Semigroups of I-type

By Tatiana Gateva-Ivanova, Michel van den Bergh. 2003

245 citations recorded by the source corpus.

Assume that $S$ is a semigroup generated by $\{x_1,...,x_n\}$, and let $\Uscr$ be the multiplicative free commutative semigroup generated by $\{u_1,...,u_n\}$. We say that $S$ is of \emph{$I$-typ}e if there is a bijection $v:\Uscr\r S$ such that for all $a\in\Uscr$, $\{v(u_1a),... v(u_na)\}=\{x_1v(a),...,x_nv(a)\}$. This condition appeared naturally in the work on Sklyanin algebras by John Tate and the second author. In this paper we show that the condition for a semigroup to be of $I$-type is related to various other mathematical notions found in the literature. In particular we show that semigroups of $I$-type appear in the study of the settheoretic solutions of the Yang-Baxter equation, in the theory of Bieberbach groups and in the study of certain skew binomial polynomial rings which were introduced by the first author.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/0285fb36b29fbd3f38b2d6685068dc48247b1ba9

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Citations in the corpus

Trusses: Between braces and rings (2017, background): “Though attracting a rather limited attention at the beginning, notwithstanding the ground-breaking contributions such as in [8], [9], [10], the set theoretic Yang-Baxter equation has become a subject of intensive study more recently after the discovery of richness of the algebraic structure of sets ”
Matched pairs approach to set theoretic solutions of the Yang–Baxter equation (2005, methodology): “(1) S(X, r) is a semigroup of I -type, (see [10] for the definition).”
Solutions of the Yang-Baxter equation associated with a left brace (2015, background): “the Yang-Baxter equation in physics, but also for its connection with other topics in mathematics of recent interest: semigroups of I-type and Bieberbach groups [16], bijective 1-cocycles [11], radical rings [23], triply factorized groups [25], construction of semisimple minimal triangular Hopf alge”
Skew left braces of nilpotent type (2018, background): “The first papers where non-degenerate set-theoretic solutions were studied are [11, 15, 21, 31].”
Braces and the Yang–Baxter Equation (2012, methodology): “We now recall the definition of a monoid of I-type introduced by GatevaIvanova and Van den Bergh in [16].”
A combinatorial approach to the set-theoretic solutions of the Yang–Baxter equation (2004, methodology): “The rings of I-type were introduced and studied by J.Tate, and M. Van den Bergh in their work on the homological properties of Sklyanin Algebras, [31].”
Garside Groups and Yang–Baxter Equation (2009, unknown): “So, the necessity of being Garside can be derived from the combination of the results from [14, 15].”
Foundations of Garside Theory (2013, result): “van den Bergh [125], building on earlier results of J.”
On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation (2015, background): “Then the adjoint group Ao of A is a nilpotent if and only if An = 0 for some natural number n. Gateva- Ivanova and Van den Bergh [28] and independently Etingof, Schedler and Soloviev [22] gave a group theoretical interpretation of the set theoretic invo- lutive non-degenerate solutions of the Yang-B”
Quadratic algebras, Yang-Baxter equation, and Artin-Schelter regularity (2010, methodology): “A class of PBW Arin Schelter regular rings of arbitrarily high global dimension n, were introduced and studied in [15], [25], [18], [17].”
Retractability of set theoretic solutions of the Yang-Baxter equation (2009, unknown): “G R ] 2 3”
Involutive Yang-Baxter groups (2008, background): “In [7] it is shown that these monoids S have a presentation”
Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation (2016, background): “The structure group G(X, r) was introduced and studied in [17, 21] and it is defined as the group with the following presentation”
Counterexample to a conjecture about braces (2015, background): “The study of this class of solutions was initiated in [11] and [15].”
Problems on skew left braces (2018, background): “Gateva–Ivanova and Van den Bergh proved in [42] that in the case of involutive solutions the multiplicative group of G(X, r) is always torsion-free.”