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22
Matched pairs approach to set theoretic solutions of the YangBaxter equation
 J. Algebra
"... Abstract. We study settheoretic solutions (X, r) of the YangBaxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterization of involutive squarefree solutions in terms of cyclicity conditions. We characterise general solutions in terms of an i ..."
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Abstract. We study settheoretic solutions (X, r) of the YangBaxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterization of involutive squarefree solutions in terms of cyclicity conditions. We characterise general solutions in terms of an induced matched pair of unital semigroups S(X, r) and construct (S, rS) from the matched pair. Finally, we study extensions of solutions in terms of matched pairs of their associated semigroups. We also prove several general results about matched pairs of unital semigroups of the required type, including iterated products S ⊲ ⊳ S ⊲ ⊳ S underlying the proof that rS is a solution, and extensions (S ⊲ ⊳ T, rS⊲⊳T). Examples include a general ‘double ’ construction (S ⊲ ⊳ S, rS⊲⊳S) and some concrete extensions, their actions and graphs based on small sets. 1.
QUANTUM SPACES ASSOCIATED TO MULTIPERMUTATION SOLUTIONS OF LEVEL TWO
"... Abstract. We study finite settheoretic solutions (X, r) of the YangBaxter equation of squarefree multipermutation type. We show that each such solution over C with multipermutation level two can be put in diagonal form with the associated YangBaxter algebra A(C, X, r) having a qcommutation form ..."
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Cited by 7 (5 self)
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Abstract. We study finite settheoretic solutions (X, r) of the YangBaxter equation of squarefree multipermutation type. We show that each such solution over C with multipermutation level two can be put in diagonal form with the associated YangBaxter algebra A(C, X, r) having a qcommutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots of a prescribed form appear as determined by the representation theory of the finite abelian group G of left actions on X. We study the structure of A(C, X, r) and show that they have a •product form ‘quantizing ’ the commutative algebra of polynomials in X  variables. We obtain the •product both as a Drinfeld cotwist for a certain canonical 2cocycle and as a braidedopposite product for a certain crossed Gmodule (over any field k). We provide first steps in the noncommutative differential geometry of A(k, X, r) arising from these results. As a byproduct of our work we find that every such level 2 solution (X, r) factorises as r = f ◦ τ ◦ f −1 where τ is the flip map and (X, f) is another solution coming from X as a crossed Gset. 1.
YangBaxter maps: dynamical point of view
, 2006
"... A review of some recent results on the dynamical theory of the YangBaxter maps (also known as settheoretical solutions to the quantum YangBaxter equation) is given. The central question is the integrability of the transfer dynamics. The relations with matrix factorisations, matrix KdV solitons, ..."
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Cited by 7 (0 self)
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A review of some recent results on the dynamical theory of the YangBaxter maps (also known as settheoretical solutions to the quantum YangBaxter equation) is given. The central question is the integrability of the transfer dynamics. The relations with matrix factorisations, matrix KdV solitons, Poisson Lie groups, geometric crystals and tropical combinatorics are discussed and demonstrated on several concrete examples.
GARSIDE STRUCTURE ON MONOIDS WITH QUADRATIC SQUAREFREE RELATIONS
, 2009
"... We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a “nice” Garside element, certain monoids S with quadratic relations, whose monoidal algebra A = kS has a Frobenius Koszul dual A! with regular socle, ..."
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Cited by 6 (2 self)
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We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a “nice” Garside element, certain monoids S with quadratic relations, whose monoidal algebra A = kS has a Frobenius Koszul dual A! with regular socle, the monoids of skewpolynomial type (or equivalently, binomial skewpolynomial rings) which were introduced and studied by the author and in 1995 provided a new class of Noetherian ArtinSchelter regular domains, and the squarefree settheoretic solutions of the YangBaxter equation. There is a beautiful symmetry in these objects due to their nice combinatorial and algebraic properties.
Communication Introduction to the YangBaxter Equation with Open Problems
, 2012
"... axioms ..."
