Hochschild (Co)Homology of Hopf Crossed Products*

For a general crossed product E = A#_f H, of an algebra A by a Hopf algebra H, we obtain complexes smaller than the canonical ones, giving the Hochschild homology and cohomology of E. These complexes are equipped with natural filtrations. The spectral sequences associated to them coincide with the ones obtained using a natural generalization of the direct method introduced in Trans. Amer. Math. Soc. 74 (1953) 110–134. We also get that if the 2-cocycle f takes its values in a separable subalgebra of A, then the Hochschild (co)homology of E with coefficients in M is the (co)homology of H with coefficients in a (co)chain complex.     

Leibniz and Hochschild homology

We construct and study the map from Leibniz homology HL_?(𝔥) of an abelian extension 𝔥 of a simple real Lie algebra 𝔤 to the Hochschild homology HH_??1(U(𝔥)) of the universal envelopping algebra U(𝔥). To calculate some homology groups, we use the Hochschild-Serre spectral sequences and Pirashvili spectral sequences. The result shows what part of the non-commutative Leibniz theory is detected by classical Hochschild homology, which is of interest today in string theory.     

Compact DG modules and Gorenstein DG algebras

When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras. When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness. For a finite connected DG algebra A, we prove that D^c(A) and D^c(A ^op) admit Auslander-Reiten triangles if and only if A and A ^op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, D^c(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D_lf^b(A) and D_lf^b (A ^op) instead, when A is a regular DG algebra. This work was supported by the National Natural Science Foundation of China (Grant No. 10731070) and the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)     

无限箭图的同调群

对任意箭图Q,我们研究路代数A=kQ的Hochschild同调群H_n(A)和同调群Tor_n^Ae(A,A),其中A^e是代数A的包络代数。在本文中,我们具体地给出了各次同调群和Hochschild同调群。     

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.     

More on the Hochschild and Cyclic Homologies of Crossed Modules of Algebras

We investigate the Hochschild and cyclic homologies of crossed modules of algebras in some special cases. We prove that the cotriple cyclic homology of a crossed module of algebras (I, A, ρ) is isomorphic to HC _*(ρ): HC _*(I) → HC _*(A), provided I is H-unital and the ground ring is a field with characteristic zero. We also calculate the Hochschild and cyclic homologies of a crossed module of algebras (R, 0, 0) for each algebra R with trivial multiplication. At the end, we give some applications proving a new five term exact sequence.     

On the cohomology of joins of operator algebras

By analogy with the join in topology, the join A*B for operator algebras A and B acting on Hilbert spaces H and K, respectively, was defined by Gilfeather and Smith (Amer. J. Math. 116 (1994) 541-561). Assuming that K is finite dimensional, they calculated the Hochschild cohomology groups for A*B with coefficients in L(K⊕H). We assume that A is a maximal abelian von Neumann algebra acting on H, A is a subalgebra of , and B is an ultraweakly closed subalgebra of M_n(A) containing A⊗1_n. We show that B may be decomposed into a finite sum of free modules. In this context, we redefine the join of A and B, generalize the calculations of Gilfeather and Smith, and calculate , for all m?0.     

Hochschild Cohomology of an Algebra Associated with a Circular Quiver

In this article we show that an algebra A = K Γ/(f(X ^s )) has a periodic projective bimodule resolution of period 2, where KΓ is the path algebra of the circular quiver Γ with s vertices and s arrows over a commutative ring K, f(x) is a monic polynomial over K and X is the sum of all arrows in KΓ. Moreover, by means of this projective bimodule resolution, we compute the Hochschild cohomology group of A, and we give a presentation of the Hochschild cohomology ring HH?(A) by the generators and the relations in the case K is a field.     

On Hochschild Cohomology of a Self-injective Special Biserial Algebra Obtained by a Circular Quiver with Double Arrows

We calculate the dimensions of the Hochschild cohomology groups of a self-injective special biserial algebra Λ _s obtained by a circular quiver with double arrows. Moreover, we give a presentation of the Hochschild cohomology ring modulo nilpotence of Λ _s by generators and relations. This result shows that the Hochschild cohomology ring modulo nilpotence of Λ _s is finitely generated as an algebra.     

Generic representations of the finite general linear groups and the Steenrod algebra: II

The category of generic representations over the finite fieldF _q , used in PartI to study modules over the Steenrod algebra, is here related to the modular representation theory of the groups GL _n (F _q ). This leads to a simple and elegant approach to the classic objects of study: irreducible representations, extensions of modules, homology stability, etc. Connections to current research in algebraicK-theory involving stableK-theory and Topological Hochschild Homology are also explained.Partially funded by the NSF.     

On Hochschild cohomology rings of Fibonacci algebras

Let Λ be a Fibonacci algebra over a field k. The multiplication of Hochschild cohomology ring of Λ induced by the Yoneda product is described explicitly. As a consequence, the multiplicative structure of Hochschild cohomology ring of Λ is proved to be trivial.     

