R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

State Models and the Jones Polynomial

완효성 비료와 4종복비의 혼용 시용이 팔레놉시스의 생육에 미치는 효과

We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haar-distributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson–Zuber type.

Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group

We develop a theory of module categories over monoidal categories (this is a straightforward categorization of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and we classify module categories over the fusion category of sl(2) at a positive integer level where we meet once again the ADE classification pattern.

Module categories, weak Hopf algebras and modular invariants

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,1], a family of algebras An(t), t E C, indexed by the natural numbers n = 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 2 (1, 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 0 Atn(m A(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 S3 can be represented by a (nonunique) element fl in some braid group Bn. There is an equivalence relation on B = H0= Bn known as Markov equivalence, which determines a 1-1 correspondence between equivalence classes [fl] E B and isotopy types of the associated oriented links Lfl. Jones' discovery was that with a small renormalization his trace function on A o(t) = Hln=l An(t) could be made into a function which lifted to an invariant on Markov classes Received by the editors December 9, 1987. 1980 Mathematics Subject Classification (1985 Revision). 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. ? 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page

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Braids, link polynomials and a new algebra

On the structure of Brauer’s centralizer algebras

The main object of this paper is the study of a sequence of finite dimensional algebras, depending on 2 parameters, which appear in connection with the Kauffman link invariant and with Drinfeld's and Jimbo'sq deformation of Lie algebras of typesB, C andD We determine for which parameters these algebras are semisimple Moreover, we classify all unitary representations of the infinite braid groupB∞ factoring through the inductive limit of these algebras This yields new examples of irreducible subfactors of finite depth, whose indices are squares ofq dimensions of irreducible representations of sympletic and orthogonal groups In the combinatorial description of these subfactors one naturally obtains truncated Weyl chambers (as for loop groups for a given level) and multiplicity coefficients of fusion rules for Wess-Zumino-Witten models

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Hans Wenzl

Papers

Braids, link polynomials and a new algebra

On the structure of Brauer’s centralizer algebras

Quantum groups and subfactors of type $B$, $C$, and $D$

Quantum invariants of 3-manifolds associated with classical simple lie algebras

Reconstructing monoidal categories