Lie algebras A non-perverse Soergel bimodule in type A Un bimodule de Soergel non pervers de type A (original) (raw)

Article history: Received 21 July 2017 Accepted 25 July 2017 Available online 7 August 2017 Presented by Michèle Vergne A basic question concerning indecomposable Soergel bimodules is to understand their endomorphism rings. In characteristic zero all degree-zero endomorphisms are isomorphisms (a fact proved by Elias and the second author) which implies the Kazhdan–Lusztig conjectures. More recently, many examples in positive characteristic have been discovered with larger degree zero endomorphisms. These give counter-examples to expected bounds in Lusztig’s conjecture. Here we prove the existence of indecomposable Soergel bimodules in type A having non-zero endomorphisms of negative degree. This gives the existence of a non-perverse parity sheaf in type A. © 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

A class of zero product determined Lie algebras

Journal of Algebra, 2011

Let L be a Lie algebra over a field F. We say that L is zero product determined if, for every F-linear space V and every bilinear map ϕ : L × L → V , the following condition holds. If ϕ(x, y) = 0 whenever [x, y] = 0, then there exists a linear map f from [L, L] to V such that ϕ(x, y) = f ([x, y]) for all x, y ∈ L. This article shows that every parabolic subalgebra p of a (finite-dimensional) simple Lie algebra defined over an algebraically closed field is always zero product determined. Applying this result, we present a method different from that of Wang et al. (2010) [9] to determine zero product derivations of p, and we obtain a definitive solution for the problem of describing two-sided commutativity-preserving maps on p.

Simple transitive 2-representations for some 2-subcategories of Soergel bimodules

Journal of Pure and Applied Algebra, 2017

We classify simple transitive 2-representations of certain 2-subcategories of the 2-category of Soergel bimodules over the coinvariant algebra in Coxeter types B 2 and I 2 (5). In the I 2 (5) case it turns out that simple transitive 2-representations are exhausted by cell 2-representations. In the B 2 case we show that, apart from cell 2-representations, there is a unique, up to equivalence, additional simple transitive 2-representation and we give an explicit construction of this 2-representation.

Right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras

Forum Mathematicum, 2018

In the present paper, right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras will be studied. For this purpose, first the structure of the set of all right 2-Engel elements of a Lie algebra will be examined and then, by taking advantage of it, a number of interesting results about central and commuting automorphisms of Lie algebras will be presented. Finally, a characterization of Lie algebras for which the set of central automorphisms is trivial or the set of commuting automorphisms is trivial will be given.

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