2024/08/23 by Tatiana Gateva-Ivanova, Shahn Majid, Gateva-Ivanova, Tatiana +1
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Polynomial and algebraic computation #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2409.02939
openalex publication_date 2024/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the Yang-Baxter algebras A(K,X,r) associated to finite set-theoretic solutions (X,r) of the braid relations. We introduce an equivalent set of quadratic relations \Re⊆ G, where G is the reduced Gröbner basis of (\Re). We show that if (X,r) is left-nondegenerate and idempotent then \Re= G and the Yang-Baxter algebra is PBW. We use graphical methods to study the global dimension of PBW algebras in the n-generated case and apply this to Yang-Baxter algebras in the left-nondegenerate idempotent case. We study the d-Veronese subalgebras for a class of quadratic algebras and use this to show that for (X,r) left-nondegenerate idempotent, the d-Veronese subalgebra A(K,X,r)(d) can be identified with A(K,X,r(d)), where (X,r(d)) are all left-nondegenerate idempotent solutions. We determined the Segre product in the left-nondegenerate idempotent setting. Our results apply to a previously studied class of `permutation idempotent' solutions, where we show that all their Yang-Baxter algebras for a given cardinality of X are isomorphic and are isomorphic to their d-Veronese subalgebras. In the linearised setting, we construct the Koszul dual of the Yang-Baxter algebra and the Nichols-Woronowicz algebra in the idempotent case, showing that the latter is quadratic. We also construct noncommutative differentials on some of these quadratic algebras.