Search

We develop the theory of Schur covers of finite skew braces. We prove the existence of at least one Schur cover. We also compute several examples. We prove that different Schur covers are isoclinic. Finally, we prove that Schur covers have the lifting property concerning projective representations of skew braces. [ABSTRACT FROM AUTHOR]

We study relations between different notions of nilpotency in the context of skew braces and applications to the structure of solutions to the Yang–Baxter equation. In particular, we consider annihilator nilpotent skew braces, an important class that turns out to be a brace-theoretic analog to the class of nilpotent groups. In this vein, several well-known theorems in group theory are proved in the more general setting of skew braces. [ABSTRACT FROM AUTHOR]

We use Constraint Satisfaction methods to enumerate and construct set-theoretic solutions to the Yang–Baxter equation of small size. We show that there are 321,931 involutive solutions of size nine, 4,895,272 involutive solutions of size ten and 422,449,480 non-involutive solution of size eight. Our method is then used to enumerate non-involutive biquandles. [ABSTRACT FROM AUTHOR]

Given a finite non-degenerate set-theoretic solution (X, r) of the Yang–Baxter equation and a field K, the structure K-algebra of (X, r) is A = A (K , X , r) = K ⟨ X ∣ x y = u v whenever r (x , y) = (u , v) ⟩ . Note that A = ⊕ n ≥ 0 A n is a graded algebra, where A n is the linear span of all the elements x 1 ⋯ x n , for x 1 , ⋯ , x n ∈ X . One of the known results asserts that the maximal possible value of dim (A 2) corresponds to involutive solutions and implies several deep and important properties of A(K, X, r). Following recent ideas of Gateva-Ivanova (A combinatorial approach to noninvolutive set-theoretic solutions of the Yang–Baxter equation. arXiv:1808.03938v3 [math.QA], 2018), we focus on the minimal possible values of the dimension of A 2 . We determine lower bounds and completely classify solutions (X, r) for which these bounds are attained in the general case and also in the square-free case. This is done in terms of the so called derived solution, introduced by Soloviev and closely related with racks and quandles. Several problems posed by Gateva-Ivanova (2018) are solved. [ABSTRACT FROM AUTHOR]

A variation of the Zamolodchikov–Faddeev algebra over a finite-dimensional Hilbert space H and an involutive unitary R-Matrix S is studied. This algebra carries a natural vacuum state, and the corresponding Fock representation spaces F S (H) are shown to satisfy F S ⊞ R (H ⊕ K) ≅ F S (H) ⊗ F R (K) , where S ⊞ R is the box-sum of S (on H ⊗ H ) and R (on K ⊗ K ). This analysis generalises the well-known structure of Bose/Fermi Fock spaces and a recent result of Pennig. These representations are motivated from quantum field theory (short-distance scaling limits of integrable models). [ABSTRACT FROM AUTHOR]

This paper presents an explicit correspondence between two different types of integrable equations: the quantum Yang–Baxter equation in its star–triangle relation form and the classical 3D-consistent quad equations in the Adler–Bobenko–Suris (ABS) classification. Each of the 3D-consistent ABS quad equations of H-type is, respectively, derived from the quasi-classical expansion of a counterpart star–triangle relation. Through these derivations, it is seen that the star–triangle relation provides a natural path integral quantization of an ABS equation. The interpretation of the different star–triangle relations is also given in terms of (hyperbolic/rational/classical) hypergeometric integrals, revealing the hypergeometric structure that links the two different types of integrable systems. Many new limiting relations that exist between the star–triangle relations/hypergeometric integrals are proven for each case. [ABSTRACT FROM AUTHOR]

In a previous paper (Kels in J Phys A 50(49):495202, 2017), the author has established an extension of the Z-invariance property for integrable edge-interaction models of statistical mechanics, that satisfy the star–triangle relation (STR) form of the Yang–Baxter equation (YBE). In the present paper, an analogous extended Z-invariance property is shown to also hold for integrable vector models and interaction-round-a-face (IRF) models of statistical mechanics respectively. As for the previous case of the STR, the Z-invariance property is shown through the use of local cubic-type deformations of a 2-dimensional surface associated to the models, which allow an extension of the models onto a subset of next nearest neighbour vertices of Z 3 , while leaving the partition functions invariant. These deformations are permitted as a consequence of the respective YBE's satisfied by the models. The quasi-classical limit is also considered, and it is shown that an analogous Z-invariance property holds for the variational formulation of classical discrete Laplace equations which arise in this limit. From this limit, new integrable 3D-consistent multi-component quad equations are proposed, which are constructed from a degeneration of the equations of motion for IRF Boltzmann weights. [ABSTRACT FROM AUTHOR]

