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Cris Negron

Publications and source records attributed to Cris Negron.

28 records · Page 2Linked to original sources
arXiv2017

Twists of quantum Borel algebras

We classify Drinfeld twists for the quantum Borel subalgebra u_q(b) in the Frobenius-Lusztig kernel u_q(g), where g is a simple Lie algebra over C and q an odd root of unity. More specifically, we show that alternating forms on the character group of the group of grouplikes for u_q(b) generate all twists for u_q(b), under a certain algebraic group action. This implies a simple classification of Hopf algebras whose categories of representations are tensor equivalent to that of u_q(b). We also show that cocycle twists for the corresponding De Concini-Kac algebra are in bijection with alternating forms on the aforementioned character group.

math.QA↗
arXiv2017

The Gerstenhaber bracket as a Schouten bracket for polynomial rings extended by finite groups

We apply new techniques to Gerstenhaber brackets on the Hochschild cohomology of a skew group algebra formed from a polynomial ring and a finite group (in characteristic 0). We show that the Gerstenhaber brackets can always be expressed in terms of Schouten brackets. We obtain as consequences some conditions under which brackets are always 0, strengthening known results.

math.QA↗
arXiv2017

Gauge invariants from the powers of antipodes

We prove that the trace of the $n$th power of the antipode of a Hopf algebra with the Chevalley property is a gauge invariant, for each integer $n$. As a consequence, the order of the antipode, and its square, are invariant under Drinfeld twists. The invariance of the order of the antipode is closely related to a question of Shimizu on the pivotal covers of finite tensor categories, which we affirmatively answer for representation categories of Hopf algebras with the Chevalley property.

math.QA↗
arXiv2017

Small quantum groups associated to Belavin-Drinfeld triples

For a simple Lie algebra L of type A, D, E we show that any Belavin-Drinfeld triple on the Dynkin diagram of L produces a collection of Drinfeld twists for Lusztig's small quantum group u_q(L). These twists give rise to new finite-dimensional factorizable Hopf algebras, i.e. new small quantum groups. For any Hopf algebra constructed in this manner, we identify the group of grouplike elements, identify the Drinfeld element, and describe the irreducible representations of the dual in terms of the representation theory of the parabolic subalgebra(s) in L associated to the given Belavin-Drinfeld triple. We also produce Drinfeld twists of u_q(L) which express a known algebraic group action on its category of representations, and pose a subsequent question regarding the classification of all twists.

math.RT↗
arXiv2016

The derived Picard group of an affine Azumaya algebra

We describe the derived Picard group of an Azumaya algebra A on an affine scheme X in terms of global sections of the constant sheaf of integers on X, the Picard group of X, and the stabilizer of the Brauer class of A under the action of Aut(X). In particular, we find that the derived Picard group of an Azumaya algebra is generally not isomorphic to that of the underlying scheme. In the case of the trivial Azumaya algebra, our result refines Yekutieli's description of the derived Picard group of a commutative algebra. We also get, as a corollary, an alternate proof of a result of Antieau which relates derived equivalences to Brauer equivalences for affine Azumaya algebras. The example of a Weyl algebra in finite characteristic is examined in some detail.

math.RA↗
arXiv2016

The cup product on Hochschild cohomology via twisting cochains and applications to Koszul rings

Given an acyclic twisting cochain $π:C\to A$, from a curved dg coalgebra $C$ to a dg algebra $A$, we show that the associated twisted hom complex $\mathrm{Hom}^π_k(C,A)$ has cohomology equal to the Hochschild cohomology of $A$, as a graded ring. As a corollary we find that the Hochschild cohomology of a Koszul algebra $A$, along with its cup product, is a subquotient of the tensor product algebra $A^!\otimes A$ of $A$ with its Koszul dual.

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arXiv2015

Braided Hochschild cohomology and Hopf actions

We show that the braided Hochschild cohomology, of an algebra in a suitably algebraic braided monoidal category, admits a graded ring structure under which it is braided commutative. We then give a canonical identification between the usual Hochschild cohomology ring of a smash product and the (derived) invariants of its braided Hochschild cohomology ring. We apply our results to identify the associative formal deformation theory of a smash product with its formal deformation theory as a module algebra over the given Hopf algebra (when the Hopf algebra is sufficiently semisimple). As a second application we deduce some structural results for the usual Hochschild cohomology of a smash product, and discuss specific implications for finite group actions on smooth affine schemes.

math.QA↗
arXiv2015

An Alternate Approach to the Lie Bracket on Hochschild Cohomology

We define Gerstenhaber's graded Lie bracket directly on complexes other than the bar complex, under some conditions. The Koszul complex of a Koszul algebra in particular satisfies our conditions. As examples we recover the Schouten-Nijenhuis bracket for a polynomial ring and the Gerstenhaber bracket for a group algebra of a cyclic group of prime order.

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arXiv2014

Spectral Sequences for the Cohomology Rings of a Smash Product

Stefan and Guichardet have provided Lyndon-Hochschild-Serre type spectral sequences which converge to the Hochschild cohomology and Ext groups of a smash product. We show that these spectral sequences carry natural multiplicative structures, and that these multiplicative structures can be used to calculate the cup product on Hochschild cohomology and the Yoneda product on an Ext algebra.

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