arXiv subjects

Cristian Chaparro

Publications and source records attributed to Cristian Chaparro.

3 recordsLinked to original sources

arXiv2024

The Hochschild cohomology and the Tamarkin-Tsygan calculus of gentle algebras

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.

math.RT↗

arXiv2020

On the Lie algebra structure of the first Hochschild cohomology of gentle algebras and Brauer graph algebras

In this paper we determine the first Hochschild homology and cohomology with different coefficients for gentle algebras and we give a geometrical interpretation of these (co)homologies using the ribbon graph of a gentle algebra as defined in earlier work by the second author. We give an explicit description of the Lie algebra structure of the first Hochschild cohomology of gentle and Brauer graph algebras (with multiplicity one) based on trivial extensions of gentle algebras and we show how the Hochschild cohomology is encoded in the Brauer graph. In particular, we show that except in one low-dimensional case, the resulting Lie algebras are all solvable.

math.RT↗

arXiv2013

Ore and Goldie theorems for skew PBW extensions

Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW (Poincaré-Birkhoff-Witt) extensions. Indeed, Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra of quantum matrices, among many others, are examples of skew PBW extensions. In this paper we extend the classical Ore and Goldie theorems, known for skew polynomial rings, to this wide class of non-commutative rings. As application, we prove the quantum version of the Gelfand-Kirillov conjecture for the skew quantum polynomials.

math.RA↗