Braids, link polynomials and a new algebra

Abstract

Abstract.A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on 2 parameters. The decomposition of the corresponding algebras into irreducible components is given and it is shown how they are related to Jones' algebras and to Brauer's centralizer algebras.In [J,3] Vaughan Jones announced the discovery of a new polynomial invariant of knots and links, which bore many similarities to the classical Alexander polynomial, but was seen to detect properties of a link which could not be detected by the Alexander invariants. The discovery was a real surprise, one of those exciting moments in mathematics when two seemingly unrelated disciplines turn out to have deep interconnections. The discovery came about in the following way. Jones' earlier contributions in the area of Operator Algebras had produced, in [J,l], a family of algebras An(t), t G C, indexed by the natural numbers «=1,2,3,..., and equipped with a trace function t : An(t) -* C. His algebra An(t) was a quotient of the well-known Hecke algebra of the symmetric group, which we denote by %?n(l,m) to delineate our particular 2-parameter version of it. Jones had discovered, in [J,2], that there were representations of Artin's braid group Bn in the algebra An(t), in fact there were maps Bn±.S?n(l,m)^An(t) from Bn into the multiplicative group of An(t) which factored through %,(!>**).Links enter the picture via braids. Each oriented link L in oriented S can be represented by a (nonunique) element ß in some braid group Bn . There is an equivalence relation on P^ = LI^I, P" , known as Markov equivalence, which determines a 1-1 correspondence between equivalence classes [ß] G B and isotopy types of the associated oriented links L". Jones' discovery was that with a small renormalization his trace function on A^t) = ]X?=XA (t) could be made into a function which lifted to an invariant on Markov classesReceived by the editors December 9, 1987. 1980 Mathematics Subject Classification (1985. Primary 57M25; Secondary 20F29, 20C07.The work of the first author was supported in part by NSF grant #DMS-8503758. The work of the second author was supported in part by NSF grant #DMS-8510816. in P^ . That modified trace, described in [J,3] and in more detail in [J,4], is the Jones polynomial VL(t). (It becomes a polynomial when the parameter t is regarded as an indeterminate.)The polynomial VL(t) was quickly generalized in a six-author paper [FYHLMO], to a 2-variable polynomial PL(l,m).One of the authors was A. Ocneanu. Ocneanu's interpretation of Pjj, m), as described in [FYHLMO] and [J,4] The purpose of this note is to reverse the process begun by Jones. We will use the existence of KL(l ,m), and apply the methods used to construct it in [K,l] to construct a new two-parameter family of finite-dimensional algebras, {Wn(l,m); n = 1,2,3, ...}, complete with trace, such that KL(l,m) is, after appropriate renormalization, that trace, just as PL(l,m), renormalized, was s...

Skein construction of idempotents in Birman-Murakami-Wenzl algebras

“…We need that the quantum dimensions λ , with |λ| = n − 1 are not zero. This result follows from [7,Theorem 3.7], and will also be proved, by considering the specializations corresponding to Brauer algebras in the next section. …”

“…The algebra K n = End K (n) is isomorphic to the Birman-Murakami-Wenzl algebra which is the quotient of the braid group algebra k[B n ] by the Kauffman skein relations [7,13]. For a proof of the above isomorphism, see [12] or [17].…”

“…The Birman-Murakami-Wenzl algebras are deformations of the Brauer centralizer algebras [4,13]. They are quotients of the Artin braid groups algebras, and have appeared in connection with the Kauffman link invariant and the quantum groups of types B, C and D. The Birman-Murakami-Wenzl algebras are generically semisimple, and their structure was given by Wenzl [20].…”

BMW algebra, quantized coordinate algebra and type 𝐶 Schur–Weyl duality

“…The following result is implicit in the proof of [34,Section 5,6]. The author is indebted to Professor Kashiwara for pointing this out to him.…”

Singly generated planar algebras of small dimension, Part III

“…In practice it seems that if the relations are powerful enough to calculate the value of a labelled planar diagram with no boundary points, then we can calculate the entire structure just from these relations. The BMW planar algebras [BW89,Mur87] are generated by an element satisfying the Yang-Baxter equation or equivalently the type III Reidemeister move. The evaluation of labeled diagrams with no boundary points is known as the Kauffman polynomial, see [Kau90].…”