We introduce and investigate the properties of Hochschild cohomology of algebras in an abelian monoidal category M. We show that the second Hochschild cohomology group of an algebra in M classifies extensions of A up to an equivalence. We characterize algebras of Hochschild dimension 0 (separable algebras), and of Hochschild dimension ≤1 (formally smooth algebras). Several particular cases and applications are included in the last section of the paper.Let k be a field and let A be a k-algebra. Recall that an A-bimodule is, by definition, a left module over the enveloping algebra A e := A ⊗ A op . Hochschild cohomology H * (A, M) of A with coefficients in M was introduced in [8] in order to classify, up to equivalence, all extensions of A with kernel M. Many other applications of this cohomology have been discovered since then. Let us mention here a few of them.The algebra A is called separable if A is projective as an A-bimodule. Separable algebras are characterized by the fact that their Hochschild dimension is zero, that is H 1 (A, M) = 0, for every bimodule M. Other homological characterizations of separable algebras can be found for example in [17] and [5].The set of equivalence classes of extensions of A with kernel M is in one-to-one correspondence with H 2 (A, M). In particular, an algebra A has no non-trivial extensions if and only if H 2 (A, M) = 0, for any bimodule M, i.e. its Hochschild dimension is less than or equal to 1. These algebras were introduced by Cuntz and Quillen in [5], where they are called quasi-free and play the role of "functions algebras" of a "noncommutative smooth affine variety". One can prove that an algebra A is quasi-free if and only if it has the following "lifting property": for any algebra E and
Let $H$ be a Hopf algebra and let $\mathcal{M}_s (H)$ be the category of all left $H$-modules and right $H$-comodules satisfying appropriate compatibility relations. An object in $\mathcal{M}_s (H)$ will be called a stable anti-Yetter–Drinfeld module (over $H$) or a SAYD module, for short. To each $M \in \mathcal{M}_s (H)$ we associate, in a functorial way, a cyclic object $\mathrm{Z}_\ast (H, M)$. We show that our construction can be used to compute the cyclic homology of the underlying algebra structure of $H$ and the relative cyclic homology of $H$-Galois extensions.Let $K$ be a Hopf subalgebra of $H$. For an arbitrary $M \in \mathcal{M}_s (K)$ we define a right $H$-comodule structure on $\mathrm{Ind}_K^H M := H \otimes_K M$ so that $\mathrm{Ind}_K^H M$ becomes an object in $\mathcal{M}_s (H)$. Under some assumptions on $K$ and $M$ we compute the cyclic homology of $\mathrm{Z}_\ast (H, \mathrm{Ind}_K^H M)$. As a direct application of this result, we describe the relative cyclic homology of strongly graded algebras. In particular, we calculate the cyclic homology of group algebras and quantum tori.Finally, when $H$ is the enveloping algebra of a Lie algebra $\mathfrak{g}$, we construct a spectral sequence that converges to the cyclic homology of $H$ with coefficients in a given SAYD module $M$. We also show that the cyclic homology of almost symmetric algebras is isomorphic to the cyclic homology of $H$ with coefficients in a certain SAYD module.
Abstract. For a (co)monad T l on a category M, an object X in M, and a functor Π : M → C, there is a (co)simplex Z * := ΠT l * +1 X in C. The aim of this paper is to find criteria for para-(co)cyclicity of Z * . Our construction is built on a distributive law of T l with a second (co)monad Tr on M, a natural transformation i : ΠT l → ΠTr , and a morphism w : Tr X → T l X in M. The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun's axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads T l = T ⊗ R (−) and Tr = (−) ⊗ R T on the category of R-bimodules. The functor Π can be chosen such that Z n = T b ⊗ R . . . b ⊗ R T b ⊗ R X is the cyclic R-module tensor product. A natural transformation i : T b ⊗ R (−) → (−) b ⊗ R T is given by the flip map and a morphism w : X ⊗ R T → T ⊗ R X is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. The notion of a stable anti Yetter-Drinfel'd module over certain bialgebroids, so called × R -Hopf algebras, is introduced. In the particular example when T is a module coring of a × R -Hopf algebra B and X is a stable anti Yetter-Drinfel'd B-module, the para-cyclic object Z * is shown to project to a cyclic structure on T ⊗ R * +1 ⊗ B X. For a B-Galois extension S ⊆ T , a stable anti Yetter-Drinfel'd B-module T S is constructed, such that the cyclic objects B ⊗ R * +1 ⊗ B T S and T b ⊗ S * +1 are isomorphic. This extends a theorem by Jara and Ştefan for Hopf Galois extensions. As an application, we compute Hochschild and cyclic homologies of a groupoid with coefficients in a stable anti Yetter-Drinfel'd module, by tracing it back to the group case. In particular, we obtain explicit expressions for (coinciding relative and ordinary) Hochschild and cyclic homologies of a groupoid. Latter extends results of Burghelea on cyclic homology of groups. Contents Introduction = H⊗n , associated to the coalgebra underlying a Hopf algebra H over a field K. The cocyclic operator was given in terms of a so called modular pair in involution.In the subsequent years the Connes-Moscovici cocyclic module was placed in a broader and broader context. In [KR03] (see also [HKRS2]) to any (co)module algebra T of a Hopf algebra H, and any H-(co)module X, there was associated a para-cyclic module with components T ⊗ * +1 ⊗ X. Dually, for any (co)module coalgebra T of a Hopf algebra H, and any H-(co)module X, there is a para-cocyclic module with components T ⊗ * +1 ⊗ X. The Connes-Moscovici cosimplex Z CM * turns out to be isomorphic to a quotient of the para-cocyclic module associated to the regular module coalgebra T := H and an H-comodule defined on K. , a modular pair in involution was proven to be equivalent to a stable anti Yetter-Drinfel'd module structure on the ground field K. In [HKRS2], the para-cocyclic module T ⊗ * +1 ⊗ X, associated to an H-module coalgebra T and a stable anti Yetter-Drinfel'd H-module X, was shown to project to a cocyclic obj...
The main aim of this paper is to classify all types of Hopf algebras of dimension less thn or equal to 11 over an algebraically closed field of characteristic 0. If A is such a Hopf algebra that is not semisimple, then we shall prove that A or A* is pointed. This property will result from the fact that, under some assumptions, any Hopf algebra that is generated as an algebra by a four-dimensional simple subcoalgebra is a Hopf quotient of the coordinate ring of quantum SL2(k). The first result allows us to reduce the classification to the case of pointed Hopf algebras of dimension 8. We shall describe their types in the last part of the paper. ~c~ 1999 Academic Press THEOREM 2.8. Let A be a Hopf algebra of dimension _< 11 that is not semisimple. Then A or A* is pointed.Therefore, if we are interested in Hopf algebras of dimension _< 11, we deal with either semisimple or pointed Hopf algebras and duals of them. First, the classification of semisimple Hopf algebras of dimension p2 p3,
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