Batalin-Vilkovisky Algebras and Cyclic Cohomology of Hopf Algebras

We show that the Connes–Moscovici negative cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree -2. More generally, we show that a cyclic operad with multiplication is a cocyclic module whose simplicial cohomology is a Batalin–Vilkovisky algebra and whose negative cyclic cohomology is a graded Lie algebra of degree -2. This generalizes the fact that the Hochschild cohomology algebra of a symmetric algebra is a Batalin–Vilkovisky algebra.     

Fourier Invariant Partially Approximating Subalgebras of the Irrational Rotation C*-Algebra

For a dense G-set of parameters, the irrational rotation algebrais shown to contain infinitely many C*-subalgebras satisfyingthe following properties. Each subalgebra is isomorphic to adirect sum of two matrix algebras of the same (perfect square)dimension; the Fourier transform maps each summand onto theother; the corresponding unit projection is approximately central;the compressions of the canonical generators of the irrationalrotation algebra are approximately contained in the subalgebra.2000 Mathematics Subject Classification 46L80, 46L40, 46L35.     

On a Theorem of Høegh-Krohn

A theorem proved by R. Høegh-Krohn in Comm. Math. Phys. 38(1974), 195–224, which yields a possibility to define states of systems of quantum particles by their values on the products , where \mathfraka _t , t are time automorphisms and F _j are multiplication operators, is generalized and extended. In particular, it is shown that the algebras generated by such products with F _j taken from the families of multiplication operators satisfying certain conditions are dense in the algebras of observables in the -weak topology, in which normal states are continuous. This result was obtained for the systems with two types of kinetic energy: the usual one expressed by means of the Laplacian; the relativistic kinetic energy defined by a pseudo-differential operator.     

nvolutorial Płonka sums

The construction of the sum of a direct (semilattice ordered) system of algebras introduced by J. Plonka – later known as the Plonka sum – is one of the most important methods of composition in universal algebra, having a number of applications in different algebraic theories, such as semigroup theory, semiring theory, etc. In this paper we present a more general way for constructing algebras with involution, that is, algebraic systems equipped with a unary involutorial operation which is at the same time an antiautomorphism of the underlying algebra. It is the sum – involutorial Plonka sum, as we call it – of an involution semilattice ordered system of algebras. We investigate its basic properties, as well as the problem of its subdirect decomposition.     

Induced product representations of extended Cuntz algebras

Let _H be the extended Cuntz algebra over the Hilbert space H. Since its zero grade part _H⁰ is the C^*-inductive limit of B(H^r), we look for some family of representations on an inductive limit of H^r as r. When such construction is shaped according to the structure of _H⁰, von Neumanns notion of a reference sequence of unit vectors for Hilbert infinite tensor products emerges; after a further Rieffel induction step, a class IPR[H] of representations of _H arises. For any two such representations, we describe explicitly their associated intertwiners. Any two representations in IPR[H] are either disjoint or unitarily equivalent. Actions of the group by translation on sequences of unit vectors are involved, as well as the ideals of .     

On a Lift of an Individual Stable Equivalence to a Standard Derived Equivalence for Representation-Finite Self-injective Algebras

We shall show that every stable equivalence (functor) between representation-finite self-injective algebras not of type (D _3m ,s/3,1) with m2, 3s lifts to a standard derived equivalence. This implies that all stable equivalences between these algebras are of Morita type.     

On Semisimple Hopf Algebras of Dimension 2 m

In this paper we classify all nontrivial semisimple Hopf algebras of dimension 2 ^{n +1} with the group of grouplikes isomorphic to _{2 n–1}×₂. Moreover, we extend some results on irreducible representations from groups to semisimple Hopf algebras and prove that certain semisimple Hopf algebras, including the ones classified in this paper, satisfy the generalized power map property.     

Galois Stability, Integrality and Realization Fields for Representations of Finite Abelian Groups

For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups GGL _n (E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E=F(G) that are obtained via adjoining all matrix coefficients of all matrices gG to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E=F(G) for prescribed integers n and t or prescribed n and d.     

On ＂Problems on von Neumann Algebras by R. Kadison, 1967＂

A brief summary of the development on Kadison‘s famous problems (1967) is given. A new set of problems in von Neumann algebras is listed.     

Anick's spaces and the double loops of odd primary Moore spaces

Several properties of Anick's spaces are established which give a retraction of Anick's off if and . The proof is alternate to and more immediate than the two proofs of Neisendorfer's.