This content will become publicly available on March 4, 2025
- PAR ID:
- 10497794
- Publisher / Repository:
- Linear and Multilinear Algebra
- Date Published:
- Journal Name:
- Linear and Multilinear Algebra
- ISSN:
- 0308-1087
- Page Range / eLocation ID:
- 1 to 13
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We describe the structure and different features of Lie algebras in the Verlinde category, obtained as semisimplification of contragredient Lie algebras in characteristic p with respect to the adjoint action of a Chevalley generator. In particular, we construct a root system for these algebras that arises as a parabolic restriction of the known root system for the classical Lie algebra. This gives a lattice grading with simple homogeneous components and a triangular decomposition for the semisimplified Lie algebra. We also obtain a non-degenerate invariant form that behaves well with the lattice grading. As an application, we exhibit concrete new examples of Lie algebras in the Verlinde category.more » « less
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An extended derivation (endomorphism) of a (restricted) Lie algebra
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We use the Darboux coordinate representation found by two of the authors (L.Ch. and M.Sh.) for entries of general symplectic leaves of the A_n-groupoid of upper-triangular matrices to express roots of the characteristic equation det(A−λA^T)=0, with A∈A_n, in terms of Casimirs of this Darboux coordinate representation, which is based on cluster variables of Fock--Goncharov higher Teichmüller spaces for the algebra sl_n. We show that roots of the characteristic equation are simple monomials of cluster Casimir elements. This statement remains valid in the quantum case as well. We consider a generalization of A_n-groupoid to a A_{Sp_2m}-groupoid.more » « less
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Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac–Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every symmetric Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements.
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Abstract Building on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra forms a Lie algebra, and a restricted Lie algebra if contains a field of characteristic . We deduce that the space of integrable classes in forms a (restricted) Lie algebra that is invariant under derived equivalences, and under stable equivalences of Morita type between self‐injective algebras. We also provide negative answers to questions about integrable derivations posed by Linckelmann and by Farkas, Geiss and Marcos. Along the way, we compute the first Hochschild cohomology of the group algebra of any symmetric group.