The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation

Abstract

For a finite involutive non-degenerate solution ( X , r ) (X,r) of the Yang–Baxter equation it is known that the structure monoid M ( X , r ) M(X,r) is a monoid of I-type, and the structure algebra K [ M ( X , r ) ] K[M(X,r)] over a field K K shares many properties with commutative polynomial algebras; in particular, it is a Noetherian PI-domain that has finite Gelfand–Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid M ( X , r ) M(X,r) and the algebra K [ M ( X , r ) ] K[M(X,r)] is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of M ( X , r ) M(X,r) as a regular submonoid in the semidirect product A ( X , r ) ⋊ Sym ⁡ ( X ) A(X,r)\rtimes \operatorname {Sym} (X) , where A ( X , r ) A(X,r) is the structure monoid of the rack solution associated to ( X , r ) (X,r) , we prove that K [ M ( X , r ) ] K[M(X,r)] is a finite module over a central affine subalgebra. In particular, K [ M ( X , r ) ] K[M(X,r)] is a Noetherian PI-algebra of finite Gelfand–Kirillov dimension bounded by | X | |X| . We also characterize, in ring-theoretical terms of K [ M ( X , r ) ] K[M(X,r)] , when ( X , r ) (X,r) is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of M ( X , r ) M(X,r) . These results allow us to control the prime spectrum of the algebra K [ M ( X , r ) ] K[M(X,r)] and to describe the Jacobson radical and prime radical of K [ M ( X , r ) ] K[M(X,r)] . Finally, we give a matrix-type representation of the algebra K [ M ( X , r ) ] / P K[M(X,r)]/P for each prime ideal P P of K [ M ( X , r ) ] K[M(X,r)] . As a consequence, we show that if K [ M ( X , r ) ] K[M(X,r)] is semiprime, then there exist finitely many finitely generated abelian-by-finite groups, G 1 , … , G m G_1,\dotsc ,G_m , each being the group of quotients of a cancellative subsemigroup of M ( X , r ) M(X,r) such that the algebra K [ M ( X , r ) ] K[M(X,r)] embeds into M v 1 ⁡ ( K [ G 1 ] ) × ⋯ × M v m ⁡ ( K [ G m ] ) \operatorname {M}_{v_1}(K[G_1])\times \dotsb \times \operatorname {M}_{v_m}(K[G_m]) , a direct product of matrix algebras.