We show that every finite-dimensional complex pointed Hopf algebra with group of group-likes isomorphic to a sporadic group is a group algebra, except for the Fischer group
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, the Baby Monster ...and the Monster. For these three groups, we give a short list of irreducible Yetter–Drinfeld modules whose Nichols algebra is not known to be finite-dimensional.
We classify Nichols algebras of irreducible Yetter–Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies ...a given inequality. This assumption turns out to be equivalent to a factorization assumption on the Hilbert series. Besides the known Nichols algebras we obtain a new example. Our method is based on a combinatorial invariant of the Hurwitz orbits with respect to the action of the braid group on three strands.
We use Cartier's preadditive symmetric monoidal categories to study Lie bialgebras. We prove that bosonization can be done consistently in this framework. In the last part of the paper we present ...explicit examples and indicate a deep relationship between certain curved Lie bialgebras and Nichols algebras over abelian groups.
Proc. Amer. Math. Soc. 152 (2024), 3197-3207 We prove that the lattice of ideals of an arbitrary $L$-algebra is
distributive. As a consequence, a spectral theory applies with no restriction.
We also ...study the spectrum (i.e. the set of prime ideals) of $L$-algebras and
characterize prime ideals in topological terms.
Motivated by the proof of Rump of a conjecture of Gateva-Ivanova on the decomposability of square-free solutions to the Yang-Baxter equation, we present several other decomposability theorems based ...on the cycle structure of a certain permutation associated with the solution.
Given a right-non-degenerate set-theoretic solution \((X,r)\) to the Yang-Baxter equation, we construct a whole family of YBE solutions \(r^{(k)}\) on \(X\) indexed by its reflections \(k\) (i.e., ...solutions to the reflection equation for \(r\)). This family includes the original solution and the classical derived solution. All these solutions induce isomorphic actions of the braid group/monoid on \(X^n\). The structure monoids of \(r\) and \(r^{(k)}\) are related by an explicit bijective \(1\)-cocycle-like map. We thus turn reflections into a tool for studying YBE solutions, rather than a side object of study. In a different direction, we study the reflection equation for non-degenerate involutive YBE solutions, show it to be equivalent to (any of the) three simpler relations, and deduce from the latter systematic ways of constructing new reflections.
Over fields of characteristic zero, we determine all absolutely irreducible Yetter-Drinfeld modules over groups that have prime dimension and yield a finite-dimensional Nichols algebra. To achieve ...our goal, we introduce orders of braided vector spaces and study their degenerations and specializations.
We develop new machinery for producing decomposability tests for involutive solutions to the Yang-Baxter equation. It is based on the seminal decomposability theorem of Rump, and on "cabling" ...operations on solutions and their effect on the diagonal map. Our machinery yields an elementary proof of a recent decomposability theorem of Camp-More and Sastriques, as well as original decomposability results. It also provides a conceptual interpretation (using the braces language) of the Dehornoy class, a combinatorial invariant naturally appearing in the Garside-theoretic approach to involutive solutions.
In this paper we give details of the proofs performed with GAP of the theorems of our paper N. Andruskiewitsch, F. Fantino, M. Graña, L. Vendramin, Pointed Hopf algebras over the sporadic simple ...groups, J. Algebra 325 (1) (2011) 305–320.
We prove that the lattice of ideals of an arbitrary \(L\)-algebra is distributive. As a consequence, a spectral theory applies with no restriction. We also study the spectrum (i.e. the set of prime ...ideals) of \(L\)-algebras and characterize prime ideals in topological terms.
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