Hochschild (co)homology of exterior algebras using algebraic Morse theory

In [1 Han, Y., Xu, Y. (2007). Hochschild (co)homology of exterior algebras. Comm. Algebra 35(1):115–131.[Web of Science ®] , [Google Scholar]], the authors computed the additive and multiplicative structure of HH*(A;A), where A is the n-th exterior algebra over a field. In this paper, we derive all their results using a different method (AMT) as well as calculate the additive structure of HH_k(A;A) and HH^k(A;A) over ?. We provide concise presentations of algebras HH_?(A;A) and HH*(A;A) as well as determine their generators in the Hochschild complex. Finally, we compute an explicit free resolution (spanned by multisets) of the A^e-module A and describe the homotopy equivalence to its bar resolution.     

On Maximal Diagonalizable Lie Subalgebras of the First Hochschild Cohomology

Let A be a basic connected finite dimensional algebra over an algebraically closed field, with ordinary quiver without oriented cycles. Given a presentation of A by quiver and admissible relations, Assem and de la Peña have constructed an embedding of the space of additive characters of the fundamental group of the presentation into the first Hochschild cohomology group of A. We compare the embeddings given by the different presentations of A. In some situations, we characterise the images of these embeddings in terms of (maximal) diagonalizable subalgebras of the first Hochschild cohomology group (endowed with its Lie algebra structure).     

On the Cyclic Homology of an Algebra Associated with a Cyclic Quiver and a Monic Polynomial

In this article, we compute the Hochschild homology group of A = KΓ/(f(X ^s )), where KΓ is the path algebra of the cyclic quiver Γ with s vertices and s arrows over a commutative ring K, f(x) is a monic polynomial over K, and X is the sum of all arrows in KΓ. Moreover, we compute the cyclic homology group of A in the case f(x) = (x ? a) ^m , where a ∈ K, so that we can determine the cyclic homology of A in general when K is an algebraically closed field.     

On Trivial Extensions of 2-Fundamental Algebras #

There are considered trivial extensions of minimal 2-fundamental algebras. It is shown that if the Auslander–Reiten quiver Γ _A of a minimal 2-fundamental algebra A contains a starting component or an ending component which is not generalized standard, then the Auslander–Reiten quiver Γ _T(A) of the trivial extension T(A) of A contains also a component that is not a generalized standard.     

The Simple Connectedness of a Tame Weakly Shod Algebra

Abstract

We prove that a tame weakly shod algebra A which is not quasi-tilted is simply connected if and only if the orbit graph of its pip-bounded component is a tree, or if and only if its first Hochschild cohomology group H¹(A) with coefficients in _A A _A vanishes. We also show that it is strongly simply connected if and only if the orbit graph of each of its directed components is a tree, or if and only if H¹(A) = 0 and it contains no full convex subcategory which is hereditary of type 𝔸?, or if and only if it is separated and contains no full convex subcategory which is hereditary of type 𝔸?.     

Representability and Specht problem for G-graded algebras

Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let idG(W) denote the T-ideal of G-graded identities of W. We prove: 1. [G-graded PI-equivalence] There exists a field extension K of F and a finite-dimensional Z/2Z×G-graded algebra A over K such that idG(W)=idG(A^∗) where A^∗ is the Grassmann envelope of A. 2. [G-graded Specht problem] The T-ideal idG(W) is finitely generated as a T-ideal. 3. [G-graded PI-equivalence for affine algebras] Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite-dimensional algebra A over K such that idG(W)=idG(A).     

Actions of Ore extensions and growth of polynomial H-identities

We show that if A is a finite-dimensional associative H-module algebra for an arbitrary Hopf algebra H, then the proof of the analog of Amitsur’s conjecture for H-codimensions of A can be reduced to the case when A is H-simple. (Here we do not require that the Jacobson radical of A is an H-submodule.) As an application, we prove that if A is a finite-dimensional associative H-module algebra where H is a Hopf algebra H over a field of characteristic 0 such that H is constructed by an iterated Ore extension of a finite-dimensional semisimple Hopf algebra by skew-primitive elements (e.g., H is a Taft algebra), then there exists integer PIexp^H(A). In order to prove this, we study the structure of algebras simple with respect to an action of an Ore extension.     

C*-index of observable algebras in <Emphasis Type="Italic">G</Emphasis>-spin model

In two-dimensional lattice spin systems in which the spins take values in a finite group G, one can define a field algebra F which carries an action of a Hopf algebra D(G), the double algebra of G and moreover, an action of D(G;H), which is a subalgebra of D(G) determined by a subgroup H of G, so that F becomes a modular algebra. The concrete construction of D(G;H)-invariant subspace A _H in F is given. By constructing the quasi-basis of conditional expectation γ _G of A _H onto A _G , the C*-index of γ _G is exactly the index of H in G.