Claude Cibils

Skew category, Galois covering and smash product of a k-category

In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In the case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beatties construction of a ring with local units. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery (1984), and we provide a unification with the results on coverings of quivers and relations by E. Green (1983). We obtain a confirmation in a quiver and relations-free categorical setting that both constructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representation theories of a category and of a Galois cover of it.

A quiver quantum group

We construct quantum groups at a root of unity and we describe their monoidal module category using techniques from the representation theory of finite dimensional associative algebras.

Cohomology of incidence algebras and simplicial complexes

Let Q be a finite quiver (i.e. a finite oriented graph) without oriented cycles and such that for each arrow se -% t there is no oriented path other than a joining s to t. These quivers are called ordered. The set Q. of vertices of Q is then a finite partially ordered set: s> t if and only if there exists an oriented path form s to t. Conversely, if Q. is a finite poset, we construct a quiver Q with Q. as set of vertices and with an arrow from s to t if and only if s > t and there is no u E QO such that s > u > t. Clearly we obtain in this way an ordered quiver and this gives a bijection between finite posets and ordered quivers. Moreover, if R is a commutative ring, consider RQ the path algebra of Q. Let I be the two-sided ideal of RQ generated by all the differences y-6 where y and 6 are parallel paths (i.e. y and 6 have the same source and the same end vertices). The algebra A = RQ/Z is the incidence algebra of the poset associated to the ordered quiver Q. We call such algebras incidence R-algebras. To a finite poset QO (and Q its ordered quiver) we can associate the simplicial complex _Ze whose i-simplices are the chains of length i. Gerstenhaber and Schack [7, 81 proved that the cohomology of ZQ with coefficients in R is isomorphic to the Hochschild cohomology of the incidence algebra RQ/I. We begin this note with the proof of this result using a smaller (but still standard) resolution of RQ/Z as a bimodule over itself. It is interesting to note that this smaller resolution works for any finite quiver Q and any two-sided admissible ideal I of RQ. In the second section we consider Q which is an ordered quiver obtained from the ordered quiver Q by adding a biggest and a smallest vertex. Let A and B be the standard RQ/Z-modules corresponding to these new vertices. We prove that the reduced cohomology of ZQ with coefficients in R is isomorphic to the Ext groups over RQ/I between A and B, with the degrees shifted by two. This is done using the smaller resolution of RQ/Z of the first section.

Rigidity of truncated quiver algebras

A quiver Q is a finite oriented graph which can contain more than one arrow between two vertices, as well as loops and oriented cycles. For n a positive integer, Q, is the set of oriented paths of length n of Q, where the length is the number of arrows of the oriented path. Notice that Q, is the set of vertices and Q, the set of arrows. Let k be a field and kQo = x St po ks be the commutative semi-simple algebra with Q0 as a k-basis of idempotents. For each arrow a E Q, with source vertex s(a) = s and end vertex r(u) = t, the one dimensional vector space k, has an evident kQ,-bimodule structure: for u E Q,, we have au=6,,sa and ua=d,.,a, where 6 is the Kronecker symbol. In this way kQ, = @,, o, ka is a kQ,-bimodule and the quiver algebra kQ is the tensor algebra over kQo of the kQ,-bimodule kQt. Of course, kQ can also be described as the vector space kQo@ kQl 0 kQ, 0 ... where the multiplication of /?E Qj and CI E Qi is /3~ E Qj+, if t(a) = s(p) and 0 otherwise. Let now n be a finite dimensional k-algebra. We suppose that /i is Morita reduced and that the endomorphism ring of each simple n-module is k. This is equivalent to A/r = k x . . . x k, were r is the Jacobson radical of /i. By definition, the set of vertices of the Gabriel’s quiver Q of /i is the set of isomorphism classes of simple /i-modules. If S and T are simple n-modules, the number of arrows from S to T is dim,Exti(S, T). By an observation of Gabriel [6; 7,4.3] every k-algebra n such that A/r = k x . .. x k admits a presentation, that is, an algebra surjection cp: kQ,, -+ A whose kernel Z verifies F” c Zc F2, where F is the two-sided ideal of kQ,, generated by Ql and m is some positive integer. Such ideals Z of a quiver algebra are called admissible. In general an algebra n has not a unique presentation, that is, two different admissible two-sided ideals Z and J of a quiver algebra kQ can give isomorphic k-algebras kQ/Z and kQ/J By definition, we say that a k-algebra /i is a truncated quiver algebra if

Hochschild cohomology algebra of abelian groups

In this paper we present a direct proof of what is suggested by Holm’s results (T. Holm, The Hochschild cohomology ring of a modular group algebra: the commutative case, Comm. Algebra 24, 1957–1969 (1996)): there is an isomorphism of algebras HH*(kG,kG) → kG ⊗ H*(G,k) where G is a finite abelian group, k a ring, HH*(kG,kG) is the Hochschild cohomology algebra and H*(G,k) the usual cohomology algebra.This result agrees with the well-known additive structure result in force for any group G; we remark that the multiplicative structure result we have obtained is quite similar to the description of the monoidal category of Hopf bimodules over kG given in “C. Cibils, Tensor product of Hopf bimodules, to appear in Proc. Amer. Math. Soc”. This similarity leads to conjecture the structure of HH*(kG,kG) for G a finite group.

Hopf bimodules are modules

Abstract We construct an algebra X associated to a finite-dimensional Hopf algebra A , such that there exists a vector space-preserving equivalence of categories between the categories of Hopf bimodules over A and of left X -modules. We show that X is isomorphic to the direct tensor product of the Heisenberg double of, A and the opposite of its Drinfeld double.

HOPF QUIVERS AND NICHOLS ALGEBRAS IN POSITIVE CHARACTERISTIC

We apply a combinatorial formula of the first author and Rosso for products in Hopf quiver algebras to determine the structure of Nichols algebras. We illustrate this technique by explicitly constructing new examples of Nichols algebras in positive characteristic. We further describe the corresponding Radford biproducts and some liftings of these biproducts, which are new finite-dimensional pointed Hopf algebras.

Connected gradings and the fundamental group

The main purpose of this paper is to provide explicit computations of the fundamental group of several algebras. For this purpose, given a

Cohomology of split algebras and of trivial extensions

k

The Intrinsic Fundamental Group of a Linear Category

-algebra