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Dynamical YangBaxter maps with an Invariance Condition
, 2007
"... By means of left quasigroups L = (L, ·) and ternary systems, we construct dynamical YangBaxter maps associated with L, L, and (·) satisfying an invariance condition that the binary operation (·) of the left quasigroup L defines. Conversely, this construction characterize such dynamical YangBaxter ..."
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Cited by 3 (0 self)
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By means of left quasigroups L = (L, ·) and ternary systems, we construct dynamical YangBaxter maps associated with L, L, and (·) satisfying an invariance condition that the binary operation (·) of the left quasigroup L defines. Conversely, this construction characterize such dynamical YangBaxter maps. The unitary condition of the dynamical YangBaxter map is discussed. Moreover, we establish a correspondence between two dynamical YangBaxter maps constructed in this paper. This correspondence produces a version of the vertexIRF correspondence.
BINOMIAL SKEW POLYNOMIAL RINGS, ARTINSCHELTER REGULARITY, AND BINOMIAL SOLUTIONS OF THE YANGBAXTER EQUATION
, 2009
"... Let k be a field and X be a set of n elements. We introduce and study a class of quadratic kalgebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical YangBaxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dime ..."
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Let k be a field and X be a set of n elements. We introduce and study a class of quadratic kalgebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical YangBaxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k \ {0}, and z, t ∈ X is satisfied in A. We prove the equivalence of the notions a binomial skew polynomial ring and a binomial solution of YBE. This implies that the YangBaxter algebra of such a solution is of PoincaréBirkhoffWitt type, and possesses a number of other nice properties such as being Koszul, Noetherian, and an ArtinSchelter regular domain.
MULTIPERMUTATION SOLUTIONS OF THE YANG–BAXTER EQUATION
, 907
"... Abstract. Settheoretic solutions of the Yang–Baxter equation form a meetingground of mathematical physics, algebra and combinatorics. Such a solution consists of a set X and a function r: X ×X → X ×X which satisfies the braid relation. We examine solutions here mainly from the point of view of fini ..."
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Abstract. Settheoretic solutions of the Yang–Baxter equation form a meetingground of mathematical physics, algebra and combinatorics. Such a solution consists of a set X and a function r: X ×X → X ×X which satisfies the braid relation. We examine solutions here mainly from the point of view of finite permutation groups: a solution gives rise to a map from X to the symmetric group Sym(X) on X satisfying certain conditions. Our results include many new constructions based on strong twisted union and wreath product, with an investigation of retracts and the multipermutation level and the solvable length of the groups defined by the solutions; and new results about decompositions and factorisations of the groups defined by invariant subsets of the solution. Contents
Quadratic algebras, YangBaxter equation, and ArtinSchelter regularity, Adv
 in Mathematics 230
, 2012
"... Abstract. We study quadratic algebras over a field k. We show that an ngenerated PBW algebra A has finite global dimension and polynomial growth iff its Hilbert series is HA(z) = 1/(1−z)n. Surprising amount can be said when the algebra A has quantum binomial relations, that is the defining relatio ..."
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Abstract. We study quadratic algebras over a field k. We show that an ngenerated PBW algebra A has finite global dimension and polynomial growth iff its Hilbert series is HA(z) = 1/(1−z)n. Surprising amount can be said when the algebra A has quantum binomial relations, that is the defining relations are nondegenerate squarefree binomials xy − cxyzt with nonzero coefficients cxy ∈ k. In this case various good algebraic and homological properties are closely related. The main result shows that for an ngenerated quantum binomial algebra A the following conditions are equivalent: (i) A is a PBW algebra with finite global dimension; (ii) A is PBW and has polynomial growth; (iii) A is an ArtinSchelter regular PBW algebra; (iv) A is a YangBaxter algebra; (v) HA(z) = 1/(1 − z)n; (vi) The dual A! is a quantum Grassman algebra; (vii) A is a binomial skew polynomial ring. So for quantum binomial algebras the problem of classification of ArtinSchelter regular PBW algebras of global dimension n is equivalent to the classification of squarefree settheoretic solutions of the YangBaxter equation (X, r), on sets X of order n. 1.