Mathematics and Science

The paper considers computer algebra in a non-commutative setting. So far, such investigations have been centred on the use of algorithms for equality and of universal properties of algebras. Here, the foundation of all computations is the presentation of the algebra under investigation by a finite number of generators subject to a finite number of defining relations, which satisfy the additional property of forming a Gr6bner (or standard) basis. Such algebras are called s.f.p.--for standard finite presentation. It is shown that various algebraic properties, such as being finite-dimensional, nilpotent, nil, algebraic are algorithmically recognisable. When the defining relations are words in the generators, this is also shown to be the case for the properties of being semi-simple, prime, semi-prime, etc.

We study finite set-theoretic solutions (X, r) of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over C with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra A(C, X, r) having a q-commutation 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 2-cocycle and as a braided-opposite product for a certain crossed G-module (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 G-set.

We study finite set-theoretic solutions $(X,r)$ of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over $\C$ with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra $\Acal(\C,X,r)$ having a $q$-commutation 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 finite abelian group $\Gcal$ of left actions on $X$. We study the structure of $\Acal(\C,X,r)$ and show that they have a $\bullet$-product form `quantizing' the commutative algebra of polynomials in $|X|$ variables. We obtain the $\bullet$-product both as a Drinfeld cotwist for a certain canonical 2-cocycle and as a braided-opposite product for a certain crossed $\Gcal$-module (over any field $k$). We provide first steps in the noncommutative differential geometry of $\Acal(k,X,r)$ arising from these results. As a byproduct of our work we find that every such level 2 solution $(X,r)$ factorises as $r=f\circ\tau\circ f^{-1}$ where $\tau$ is the flip map and $(X,f)$ is another solution coming from $X$ as a crossed $\Gcal$-set.

We study set-theoretic solutions (X, r) of the Yang-Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of abstract matched pair properties of the associated monoid S(X, r) and we show that r extends as a solution (S(X, r), r S ). Finally, we study extensions of solutions both directly and in terms of matched pairs of their associated monoids. We also prove several general results about matched pairs of monoids S of the required type, including iterated products S S S equivalent to r S a solution, and extensions (S T, r S T ). Examples include a general 'double' construction (S S, r S S ) and some concrete extensions, their actions and graphs based on small sets.

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.

A bijective map r : X 2 −→ X 2 , where X = {x 1 , · · · , xn} is a finite set, is called a set-theoretic solution of the Yang-Baxter equation (YBE) if the braid relation r 12 r 23 r 12 = r 23 r 12 r 23 holds in X 3 . A non-degenerate involutive solution (X, r) satisfying r(xx) = xx, for all x ∈ X, is called squarefree solution. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh.

We study set-theoretic solutions (X, r) of the Yang-Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of abstract matched pair properties of the associated monoid S(X, r) and we show that r extends as a solution (S(X, r), r S ). Finally, we study extensions of solutions both directly and in terms of matched pairs of their associated monoids. We also prove several general results about matched pairs of monoids S of the required type, including iterated products S S S equivalent to r S a solution, and extensions (S T, r S T ). Examples include a general 'double' construction (S S, r S S ) and some concrete extensions, their actions and graphs based on small sets.

Let $k$ be a field and $X$ be a set of $n$ elements. We introduce and study a class of quadratic $k$-algebras called \emph{quantum binomial algebras}. Our main result shows that such an algebra $A$ defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual $A^{!}$ is Frobenius of dimension $n,$ with a \emph{regular socle} and for each $x,y \in X $ an equality of the type $xyy=\alpha zzt,$ where $\alpha \in k \setminus\{0\},$ and $z,t \in X$ is satisfied in $A$. We prove the equivalence of the notions \emph{a binomial skew polynomial ring} and \emph{a binomial solution of YBE}. This implies that the Yang-Baxter algebra of such a solution is of Poincar\'{e}-Birkhoff-Witt type, and possesses a number of other nice properties such as being Koszul, Noetherian, and an Artin-Schelter regular domain.

We extend our recent work on set-theoretic solutions of the Yang-Baxter or braid relations with new results about their automorphism groups, strong twisted unions of solutions and multipermutation solutions. We introduce and study graphs of solutions and use our graphical methods for the computation of solutions of finite order and their automorphisms. Results include a detailed study of solutions of multipermutation level 2.

