Our goal in this paper is to define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called “module algebras” (Definition 2.1). Our motivation lies in the following problem: for an algebra A which admits a module structure over an arbitrary bialgebra B compatible with its product structure, the Hochschild or the cyclic ...

Cλ-extended oscillator algebras generalizing the Calogero-Vasiliev algebra, where Cλ is the cyclic group of order λ, are studied both from mathematical and applied viewpoints. Casimir operators of the algebras are obtained, and used to provide a complete classification of their unitary irreducible representations under the assumption that the number operator spectrum is nondegenerate. Deformed ...

We investigate the higher-dimensional amenability of tensor products A ⊗̂ B of Banach algebras A and B. We prove that the weak bidimension dbw of the tensor product A⊗̂B of Banach algebras A and B with bounded approximate identities satisfies dbwA ⊗̂ B = dbwA+ dbwB. We show that it cannot be extended to arbitrary Banach algebras. For example, for a biflat Banach algebra A which has a left or right...

We consider algebras over a field k of characteristic zero. The article is concerned with the isomorphism of graded vectorspaces H(gl(A)) ∼ → ∧(HC(A)[−1]) between the Lie algebra homology of matrices and the free graded commutative algebra on the cyclic homology of the k-algebra A, shifted down one degree. For unital algebras this isomorphism is a classical result obtained by Loday and Quillen ...

1. Let A1, . . . ,An be central simple disjoint algebras over a field F . Let also li|exp(Ai), mi|ind(Ai), li|mi, and for each i = 1, . . . , n, let li and mi have the same sets of prime divisors. Then there exists a field extension E/F such that exp(AiE) = li and ind(AiE) = mi, i = 1, . . . , n. 2. Let A be a central simple algebra over a field K with an involution τ of the second kind. We pro...

Let $\A$ be a finitary hereditary abelian category with enough projectives. We study the Hall algebra of complexes fixed size over Explicitly, we first give relation between algebras and cyclic complexes. Secondly, characterize by generators relations, relate it to derived $\A$. Finally, integration map on $2$-term

where g±d 6= 0. The positive integer d is called the depth of this Z-grading, and of the nilpotent element e. This notion was previously studied e.g. in [P1]. An element of g of the form e+ F , where F is a non-zero element of g−d, is called a cyclic element, associated with e. In [K1] Kostant proved that any cyclic element, associated with a principal (= regular) nilpotent element e, is regula...

In this paper we give a complete classiication of the minimal cyclic M q (n)-modules and construct them explicitly. Also, we give a complete classiica-tion of the minimal cyclic modules of the so-called Dipper-Donkin quantum matrix algebra as well as of two other natural quantized matrix algebras. In the last part of the paper we relate the results to the De Concini { Procesi conjecture. 1. int...

Borrowing from a classical construction for counterexamples to the Hasse principle, we show that for a certain family of affine varieties which do not satisfy the Hasse principle, the Brauer-Manin obstruction is not satisfied. The approach is elementary and requires little algebraic geometry.

We define an equivariant K0-theory for Yetter-Drinfeld algebras over a Hopf algebra with an invertible antipode. We show that there exists a pairing, generalizing Connes’ pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.