arXiv subjects
Publications and source records attributed to Mariano Suárez-Álvarez.
We prove that the Nakayama automorphism of a Frobenius algebra acts trivially on the Hochschild cohomology of the algebra. As an application of this fact, we show how to construct certain invariants attached to such algebras, and to their automorphisms and derivations.
The Tamarkin Tsygan calculus of a finite dimensional algebra is a differential calculus given by the comprehensive data of the Hochschild cohomology, its structure both as a graded commutative algebra under the cup product and as a graded Lie algebra under the Gerstenhaber bracket, together with the Hochschild homology and its module structure over the Hochschild cohomology given by the cap product as well as the Connes differential. In this paper, we calculate the whole of the Tamarkin Tsygan calculus for the class of gentle algebras. Apart from some isolated calculations, this is, to our knowledge, the first complete calculation of this calculus for a family of finite dimensional algebras. Gentle algebras appear in many different areas of mathematics such as the theory of cluster algebras, N=2 gauge theories and homological mirror symmetry of surfaces. For the latter Haiden-Katzarkov-Kontsevich show that the partially wrapped Fukaya category of a graded surface is triangle equivalent to the perfect derived category of a graded gentle algebra. Thus the symplectic cohomology of the surface should be linked to the Hochschild cohomology of the gentle algebra. We give a description of how this connection might be visualised, in that, we show how these structures are encoded in the geometric surface model of the bounded derived category associated to a gentle algebra via its ribbon graph. Finally, we show how to recover structural results for a gentle algebra given its Tamarkin-Tsygan calculus.
We consider the algebra $A_N=k\langle x, y\rangle/(yx-xy-x^N)$, with $k$ a field of characteristic zero and $N$ a positive integer. Our main result is a complete description of the first Hochschild cohomology $\operatorname{HH}^1(A_N)$ of $A_N$ that consists both of explicit derivations of $A_N$ whose cohomology classes span it and a description of its Lie algebra structure. As we do this, we compute the automorphism group of the algebra, as well as certain characteristic subgroups thereof related to locally nilpotent derivations, classify the finite groups that act on $A_N$ and, finally, show that there are no inner-faithful actions of generalized Taft Hopf algebras on $A_N$. We establish this last result thanks to another calculation of Hochschild cohomology, now with twisted coefficients.
We prove that among the finite dimensional algebras of finite representation type those that are string algebras are precisely the ones that have the property that the middle term of an arbitrary extension of indecomposable modules has at most two direct factors. On the other hand, we show that non-domestic string algebras are very far from having that property.
Given a central arrangement of lines $\mathcal{A}$ in a $2$-dimensional vector space $V$ over a field of characteristic zero, we study the algebra $\mathcal D(\mathcal A)$ of differential operators on $V$ which are logarithmic along $\mathcal A$. Among other things we determine the Hochschild cohomology of $\mathcal D(\mathcal A)$ as a Gerstenhaber algebra, establish a connection between that cohomology and the de Rham cohomology of the complement $M(\mathcal A)$ of the arrangement, determine the isomorphism group of $\mathcal D(\mathcal A)$ and classify the algebras of that form up to isomorphism.
We prove using an extension of Saito's criterion that the algebra of differential operators tangent to a free hyperplane arrangement is generated by the derivations of the arrangement, recovering in this way a result of F. J. Calderón-Moreno.
We show that the action of the Lie algebra HH^1(A) of outer derivations of an associative algebra A on the Hochschild cohomology HH^*(A) of A given by the Gerstenhaber bracket can be computed in terms of an arbitrary projective resolution of A as an A-bimodule, without having recourse to comparison maps between the resolution and the bar resolution.
We show that finite Galois extensions with cyclic Galois group are radical.
In this article we establish an explicit link between the classical theory of deformations à la Gerstenhaber -- and a fortiori with the Hochschild cohomology-- and (weak) PBW-deformations of homogeneous algebras. Our point of view is of cohomological nature. As a consequence, we recover a theorem by R. Berger and V. Ginzburg, which gives a precise condition for a filtered algebra to satisfy the so-called PBW property, under certain assumptions.