We study quadratic algebras over a field $\textbf{k}$. We show that an $n$-generated PBW algebra $A$ has finite global dimension and polynomial growth \emph{iff} its Hilbert series is $H_A(z)= 1 /(1-z)^n$. Surprising amount can be said when the algebra $A$ has \emph{quantum binomial relations}, that is the defining relations are nondegenerate square-free binomials $xy-c_{xy}zt$ with non-zero coefficients $c_{xy}\in \textbf{k}$. In this case various good algebraic and homological properties are closely related. The main result shows that for an $n$-generated 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 Artin-Schelter regular PBW algebra; (iv) $A$ is a Yang-Baxter algebra; (v) $H_A(z)= 1/(1-z)^n;$ (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 Artin-Schelter regular PBW algebras of global dimension $n$ is equivalent to the classification of square-free set-theoretic solutions of the Yang-Baxter equation $(X,r)$, on sets $X$ of order $n$.

We consider algebras over a field K defined by a presentation K x 1 , . . . , x n | R , where R consists of n 2 square-free relations of the form x i x j = x k x l with every monomial x i x j , i = j , appearing in one of the relations. Certain sufficient conditions for the algebra to be noetherian and PI are determined. For this, we prove more generally that right noetherian algebras of finite Gelfand-Kirillov dimension defined by homogeneous semigroup relations satisfy a polynomial identity. The structure of the underlying monoid, defined by the same presentation, is described. This is used to derive information on the prime radical and minimal prime ideals. Some examples are described in detail. Earlier, Gateva-Ivanova and van den Bergh, and Jespers and Okniński considered special classes of such algebras in the contexts of noetherian algebras, Gröbner bases, finitely generated solvable groups, semigroup algebras, and set theoretic solutions of the Yang-Baxter equation.

We extend our recent work on set-theoretic solutions of the Yang-Baxter or braid relations with new results about their automorphism groups, strong twisted unions of solutions and multipermutation solutions. We introduce and study graphs of solutions and use our graphical methods for the computation of solutions of finite order and their automorphisms. Results include a detailed study of solutions of multipermutation level 2.

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 H A (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 square-free binomials xy − cxyzt with non-zero coefficients cxy ∈ k. In this case various good algebraic and homological properties are closely related. The main result shows that for an n-generated 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 Artin-Schelter regular PBW algebra; (iv) A is a Yang-Baxter algebra; (v) H A (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 Artin-Schelter regular PBW algebras of global dimension n is equivalent to the classification of square-free set-theoretic solutions of the Yang-Baxter equation (X, r), on sets X of order n.

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 skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were introduced and studied by the author and in 1995 provided a new class of Noetherian Artin-Schelter regular domains, and the square-free set-theoretic solutions of the Yang-Baxter equation. There is a beautiful symmetry in these objects due to their nice combinatorial and algebraic properties.

We study finite set-theoretic solutions (X, r) of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over C with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra A(C, X, r) having a q-commutation 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 2-cocycle and as a braided-opposite product for a certain crossed G-module (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 G-set.

Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter 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 Yang-Baxter algebra of such a solution is of Poincaré-Birkhoff-Witt type, and possesses a number of other nice properties such as being Koszul, Noetherian, and an Artin-Schelter regular domain.

Set-theoretic 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 10. More about YB permutation groups 58 References 59

We study set-theoretic solutions (X, r) of the Yang-Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of abstract matched pair properties of the associated monoid S(X, r) and we show that r extends as a solution (S(X, r), r S). Finally, we study extensions of solutions both directly and in terms of matched pairs of their associated monoids. We also prove several general results about matched pairs of monoids S of the required type, including iterated products S S S equivalent to r S a solution, and extensions (S T, r S T). Examples include a general 'double' construction (S S, r S S) and some concrete extensions, their actions and graphs based on small sets.

We consider fmitely generated aasocnttive algebras over a tixed held K of arbitrary characteristic. For such an algebra A we impose some structural restrictions (WC call A strictly ordcrcd). We arc interested in the implication of strict order on A for its noetherian properties. In particular, we prove that if A is a graded standard hmtely presented strictly ordered algebra, then .4 is left noetherian If and only if it is almost commutative. In this case A has polynomial growth.