Operator identities on Lie algebras, rewriting systems and Gr\"obner-Shirshov bases
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Publication:6390886
DOI10.1016/J.JALGEBRA.2023.01.001zbMath1517.17020arXiv2202.05914MaRDI QIDQ6390886
Publication date: 11 February 2022
Abstract: Motivated by the pivotal role played by linear operators, many years ago Rota proposed to determine algebraic operator identities satisfied by linear operators on associative algebras, later called Rota's program on algebraic operators. Recent progresses on this program have been achieved in the contexts of operated algebra, rewriting systems and Groebner-Shirshov bases. These developments also suggest that Rota's insight can be applied to determine operator identities on Lie algebras, and thus to put the various linear operators on Lie algebras in a uniform perspective. This paper carries out this approach, utilizing operated polynomial Lie algebras spanned by non-associative Lyndon-Shirshov bracketed words. The Lie algebra analog of Rota's program was formulated in terms convergent rewriting systems and equivalently in terms of Groebner-Shirshov bases. This Lie algebra analog is shown to be compatible with Rota's program for associative algebras. As applications, a classification of differential type operators and Rota-Baxter operators are presented.
Permutations, words, matrices (05A05) Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) (16S10) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) (13P10) Automorphisms, derivations, other operators for Lie algebras and super algebras (17B40) Yang-Baxter equations and Rota-Baxter operators (17B38) Gröbner-Shirshov bases (16Z10) Gröbner-Shirshov bases in nonassociative algebras (17A61)
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