Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0704.2771

Bialgebra cohomology, pointed Hopf algebras, and deformations

Mitja Mastnak·Sarah Witherspoon
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give explicit formulas for maps in a long exact sequence connecting bialgebra cohomology to Hochschild cohomology. We give a sufficient condition for the connecting homomorphism to be surjective. We apply these results to compute all bialgebra two-cocycles of certain Radford biproducts (bosonizations). These two-cocycles are precisely those associated to the finite dimensional pointed Hopf algebras in the recent classification of Andruskiewitsch and Schneider, in an interpretation of these Hopf algebras as graded bialgebra deformations of Radford biproducts.

Explore related subjects

math.RAmath.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Mitja Mastnak, Sarah Witherspoon. 2008-05-11. Bialgebra cohomology, pointed Hopf algebras, and deformations. https://arxiv.org/abs/0704.2771

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Orthogonality of Defect Ideals for Jordan Homomorphisms

Let $φ\colon A\to B$ be a Jordan homomorphism of associative algebras over a field $\mathbb{F}$ of characteristic different from $2$, with defect ideals $V_φ$ and $W_φ$. If $A=\operatorname{Alg}([[A,A],[A,A]])$, then $V_φW_φ=W_φV_φ=0$ and $V_φ\cap W_φ\subseteq\operatorname{Ann}(\operatorname{Alg}(φ(A)))$. Moreover, the induced map modulo $V_φ\cap W_φ$ is standard. The annihilator bound is sharp, even in finite dimension. As an application, when $A$ is unital and $A=K(A)$, every Jordan homomorphism defined on $A$ is splittable and standard. In this setting, the corresponding result of Brešar and Zelmanov holds without the surjectivity hypothesis.

math.RA↗
arXiv2026

On the Real Waring Rank of a Biharmonic Form

We develop an effective method for determining the real Waring rank and the real border rank of biharmonic binary forms. By means of real apolarity, we convert both of these problems into membership questions for regions bounded by a hypocycloid and an epicycloid. We further provide a computable algebraic criterion for the relevant memberships, which involves only the signs of certain real-valued resultants.

math.RA↗
arXiv2026

Actions of central Morita monoidal groupoids, and seven term exact sequences of all types

We introduce a strict monoidal groupoid associated with a given commutative ring $K$. It acts on the Clifford extension systems of $K$-central algebras which are graded by a group $G$, as well as on the generalized collective characters, both of type $ϕ\in$ Hom$(G,$ Aut$(K))$. These actions partition the two sets above with respect to graded Morita equivalence. By that we establish a seven term exact sequence of type $ϕ$, which boils down to the celebrated Chase-Harrison-Rosenberg sequence when $ϕ$ is Galois.

math.RA↗