Constructions of stable equivalences of Morita type for finite dimensional algebras II

Suppose k is a field. Let A and B be two finite dimensional k-algebras such that there is a stable equivalence of Morita type between A and B. In this paper, we prove that (1) if A and B are representation-finite then their Auslander algebras are stably equivalent of Morita type; (2) The n-th Hochschild homology groups of A and B are isomorphic for all n≥1. A new proof is also provided for Hochschild cohomology groups of self-injective algebras under a stable equivalence of Morita type.     

Cohomology Algebras of Blocks of Finite Groups and Brauer Correspondence II

Let k be an algebraically closed field of characteristic p. We shall discuss the cohomology algebras of a block ideal B of the group algebra kG of a finite group G and a block ideal C of the block ideal of kH of a subgroup H of G which are in Brauer correspondence and have a common defect group, continuing (Kawai and Sasaki, Algebr Represent Theory 9(5):497–511, 2006). We shall define a (B,C)-bimodule L. The k-dual L ^* induces the transfer map between the Hochschild cohomology algebras of B and C, which restricts to the inclusion map of the cohomology algebras of B into that of C under some condition. Moreover the module L induces a kind of refinement of Green correspondence between indecomposable modules lying in the blocks B and C; the block varieties of modules lying in B and C which are in Green correspondence will also be discussed.     

Dynamical Systems Associated with Crossed Products

In this paper, we consider both algebraic crossed products of commutative complex algebras A with the integers under an automorphism of A, and Banach algebra crossed products of commutative C ^*-algebras A with the integers under an automorphism of A. We investigate, in particular, connections between algebraic properties of these crossed products and topological properties of naturally associated dynamical systems. For example, we draw conclusions about the ideal structure of the crossed product by investigating the dynamics of such a system. To begin with, we recall results in this direction in the context of an algebraic crossed product and give simplified proofs of generalizations of some of these results. We also investigate new questions, for example about ideal intersection properties of algebras properly between the coefficient algebra A and its commutant A′. Furthermore, we introduce a Banach algebra crossed product and study the relation between the structure of this algebra and the topological dynamics of a naturally associated system.     

Cyclic Homology of Hopf Comodule Algebras and Hopf Module Coalgebras

Abstract

In this paper we construct a cylindrical module A ? ? for an ?-comodule algebra A, where the antipode of the Hopf algebra ? is bijective. We show that the cyclic module associated to the diagonal of A ? ? is isomorphic with the cyclic module of the crossed product algebra A ? ?. This enables us to derive a spectral sequence for the cyclic homology of the crossed product algebra. We also construct a cocylindrical module for Hopf module coalgebras and establish a similar spectral sequence to compute the cyclic cohomology of crossed product coalgebras.     

Division algebras with a projective basis

Letk be any field andG a finite group. Given a cohomology class α∈H ²(G,k ^*), whereG acts trivially onk ^*, one constructs the twisted group algebrak ^αG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Z _n×Z_n). This paper has two main results: First we prove that ifD=k ^α G is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and , the division algebraD=k ^α G is a product of cyclic algebras. Furthermore, ifD _p is ap-primary factor ofD, thenD _p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and , the same result holds forD _p, p odd. Ifp=2 we show thatD ₂ is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z ₂×Z₂n.     

On Krull and Valuative Dimension of Tensor Products of Algebras

This article is concerned with the dimension theory of tensor products of algebras over a field k. In fact, we provide formulas for the Krull and valuative dimension of A? _k B when A and B are k-algebras such that the polynomial ring A[n] is an AF-domain for some positive integer n. Also, we compute dim _v (A? _k B) in the case where A ? B.     

Twisted Bimodules and Hochschild Cohomology for Self-injective Algebras of Class A n , II

Up to derived equivalence, the representation-finite self-injective algebras of class A _n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Möbius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Möbius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Möbius algebras.     

Invariant Cyclic Homology

We define a noncommutative analogue of invariant de Rham cohomology. More precisely, for a triple (A, H, M) consisting of a Hopf algebra H, an H-comodule algebra A, an H-module M, and a compatible grouplike element in H, we define the cyclic module of invariant chains on A with coefficients in M and call its cyclic homology the invariant cyclic homology of A with coefficients in M. We also develop a dual theory for coalgebras. Examples include cyclic cohomology of Hopf algebras defined by Connes–Moscovici and its dual theory. We establish various results and computations including one for the quantum group SL(q,2).     

Hochschild Cohomology of Quiver Algebras Defined by Two Cycles and a Quantum-Like Relation II

We consider quiver algebras A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of A _q and give necessary and sufficient conditions for A _q to have the finitely generated Hochschild cohomology ring.     

Hochschild Cohomology and the Coradical Filtration of Pointed Coalgebras: Applications

In this paper we show that there is a close connection between the coradical filtration of a pointed coalgebra and the Hochschild cohomology of that coalgebra with coefficients in some one-dimensional bicomodules. As an application, for a given prime numberpand an algebraically closed fieldkof characteristic 0, we classify all pointed Hopf algebras of dimensionp³overk.     

