If H is a commutative connected graded Hopf algebra over a commutative ring k, then a certain canonical k-algebra homomorphism H → H⊗QSymk is defined, where QSymk denotes the Hopf algebra of quasisymmetric functions. This homomorphism generalizes the “internal comultiplication” on QSymk, and extends what Hazewinkel (in §18.24 of his “Witt vectors”) calls the Bernstein homomorphism. We construct...

Let H be a finite dimensional quasi-Hopf algebra over a field k and A a right H-comodule algebra in the sense of [12]. We first show that on the k-vector space A⊗H∗ we can define an algebra structure, denoted by A # H∗, in the monoidal category of left H-modules (i.e. A # H∗ is an Hmodule algebra in the sense of [2]). Then we will prove that the category of two-sided (A,H)bimodules HM H A is is...

Miriam Cohen raised the question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. In this paper we show that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. In particular we show that the central H-invariant elements of the Martindale ring of quotients of a module algebra for...

Following [3] we confirm a conjecture of Yde Venema by piecing together results from various authors. Specifically, we show that if A is a residually finite, finitely generated modal algebra such that HSP(A) has equationally definable principal congruences, then the profinite completion of A is the MacNeille completion of A, and ♦ is smooth. Specific examples of such modal algebras are the free...

Miriam Cohen raised the question whether the smash product of a semisimple Hopf algebra and a semiprime module algebra is semiprime. In this paper we show that the smash product of a commutative semiprime module algebra over a semisimple cosemisimple Hopf algebra is semiprime. In particular we show that the central H-invariant elements of the Martindale ring of quotients of a module algebra for...

We study a finite-dimensional quotient of the Hecke algebra of type H n for general n, using a calculus of diagrams. This provides a basis of monomials in a certain set of generators. Using this, we prove a conjecture of C.K. Fan about the semisimplicity of the quotient algebra. We also discuss the cellular structure of the algebra, with certain restrictions on the ground ring. To appear in " M...

We introduce the cylindrical module A♮H, where H is a Hopf algebra with S2 = idH and A is a Hopf module algebra over H. We show that there exists a cyclic map between the cyclic module of the crossed product algebra A⋊H and ∆(A♮H), the cyclic module related to the diagonal of A♮H. In the cocommutative case, ∆(A♮H) ∼= C•(A ⋊H). Finally we approximate ∆(A♮H) by a spectral sequence and we give an ...

A current Lie algebra is contructed from a tensor product of a Lie algebra and a commutative associative algebra of dimension greater than 2. In this work we are interested in deformations of such algebras and in the problem of rigidity. In particular we prove that a current Lie algebra is rigid if it is isomorphic to a direct product g× g × ...× g where g is a rigid Lie algebra. 1 Current Lie ...

Let H be a finite Hopf algebra with CH,H = C −1 H,H . The duality theorem is shown for H, i.e., (R#H)#H ∗̂ ∼= R ⊗ (H⊗̄H ) as algebras in C. Also, it is proved that the Drinfeld double (D(H), [b]) is a quasi-triangular Hopf algebra in C. 2000 Mathematics subject Classification: 16w30.

The R12 h matrices of the Jordanian Uh(sl(2)) algebra at arbitrary dimensions may be obtained from the corresponding Rj1;j2 q matrices of the standard q-deformed Uq(sl(2)) algebra through a contraction technique. By extending this method, the coloured two-parametric (h, α) Jordanian R1122 h,α matrices of the Uh,α(gl(2)) algebra may be derived from the corresponding coloured R j1,z1;j2,z2 q,λ ma...