Yang-Baxter Operator Research Articles

Quasi-associativity and flatness criteria for quadratic algebra deformations

LetRh be the quantumR-matrix corresponding to a Drinfeld-Jimbo quantum groupUh(G). Suppose a finite dimensional representationMh ofUh(G) is given. TheRh induces an operator onMh⊗2 andSh, its composition with the standard transposition, is the Yang-Baxter operator. It turns out that the spaceMh⊗2 admits the decompositionMh=⊕inJih whereJih are the eigensubspaces ofSh. Consider the quadratic algebras (Mh, Ehk) whereEhk=⊕i≠kJih. We prove that all (Mh,Ehk) are flat deformations of the quadratic algebras (V0,E0k). Let End(Mh;J1h, …,Jnh) be the quantum semigroup corresponding to this decomposition. Our second result is that this gives a flat deformation of the quantum semigroup End(M0;J1,0, …,Jn,0).

On tensor representations of knot algebras

The fact that a Yang-Baxter operator defines tensor representations of the Artin braid group has been used to construct knot invariants. The main purpose of this note is to extend the tensor representations of the Artin braid group to representations of the braid groupZ Bk associated to the Coxeter graphBk. This extension is based on some fundamental identities for the standardR-matrices of quantum Lie theory, here called four braid relations. As an application, tensor representations of knot algebras of typeB (Hecke, Temperley-Lieb, Birman-Wenzl-Murakami) are derived.

Braided antisymmetrizer as bialgebra homomorphism

For a Yang-Baxter operator we show that a bialgebra homomorphism from a free braided tensor bialgebra to a cofree braided shuffle bialgebra is the Woronowicz braided antisymmetrizer. A cofree braided shuffle bialgebra is a braided generalization of a cofree shuffle bialgebra introduced by Sweedler. Its graded dual bialgebra is a free braided tensor bialgebra.

Multiparameter quantum groups and multiparameterR-matrices

There exists an ( 2 ) + 1 parameter quantum group deformation of GLn which has been constructed independently by several (groups of) authors. In this note, I give an explicitR-matrix for this multiparameter family. This gives additional information on the nature of this family and facilitates some calculations. This explicitR-matrix satisfies the Yang-Baxter equation. The centre of the paper is Section 3 which describes all solutions of the YBE under the restriction r cd ab =0 unlessa, b=c, d. One kind of the most general constituents of these solutions precisely corresponds to the ( 2 ) + 1 parameter quantum group mentioned above. I describe solutions which extend to an enhanced Yang-Baxter operator and, hence, define link invariants. The paper concludes with some preliminary results on these link invariants.

MULTIPARAMETER QUANTUM FORMS OF THE ENVELOPING ALGEBRA $U_{{\mathfrak g}{\mathfrak l}_N}$ RELATED TO THE FADDEEV-RESHETIKHIN-TAKHTAJAN U(R) CONSTRUCTIONS

Two quantum enveloping algebras UR and ÛR are associated in [RTF] to any Yang-Baxter operator R. These are constructed as subalgebras of A(R)* with specific generating sets. Also, [RTF] construct specific relations on the generators for UR, leaving open the question whether these generate all relations on these generators—let us say R is “perfect” when this is the case. Given an [Formula: see text] -tuple [Formula: see text] of nonzero elements qij,r in the groundfield, ([AST], [R], [S]) construct a multiparameter deformation [Formula: see text] of GLN associated with a Yang-Baxter operator [Formula: see text]. The method of ‘braiding maps’, introduced in [LT], is applied, in order to derive a PBW basis and a generators-and-relations presentation for a suitable generalization of [Formula: see text]. These results imply that [Formula: see text] is perfect, for generic [Formula: see text]. The construction [Formula: see text] is in some ways unsatisfactory if r is a root of 1. A construction [Formula: see text] is proposed, which is in some ways better behaved, coincides with [Formula: see text] if r is not a root of 1, and also makes sense over arbitrary commutative rings.

Quantum Koszul complexes for Majid’s braided Lie algebras

A Yang–Baxter operator was canonically associated to any cocommutative braided Lie algebra as defined by Majid. Applying previous results, we construct quantum Koszul complexes for braided Lie algebras with respect to this operator. In the usual case, the complexes reduce to the Chevalley–Eilenberg complexes for classical Lie algebras.

Quantum and braided-Lie algebras

We introduce the notion of a braided-Lie algebra consisting of a finite-dimensional vector space L equipped with a bracket [ , ]: L ⊗ L → L and Yang-Baxter operator Ψ: L ⊗ L → L ⊗ L obeying some axioms. We show that such an object has an enveloping braided-bialgebra U (L). We show that every generic R-matrix leads to such a braided-Lie algebra with [ , ] given by structure constants c IJ K determined from R. In this case U(L) = B(R) the braided matrices introduced previously. We also introduce the basic theory of these braided-Lie algebras, including the natural right-regular action of a braided-Lie algebra L by braided vector fields, the braided-Killing form and the quadratic Casimir associated to L. These constructions recover the relevant notions for usual, colour and super-Lie algebras as special cases. In addition, the standard quantum deformations U q(g) are understood as the enveloping algebras of such underlying braided-Lie algebras with [ , ] on L ⊂ U q(g) given by the quantum adjoint action.

Non-Birman-Wenzl algebraic properties and redundancy of exotic enhanced Yang-Baxter operator for spin model

In this paper we explicitly construct the Markov trace for the general coloured exotic braid group representations (BGR) with spin-j. It is verified that the BGR for j=1 is redundant in the sense of Murakami (1990), but no Birman-Wenzl algebra.

R-commutative geometry and quantization of Poisson algebras

An r-commutative algebra is an algebra A equipped with a Yang-Baxter operator R: A ⊗ A → A ⊗ A satisfying m = mR, where m: A ⊗ A → A is the multiplication map, together with the compatibility conditions R( a ⊗ 1) = 1 ⊗ a, R(1 ⊗ a) = a ⊗ 1, R(id ⊗ m) = ( m ⊗ id) R 2 R 1, and R( m ⊗ id) = (id ⊗ m) R 1 R 2. The basic notions of differential geometry extend from commutative (or supercommutative) algebras to r-commutative algebras. Examples of r-commutative algebras obtained by quantization of Poisson algebras include the Weyl algebra, noncommutative tori, quantum groups, and certain quantum vector spaces. In many of these cases the r-commutative de Rham cohomology is stable under quantization.

Tortile Yang-Baxter operators in tensor categories

The free tortile tensor category T ∼ on one generating object is shown here, by purely algebraic techniques, to also be the free tensor category containing an object equipped with a tortile Yang-Baxter operator. Shum has a geometric characterization of T ∼ as the category whose arrows are tangled ribbons. This alternative universal property can be used to construct representations of T ∼.

Exactly Solvable Models and New Link Polynomials. V. Yang-Baxter Operator and Braid-Monoid Algebra

Yang-Baxter operator is shown to be a fundamental object which relates theory of solvable models to theory of knots and links. First, general properties of Yang-Baxter operators are investigated. Second, a method to construct composite Yang-Baxter operators is explicitly shown. Lastly, from Yang-Baxter operators with crossing symmetry, braid-monoid algebras are derived. It is emphasized that the factorized S -matrices and their graphical illustrations link two approaches, algebraic and combinatorial, in the knot theory.