Abstract We involve simultaneously the theory of braided groups and the theory of braces to study set-theoretic solutions of the Yang–Baxter equation (YBE). We show the intimate relation between the notions of "a symmetric group" , in the sense of Takeuchi, i.e. "a braided involutive group" , and "a left brace". We find new results on symmetric groups of finite multipermutation level and the corresponding braces. We introduce a new invariant of a symmetric group (G , r) , the derived chain of ideals of G , which gives a precise information about the recursive process of retraction of G. We prove that every symmetric group (G , r) of finite multipermutation level m is a solvable group of solvable length ≤ m. To each set-theoretic solution (X , r) of YBE we associate two invariant sequences of involutive braided groups: (i) the sequence of its derived symmetric groups ; (ii) the sequence of its derived permutation groups ; and explore these for explicit descriptions of the recursive process of retraction of (X , r). We find new criteria necessary and sufficient to claim that (X , r) is a multipermutation solution. [ABSTRACT FROM AUTHOR]

We relate the Belavin-Drinfeld cohomologies (twisted and untwisted) that have been introduced in the literature to study certain families of quantum groups and Lie bialgebras over a non algebraically closed field K of characteristic 0 to the standard non-abelian Galois cohomology H1(K,H) for a suitable algebraic K-group H. The approach presented allows us to establish in full generality certain conjectures that were known to hold for the classical types of the split simple Lie algebras. [ABSTRACT FROM AUTHOR]

We study the representations of the braid group in a generalization of the Temperley-Lieb algebra. This algebra was introduced by Kloster and contains the diagrams of the TL algebra with cut strands. We address the question of existence of Markov traces on these representations and make some observations on the possibility of their Baxterizations. [ABSTRACT FROM AUTHOR]

It is shown how Yang–Baxter maps may be directly obtained from classical counterparts of the star-triangle relations and quantum Yang–Baxter equations. This is based on reinterpreting the latter equation and its solutions which are given in terms of special functions, as a set-theoretical form of the Yang–Baxter equation whose solutions are given by quadrirational Yang–Baxter maps. The Yang–Baxter maps obtained through this approach are found to satisfy two different types of Yang–Baxter equations, one that is the usual equation involving a single map, and another equation that involves a pair of maps, which is a case of what is also known as an entwining Yang–Baxter equation. Apart from the elliptic case, each of these Yang–Baxter maps are quadrirational, but only maps that solve the former type of Yang–Baxter equation are reversible. The Yang–Baxter maps are expressed in terms of two-component variables, and two-component parameters, and have a natural QRT-like composition of separate maps for each component. Through this approach, sixteen different Yang–Baxter maps are derived from known solutions of the classical star-triangle relations. • Explicit new quadrirational Yang–Baxter maps with two-component variables. • Explicit elliptic map with parameterisation by Weierstrass functions. • Yang–Baxter maps constructed from solutions of the star-triangle relation. • Yang–Baxter maps solve two different types of Yang–Baxter equation. [ABSTRACT FROM AUTHOR]

This review is an expansion of my talk at the conference on Sixty Years of Yang-Mills Theory. I review and explain the line of thoughts that lead to a recent joint work with Hu and Geer [Hu et al., arXiv:1502.03433] on the construction, exact solutions and ubiquitous properties of a class of quantum group gauge models on a honey-comb lattice. Conceptually the construction achieves a synthesis of the ideas of Yang-Baxter equations with those of Yang-Mills theory. Physically the models describe topological anyonic states in 2D systems. [ABSTRACT FROM AUTHOR]

Abstract: We show that arising out of noncommutative geometry is a natural family of edge Laplacians on the edges of a graph. The family includes a canonical edge Laplacian associated to the graph, extending the usual graph Laplacian on vertices, and we find its spectrum. We show that for a connected graph its eigenvalues are strictly positive aside from one mandatory zero mode, and include all the vertex degrees. Our edge Laplacian is not the graph Laplacian on the line graph but rather it arises as the noncommutative Laplace-Beltrami operator on differential 1-forms, where we use the language of differential algebras to functorially interpret a graph as providing a ‘finite manifold structure’ on the set of vertices. We equip any graph with a canonical ‘Euclidean metric’ and a canonical bimodule connection, and in the case of a Cayley graph we construct a metric compatible connection for the Euclidean metric. We make use of results on bimodule connections on inner calculi on algebras, which we prove, including a general relation between zero curvature and the braid relations. [Copyright &y& Elsevier]

