Related papers: Kauffman polynomial from a generalized Yang-Yang f…
We define and study a bigraded knot invariant whose Euler characteristic is the Alexander polynomial, closely connected to knot Floer homology. The invariant is the homology of a chain complex whose generators correspond to Kauffman states…
In this article we present some probably unexpected (in our opinion) properties of representations of Yang-Mills algebras. We first show that any free Lie algebra with m generators is a quotient of the Yang-Mills algebra ym(n) on n…
We study the algebras underlying solvable lattice models of the type fusion interaction round the face (IRF). We propose that the algebras are universal, depending only on the number of blocks, which is the degree of polynomial equation…
A Cartan Calculus of Lie derivatives, differential forms, and inner derivations, based on an undeformed Cartan identity, is constructed. We attempt a classification of various types of quantum Lie algebras and present a fairly general…
The generalized covariant derivative, that uses both scalar and vector bosons, is defined. It is shown how a grand unified theory of the Standard Model can be constructed using a generalized Yang-Mills theory.
We give a general surgery formula for the Casson-Walker-Lescop invariant of closed 3-manifolds, seen as the leading term of the LMO invariant, in a purely diagrammatic and combinatorial way. This provides a new viewpoint on a formula…
We have new solutions to the Yang-Baxter equation, from which we have constructed new link invariants containing more than two arbitrary parameters. This may be regarded as a generalization of the Jones' polynomial. We have also found…
We use the isomorphisms between the $R$-matrix and Drinfeld presentations of the quantum affine algebras in types $B$, $C$ and $D$ produced in our previous work to describe finite-dimensional irreducible representations in the $R$-matrix…
Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided…
The paper contains the computation of the noncommutative A-ideal of the figure-eight knot, a noncommutative generalization of the A-polynomial. We show that if a knot has the same noncommutative A-ideal as the figure-eight knot, then all…
The object of this work is the systematical study of a certain type of generalized Cartan matrices associated with the Dynkin diagrams that characterize Cartan-Lie and affine Kac-Moody algebras. These generalized matrices are associated to…
We present the classification of the most general regular solutions to the boundary Yang-Baxter equations for vertex models associated with non-exceptional affine Lie algebras. Reduced solutions found by applying a limit procedure to the…
The Yang-Baxter equation plays a fundamental role in various areas of mathematics. Its solutions, called braidings, are built, among others, from Yetter-Drinfel'd modules over a Hopf algebra, from self-distributive structures, and from…
In \cite{GS1} the notion of braided Yangians of Reflection Equation type was introduced. Each of these algebras is associated with an involutive or Hecke symmetry $R$. Besides, the quantum analogs of certain symmetric polynomials…
We establish a degeneration isomorphism between quantum toroidal algebras and untwisted affine Yangians, valid for all untwisted affine Kac-Moody Lie algebras. Specifically, we prove that the affine Yangian $Y_\hbar(\mathfrak{g})$ is…
Any solution to the Yang-Baxter equation yields a family of representations of braid groups. Under certain conditions, identified by Turaev, the appropriately normalized trace of these representations yields a link invariant. Any…
We prove two generalisations of the Binomial theorem that are also generalisations of the q-binomial theorem. These generalisations arise from the commutation relations satisfied by the components of the co-multiplications of non-simple…
In the present paper, we define the generalized Kwang-Wu Chen matrix. Basic properties of this generalization, such as explicit formulas and generating functions are presented. Moreover, we focus on a new class of generalized Fubini…
We construct a state model for the two-variable Kauffman polynomial using planar trivalent graphs. We also use this model to obtain a polynomial invariant for a certain type of trivalent graphs embedded in three-dimensional space.
We introduce the special set-theoretic Yang-Baxter algebra and show that it is a Hopf algebra subject to certain conditions. The associated universal R-matrix is also obtained via an admissible Drinfel'd twist. The structure of braces…