Homotopy and homology of simplicial abelian Hopf algebras

Let A be a simplicial bicommutative Hopf algebra over the field with the property that . We show that is a functor of the André-Quillen homology of A, where A is regarded as an algebra. Then we give a method for calculating that André-Quillen homology independent of knowledge of . Received November 15, 1996 ; in final form March 15, 1997     

Abstract convexity:examples and applications

This paper presents a survey of some results from and applications of abstract convexity based on the notions of Minkowski duality, supremal generators, subdifferentials and conjugations. The paper contains very many examples, which are its essential part     

Homotopy algebras and the inverse of the normalization functor

In this paper, we investigate multiplicative properties of the classical Dold-Kan correspondence. The inverse of the normalization functor maps commutative differential graded algebras to E_∞-algebras. We prove that it in fact sends algebras over arbitrary differential graded E_∞-operads to E_∞-algebras in simplicial modules and is part of a Quillen adjunction. More generally, this inverse maps homotopy algebras to weak homotopy algebras. We prove the corresponding dual results for algebras under the conormalization, and for coalgebra structures under the normalization resp. the inverse of the conormalization.     

Supplemental examples on complementary matrix algebras

In [1], M.-D. Choi, H. Radjavi and P. Rosenthal proved that, for each integer n4 (except 7 and 11), complementary algebras of linear operators on ⁿ do not necessarily have nontrivial complementary invariant subspaces. In this note we give counter-examples for the remaining two cases.     

Homotopy stability of orthogonal spaces in commutative banach algebras

We study the concept of real stable rank for a complex commutative Banach algebra A (rsr A). It is shown that this invariant has a behaviour completely analogous to the classical Bass stable rank; in particular, we establish a precise relation between both invariants.Further, we use this machinery to show that the connected components of the orthogonal spaces of A, O _k ⁿ (A)={ A ^k×n:^t=id_k×k}, stabilize when k and n increase in a way that depends on rsr A Finally, we prove an analogous stabilization result for the homotopy classes of maps from the j-sphere into O _k ⁿ (A), where j is an arbitrary positive integer.Research supported by a grant of the CONICET, Argentina.     

Nilpotent algebras and their applications

The work is devoted to the structure of finite nilpotent algebras and their applications, first of all, to the structure of products of groups. Some questions concerning the application of nilpotent algebras (and related structures) to the construction of recurrent sequences are touched upon.     

Homotopy G-algebra structure on the cochain complex of hom-type algebras

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. We show that the Hochschild type cochain complex of a hom-associative algebra carries a homotopy G-algebra structure. As a consequence, we get a Gerstenhaber algebra structure on the cohomology of a hom-associative algebra. We also find similar results for hom-dialgebras.     

More examples of bicrossproduct and double cross product Hopf algebras

We construct examples of bicrossproducts and double cross products of quantum groupsă(R) associated to general matrix solutionsR of the Quantum Yang-Baxter Equations. We also describe iterated double cross products of quantum groups. In the course of constructingă(R) we are led to introduce a suitable notion of mutually dual Hopf algebras and a dual quantum groupŬ(R). Work supported by SERC Research Assistantship.     

Some examples of semigroup algebras of finite representation type

The semigroup algebras over a field K of the semigroups T_n of all permutations of a set of n elements are considered. It is proved: if n≤3 and (n!)^-1∈ K then the algebra KT_n has a finite representation type. Also the finiteness of the representation type of the semigroup algebra KS is established, where S is the sub-semigroup of T_n (n is arbitrary) such that S=J_n∪G where J_n={x∈T_n|rank x=1}, while G is a doubly transitive subgroup of the symmetric group S_n, the order of G being invertible in K. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 229–238, 1987.     

Quasi-trace functions on Lie algebras and their applications to 3-Lie algebras

Czechoslovak Mathematical Journal - We introduce the notion of quasi-trace functions on Lie algebras. As applications we study realizations of 3-dimensional and 4-dimensional 3-Lie algebras. Some...     

Duality and operator algebras: automatic weak* continuity and applications

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras, and give several applications of the surprising fact that certain maps are always weak^*-continuous on dual spaces. In particular, if X is a subspace of a C^*-algebra A, and if a∈A satisfies aX⊂X, then we show that the function x?ax on X is automatically weak^* continuous if either (a) X is a dual operator space, or (b) a^*X⊂X and X is a dual Banach space. These results hinge on a generalization to Banach modules of Tomiyama's famous theorem on contractive projections onto a C^*-subalgebra. Applications include a new characterization of the σ-weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and a generalization of the theory of W^*-modules to the framework of modules over such algebras. We also give a Banach module characterization of σ-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.     

Braided algebras and their applications to Noncommutative Geometry

We introduce the notion of a braided algebra and study some examples of these. In particular, R-symmetric and R-skew-symmetric algebras of a linear space V equipped with a skew-invertible Hecke symmetry R are braided algebras. We prove the “mountain property” for the numerators and denominators of their Poincaré–Hilbert series (which are always rational functions).