Real <Emphasis Type="Italic">AW</Emphasis><Superscript>*</Superscript>-Algebras of type I

Let R be a real AW ^*-algebra, and suppose that its complexification M = R + iR is also a (complex) AW ^*-algebra. We prove that R is of type I if and only if so is M.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 79–81, 2004Original Russian Text Copyright © by Sh. A. Ayupov     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

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.     

The Hodge Structure on a Filtered Boolean Algebra

Let (B _n) be the order complex of the Boolean algebra and let B(n, k) be the part of (B _n) where all chains have a gap at most k between each set. We give an action of the symmetric group S _l on the l-chains that gives B(n, k) a Hodge structure and decomposes the homology under the action of the Eulerian idempontents. The S _n action on the chains induces an action on the Hodge pieces and we derive a generating function for the cycle indicator of the Hodge pieces. The Euler characteristic is given as a corollary.We then exploit the connection between chains and tabloids to give various special cases of the homology. Also an upper bound is obtained using spectral sequence methods.Finally we present some data on the homology of B(n, k).     

The first cohomology group of trivial extensions of special biserial algebras

Given a finite dimensional special biserial algebra A with normed basis we obtain the dimension formulae of the first Hochschild homology groups of A and the vector space Alt(DA). As a consequence, an explicit dimension formula of the first Hochschild cohomology group of trivial extension TA = A×DA in terms of the combinatorics of the quiver and relations is determined.     

Homological Properties of Modules Over Group Algebras

Let G be a locally compact group, and let L¹ (G) be the Banachalgebra which is the group algebra of G. We consider a varietyof Banach left L¹ (G)-modules over L¹ (G), and seek to determineconditions on G that determine when these modules are eitherprojective or injective or flat in the category. The answerstypically involve G being compact or discrete or amenable. Forexample, in the case where G is discrete and 1 < p < ,we find that the module ^p (G) is injective whenever G is amenable,and that, if it is amenable, then G is ‘pseudo-amenable’,a property very close to that of amenability. 2000 MathematicsSubject Classification 46H25, 43A20.     

Dialgebra cohomology as a G-algebra

It is well known that the Hochschild cohomology of an associative algebra admits a G-algebra structure. In this paper we show that the dialgebra cohomology of an associative dialgebra has a similar structure, which is induced from a homotopy G-algebra structure on the dialgebra cochain complex .

     

On Quasi-Bicrossed Products of Weak Hopf Algebras

The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi-R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.     

Equations in Dual Variables for Whittaker Functions

It is known that the Whittaker functions w(q) associated with the group GL(N) are eigenfunctions of the Hamiltonians of the open Toda chain, hence satisfy a set of differential equations in the Toda variables q _i . Using the expression of the q _i for the closed Toda chain in terms of Sklyanin variables _i , and the known relations between the open and the closed Toda chains, we show that Whittaker functions also satisfy a set of new difference equations in _i .     

Generalized Sasaki Projections and Riesz Ideals in Pseudoeffect Algebras

A noncommutative version of generalized Sasaki projections in pseudoeffect algebras is introduced. It is proved that an ideal in a pseudoeffect algebra is Riesz if and only if it is closed under the right and left Sasaki projections. In lattice ordered pseudoeffect algebras, it is shown that generalized Sasaki projections are one-element sets, and their explicit form is found. It is shown that if a supremum of a normal Riesz ideal in a lattice ordered pseudoeffect algebra exists, it is a central element. These results extend those obtained recently by Avallone and Vitolo for effect algebras.