On skew field coproducts with a finite extension

It is shown that the field coproduct of any skew fieldE with a binomial (commutative) field extensionF/k overk can be expressed as a cyclic extension of a skew fieldK (theE-socle), itself the field coproduct of [F:k] copies ofE overk. Qua vector space the coproduct may also be expressed as a tensor product ofE andK overk. To the memory of Shimshon Amitsur     

On Codimension Growth of Finitely Generated Associative Algebras

LetAbe a PI-algebra over a fieldF. We study the asymptotic behavior of the sequence of codimensionsc_n(A) ofA. We show that ifAis finitely generated overFthenInv(A)=lim_n→∞ always exists and is an integer. We also obtain the following characterization of simple algebras:Ais finite dimensional central simple overFif and only ifInv(A)=dim=A.     

Topological *-algebras withC*-enveloping algebras II

UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC _c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC ^∞-elementsC ^∞(A), the analytic elementsC ^ω(A) as well as the entire analytic elementsC ^є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI _α is constructed satisfyingA =C*-ind limI _α; and the locally convex inductive limit ind limI _α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK _a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.     

Action on flag varieties: 2-Dimensional case

LetA be a finite dimensional commutative semisimple algebra over a fieldk and letV be a finitely generatedA-module. We examine the action of the general linear group GL _A (V) on the set of flags ofk-subspaces ofV. Also, let (V, B) be a finitely generated symplectic module overA. We also investigate the action of the symplectic group Sp _A (V, B) on the set of flags ofB-isotropick-subspaces ofV, whereB=°B is thek-symplectic form induced by a nonzerok-linear map :A k. In both cases, the orbits are completely classified in terms of certain integer invariants provided that dim _{k A}=2.This work is partially supported by a KOSEF research grant.     

Finite Group Actions with Fixed Points on Homology 3-Spheres

 If a finite group acts freely on a homology 3-sphere, then it has periodic cohomology. To say that a finite group F has periodic cohomology is equivalent to say that any Sylow subgroup of F of odd order is cyclic and a Sylow 2-subgroup of F is either cyclic or a quaternion group. In this paper we consider more generally smooth actions of finite groups G on homology 3-spheres which may have fixed points. We prove that any Sylow subgroup of G of odd order is either cyclic or the direct sum of two cyclic groups. Moreover, we show that if G has odd order, then it splits as a semidirect product of a subgroup A and a normal subgroup B such that B acts freely and there exist some simple closed curves in the homology 3-sphere which are fixed pointwise by some non-trivial element of A. We discuss the relation between these algebraic results and some classical constructions of the theory of 3-manifolds. Received 25 September 1997; in revised form 2 June 1998     

On free quadratic extensions of rings

The quaternion algebraB[j] over a commutative ringB with 1 defined byS. Parimala andR. Sridharan is generalized in two directions: (1) the ringB may be non-commutative with 1, and (2)j ² may be any invertible element (not necessarily –1). LetG={} be an automorphism group ofB of order 2, andA={b inB| (b)=b}. LetB[j] be a generalized quaternion algebra such thataj (a) for eacha inB. It will be shown thatB is Galois (for non-commutative ring extensions) overA which is contained in the center ofB if and only ifB[j] is Azumaya overA. Also,A[j] is a splitting ring forB[j] such thatA[j] is Galois overA. Moreover, we shall determine which automorphism group ofA[j] is a Galois group.     

Hochschild Cohomology of Quiver Algebras Defined by Two Cycles and a Quantum-Like Relation

We consider quiver algebras A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k, and describe the minimal projective bimodule resolution of A _q . In particular, in the case q = 1, we determine the Hochschild cohomology ring of A ₁ and show that it is a finitely generated k-algebra. Moreover the Hochschild cohomology ring of A ₁ modulo nilpotence is isomorphic to the polynomial ring of two variables.     

Restriction Between Cohomology Algebras of Blocks of Finite Groups

Let k be an algebraically closed field of characteristic p and G be a finite group. Let N be a normal subgroup of G and c be a G-stable block of kN. We shall discuss the cohomology algebra of the block c, defined by M. Linckelmann and, in this case a generalized block cohomology which can be defined using some generalized Brauer pairs, denoted (c, G)-Brauer pairs, which are introduced by R. Kessar and R. Stancu. We also analyze the restriction map between these two cohomology algebras associated to the block c through transfer maps between the Hochschild cohomology algebras of kGc and of the block c.     

Cyclic cohomology and algebra extensions

We construct a spectral sequence to study cyclic cohomology for an extension A = R/I of algebras. When R is a free algebra we describe the cyclic cohomology of A in terms of traces defined on R or powers of I. Explicit cyclic cocycles representing the cyclic cohomology class belonging to the trace are constructed as analogues of Chern character and Chern-Simons forms.Dedicated to Alexander Grothendieck