Cris Negron
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View article: Cochain valued TQFTs from nonsemisimple modular tensor categories
Cochain valued TQFTs from nonsemisimple modular tensor categories Open
We show that a vector space valued TQFT constructed in work of De Renzi et al. [DGGPR23] extends naturally to a topological field theory which takes values in the symmetric monoidal category of linear cochains. Specifically, we consider a …
View article: Quantum $\operatorname{SL}(2)$ and logarithmic vertex operator algebras at $(p,1)$-central charge
Quantum $\operatorname{SL}(2)$ and logarithmic vertex operator algebras at $(p,1)$-central charge Open
We provide a ribbon tensor equivalence between the representation category of small quantum \operatorname{SL}(2) , at parameter q=e^{\pi i/p} , and the representation category of the triplet vertex operator algebra at integral parameter p>…
View article: Quantum Frobenius and modularity for quantum groups at arbitrary roots of 1
Quantum Frobenius and modularity for quantum groups at arbitrary roots of 1 Open
We consider quantum group representations Rep(G_q) for a semisimple algebraic group G at a complex root of unity q. Here we allow q to be of any order. We first show that the Tannakian center in Rep(G_q) is calculated via a twisting of Lus…
View article: Revisiting the Steinberg representation at arbitrary roots of 1
Revisiting the Steinberg representation at arbitrary roots of 1 Open
We consider quantum group representations for a semisimple algebraic group G at a complex root of unity q. Here q is allowed to be of any order. We revisit some fundamental results of Parshall-Wang and Andersen-Polo-Wen from the 90's. In p…
View article: Hypersurface support and prime ideal spectra for stable categories
Hypersurface support and prime ideal spectra for stable categories Open
We use hypersurface support to classify thick (two-sided) ideals in the stable categories of representations for several families of finite-dimensional integrable Hopf algebras: bosonized quantum complete intersections, quantum Borels in t…
View article: Support theory for Drinfeld doubles of some infinitesimal group schemes
Support theory for Drinfeld doubles of some infinitesimal group schemes Open
Consider a Frobenius kernel G in a split semisimple algebraic group, in very\ngood characteristic. We provide an analysis of support for the Drinfeld center\nZ(rep(G)) of the representation category for G, or equivalently for the\nrepresen…
View article: The half-quantum flag variety and representations for small quantum groups
The half-quantum flag variety and representations for small quantum groups Open
Consider an almost-simple algebraic group G and a choice of complex root of unity q. We study the category of quasi-coherent sheaves $\mathscr{X}_q$ on the half-quantum flag variety, which itself forms a sheaf of tensor categories over the…
View article: Support theory for the small quantum group and the Springer resolution
Support theory for the small quantum group and the Springer resolution Open
We consider the small quantum group u_q(G), for an almost-simple algebraic group G over the complex numbers and a root of unity q of sufficiently large order. We show that the Balmer spectrum for the small quantum group in type A admits a …
View article: HYPERSURFACE SUPPORT FOR NONCOMMUTATIVE COMPLETE INTERSECTIONS
HYPERSURFACE SUPPORT FOR NONCOMMUTATIVE COMPLETE INTERSECTIONS Open
We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $\operatorname {Sing}(R)$ ,…
View article: Support for Integrable Hopf Algebras via Noncommutative Hypersurfaces
Support for Integrable Hopf Algebras via Noncommutative Hypersurfaces Open
We consider finite-dimensional Hopf algebras $u$ that admit a smooth deformation $U\to u$ by a Noetherian Hopf algebra $U$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers, re…
View article: Support theory for Drinfeld doubles of some infinitesimal group schemes
Support theory for Drinfeld doubles of some infinitesimal group schemes Open
Consider a Frobenius kernel G in a split semisimple algebraic group, in very good characteristic. We provide an analysis of support for the Drinfeld center Z(rep(G)) of the representation category for G, or equivalently for the representat…
View article: Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge
Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge Open
We provide a ribbon tensor equivalence between the representation category of small quantum SL(2), at parameter q=exp($π$ i/p), and the representation category of the triplet vertex operator algebra at integral parameter p>1. We provide si…
View article: Hypersurface support and prime ideal spectra for stable categories
Hypersurface support and prime ideal spectra for stable categories Open
We use hypersurface support to classify thick (two-sided) ideals in the stable categories of representations for several families of finite-dimensional integrable Hopf algebras: bosonized quantum complete intersections, quantum Borels in t…
View article: Hypersurface support for noncommutative complete intersections
Hypersurface support for noncommutative complete intersections Open
We introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. By a noncommutative complete intersection we mean an algebra R which admits a smooth deformation $Q\to R$ by a Noetheri…
View article: Braided Hochschild cohomology and Hopf actions
Braided Hochschild cohomology and Hopf actions Open
We showt hat 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 usua…
View article: Cohomology for Drinfeld doubles of some infinitesimal group schemes
Cohomology for Drinfeld doubles of some infinitesimal group schemes Open
Consider a field k of characteristic p > 0, G_r the r-th Frobenius kernel of\na smooth algebraic group G, DG_r the Drinfeld double of G_r, and M a finite\ndimensional DG_r-module. We prove that the cohomology algebra H*(DG_r,k) is\nfinitel…
View article: Cohomology of finite tensor categories: duality and Drinfeld centers
Cohomology of finite tensor categories: duality and Drinfeld centers Open
We consider the finite generation property for cohomology of a finite tensor category C, which requires that the self-extension algebra of the unit Ext*_C(1,1) is a finitely generated algebra and that, for each object V in C, the graded ex…
View article: $A_{\infty}$-coderivations and the Gerstenhaber bracket on Hochschild cohomology
$A_{\infty}$-coderivations and the Gerstenhaber bracket on Hochschild cohomology Open
We show that Hochschild cohomology of an algebra over a field is a space of infinity coderivations on an arbitrary projective bimodule resolution of the algebra. The Gerstenhaber bracket is the graded commutator of infinity coderivations. …
View article: Gauge invariants from the powers of antipodes
Gauge invariants from the powers of antipodes Open
We prove that the trace of the $n$th power of the antipode of a Hopf algebra\nwith the Chevalley property is a gauge invariant, for each integer $n$. As a\nconsequence, the order of the antipode, and its square, are invariant under\nDrinfe…
View article: Small quantum groups associated to Belavin-Drinfeld triples
Small quantum groups associated to Belavin-Drinfeld triples Open
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-dimensio…
View article: An alternate approach to the Lie bracket on Hochschild cohomology
An alternate approach to the Lie bracket on Hochschild cohomology Open
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-Nijenhu…