Abstract: We study two classes of quadratic algebras over a field : the class of all -generated PBW algebras with polynomial growth and finite global dimension, and the class of quantum binomial algebras. We show that a PBW algebra is in iff its Hilbert series is . Furthermore, the class contains a unique (up to isomorphism) monomial algebra, . A surprising amount can be said when is a quantum binomial algebra, that is its defining relations are nondegenerate square-free binomials , . Our main result shows that for an -generated quantum binomial algebra the following conditions are equivalent: (i) is an Artin–Schelter regular PBW algebra. (ii) is a Yang–Baxter algebra, that is the set of relations defines canonically a solution of the Yang–Baxter equation. (iii) is a binomial skew polynomial ring, with respect to an enumeration of . (iv) The Koszul dual is a quantum Grassmann algebra. [Copyright &y& Elsevier]

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 $A= \textbf{k}S$ 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. [ABSTRACT FROM AUTHOR]

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 $\mathcal{A}(\mathbb{C},X,r)$ 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 $\mathcal{G}$ of left actions on X. We study the structure of $\mathcal{A}(\mathbb{C},X,r)$ 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 $\mathcal{G}$-module (over any field k). We provide first steps in the noncommutative differential geometry of $\mathcal{A}(k,X,r)$ 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 where τ is the flip map and ( X, f) is another solution coming from X as a crossed $\mathcal{G}$-set. [ABSTRACT FROM AUTHOR]

In this paper, we show how generalized quaternions including some 2 × 2 matrices, can be used to find solutions of the equation \[ [B,(A - 1)(A,B)] = 0. \] These solutions can then be used to find polynomial invariants of virtual knots and links. The remaining 2 × 2 matrices will be considered in a later paper. [ABSTRACT FROM AUTHOR]

Let A, B be invertible, non-commuting elements of a ring R. Suppose that A - 1 is also invertible and that the equation \[ [B,(A - 1)(A,B)] = 0 \] called the fundamental equation is satisfied. Then an invariant R-module is defined for any diagram of a (virtual) knot or link. Solutions in the classic quaternion case have been found by Bartholomew, Budden and Fenn. Solutions in the generalized quaternion case have been found by Fenn in an earlier paper. These latter solutions are only partial in the case of 2 × 2 matrices and the aim of this paper is to provide solutions to the missing cases. [ABSTRACT FROM AUTHOR]

Abstract: We study set-theoretic solutions 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 characterisation 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 and we show that r extends as a solution on as a set. 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 equivalent to a solution, and extensions . Examples include a general ‘double’ construction and some concrete extensions, their actions and graphs based on small sets. [Copyright &y& Elsevier]

In this paper, we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2 × 2 matrices with entries in a possibly non-commutative ring, for example, the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from any classical knot, including the unknot. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Recent research suggests that there are natural connections between quantum information theory and the Yang–Baxter equation. In this paper, in terms of the almost-complex structure and with the help of its algebra, we define the Bell matrix to yield all the Greenberger–Horne–Zeilinger (GHZ) states from the product basis, prove it to form a unitary braid representation and presents a new type of solution of the quantum Yang–Baxter equation. We also study Yang–Baxterization, Hamiltonian, projectors, diagonalization, noncommutative geometry, quantum algebra and FRT dual algebra associated with this generalized Bell matrix. [ABSTRACT FROM AUTHOR]

This is a short review of nonperturbative techniques that have been used in the past 5 years to study transport out of equilibrium in low dimensional, strongly interacting systems of condensed matter physics. These techniques include massless factorized scattering, the generalization of the Landauer Bu¨ttiker approach to integrable quaisparticles, and duality. The case of tunneling between edges in the fractional quantum Hall effect is discussed in details. To cite this article: H. Saleur, C. R. Physique 3 (2002) 685–695. [Copyright &y& Elsevier]

We define the radical and weight of a skew left brace and provide some basic properties of these notions. In particular, we obtain a Wedderburn type decomposition for Artinian skew left braces. Furthermore, we prove analogues of a theorem of Wiegold, a theorem of Schur and its converse in the context of skew left braces. Finally, we apply these results to detect torsion in the structure group of a finite bijective non-degenerate set-theoretic solution of the Yang–Baxter equation. [ABSTRACT FROM AUTHOR]

Abstract: The expected number of Yang–Baxter moves applicable to a reduced decomposition of the longest element in the symmetric group on letters is observed to be 1, independent of . [Copyright &y& Elsevier]