05

We extend the Larson–Sweedler theorem for weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We establish the autonomous monoidal category of the modules of a weak Hopf algebra A and show the semisimplicity of the unit and the invertible modules of A. We also reveal the connection of these modules to lef...

The planar dilatation operator of N = 4 supersymmetric Yang-Mills is the hamiltonian of an integrable spin chain whose length is allowed to fluctuate. We will identify the dynamics of length fluctuations of planar N = 4 Yang-Mills with the existence of an abelian Hopf algebra Z symmetry with non-trivial co-multiplication and antipode. The intertwiner conditions for this Hopf algebra will restri...

We study the basic monoidal properties of the category of Hopf modules for a coquasi Hopf algebra. In particular we discuss the so called fundamental theorem that establishes a monoidal equivalence between the category of comodules and the category of Hopf modules. We present a categorical proof of Radford’s S formula for the case of a finite dimensional coquasi Hopf algebra, by establishing a ...