Last update: 7 March 2014

The Iwahori-Hecke algebras of Type A and the Birman-Wenzl-Murakami algebras arise naturally in the following setting: Let $\U0001d518$ be a quantum group corresponding to a finite dimensional complex simple Lie algebra of Type A, B, C, or D, and let $V$ be the irreducible representation of $\U0001d518$ corresponding to the fundamental weight ${\omega}_{1}\text{.}$ Then the centralizer algebra ${\mathcal{Z}}_{m}={\text{End}}_{\U0001d518}\left({V}^{\otimes m}\right)$ is isomorphic to a quotient of either the Iwahori-Hecke algebra or the Birman-Wenzl algebra.

The purpose of this paper is to give a unified approach for determining explicit realizations of the irreducible representations of the Iwahori-Hecke algebras of Type A and the Birman-Wenzl-Murakami algebras. Indeed, the formulas for the irreducible representations which we find are equivalent to those in [Hoe1974] and [Wen1988] in the case of the Iwahori-Hecke algebras of type A and to those in [Mur1990] for the case of the Birman-Wenzl-Murakami algebras. However, we have found that in all three of these previous works the appropriate formulas are stated without derivation and then proved to be correct. In this paper we show that there is indeed a consistent method by which one may actually derive the appropriate formulas.

Our method is motivated strongly by the machinery which has developed in the context of operator algebras, quantum groups, and link invariants, in particular the work of Reshetikhin [Res1987], Drinfel'd [Dri1990], Wenzl [Wen1990], and Turaev [Tur1988]. See also the papers [RTu1991, RTu1990, TWe1217386, Wen1993, and BWe1989]. Although we have applied our methods in the particular case of the quantum groups corresponding to finite dimensional simple Lie algebras of types A, B, C, it is clear that main aspects of our approach hold in the setting of quasitriangular Hopf algebras and ribbon Hopf algebras. The following list describes the central features in our approach.

(1) From operator algebras: We have used the path model approach for towers of algebras in [GHJ1989] in order to work with infinite families of centralizer algebras all at once. In some sense the path algebra mechanism reduces all of the "difficult" parts of the derivation to simple computations with matrix units in direct sums of ordinary $n\times n$ matrix algebras.

(2) From quantum groups: The Drinfel'd-Jimbo quantum groups carry the structure of quasitriangular Hopf algebras and ribbon Hopf algebras [Dri1990]. We have been able to use this structure to get very specific information about certain elements in the centralizer algebra. The quasitriangular structure guarantees that the product ${\mathcal{R}}_{21}{\mathcal{R}}_{12},$ where $\mathcal{R}$ is the $\mathcal{R}\text{-matrix,}$ is always an element of the centralizer algebra and the ribbon structure allows us to determine the eigenvalues of this element. These eigenvalues turn out to be determined by the Casimir element from the corresponding Lie algebra. This idea is the central idea in [Res1987].

(3) Combining tools from 3-manifold invariants and operator algebras: We show that the Markov traces used to derive link invariants and 3-manifold invariants are equivalent to certain traces on towers of algebras that arise from Wenzl's approach to the Jones basic construction. This was observed in [Wen1990] for the case of quantum groups of type B using the explicit form of the $\u0158$ matrix. In our approach we have obtained this result for any ribbon Hopf algebra. This idea allows one to give an easy derivation of the framing anomalies for the Reshetikhin-Turaev 3-manifold invariants.

In the first three sections of this paper we develop these tools in the context of centralizer algebras. Although the main objects have all appeared in previous work ([Res1987, Tur1988, Dri1990, Wen1990]), we have felt it necessary to give a consistent presentation in the context of centralizer algebras since it is not necessarily clear from the previous work how these techniques apply to our situation.

Our paper is organized as follows:

In Section 1 we review the path algebra setup. In the second half of Section 1 we show that if $\U0001d518$ is a Hopf algebra such that all finite dimensional representations of $\U0001d518$ are completely reducible and if $V$ is a $\U0001d518\text{-module}$ then the centralizer algebras ${\mathcal{Z}}_{m}={\text{End}}_{\U0001d518}\left({V}^{\otimes m}\right)$ can be identified with path algebras in a natural way.

In Section 2 we begin by reviewing the definitions of quasitriangular Hopf algebras, ribbon Hopf algebras, and the Drinfel'd-Jimbo quantum groups. Then, letting $\U0001d518$ be a quasitriangular Hopf algebra and letting $V$ be a $\U0001d518\text{-module,}$ we show how to determine explicitly the image of the element ${\mathcal{R}}_{21}{\mathcal{R}}_{12}$ both as an element of the centralizer algebra ${\mathcal{Z}}_{m}={\text{End}}_{\U0001d518}\left({V}^{\otimes m}\right)$ and as an element of the corresponding path algebra.

In Section 3 we let $\U0001d518$ be a ribbon Hopf algebra and let $V$ be a $\U0001d518\text{-module.}$ Then there is a natural projection $\u011b\in {\text{End}}_{\U0001d518}(V\otimes {V}^{*})$ onto the invariants in the $\U0001d518\text{-module}$ $V\otimes {V}^{*}\text{.}$ This projection gives rise to a natural trace on the centralizer algebras ${\mathcal{Z}}_{m},$ and it turns out that this trace is always a Markov trace with respect to the corresponding $\mathcal{R}\text{-matrix.}$ We are able to determine explicit formulas for the image of the element $\u011b$ in the path algebras corresponding to the centralizer algebras ${\mathcal{Z}}_{m}\text{.}$

In Section 4 we apply the results of the first three sections to compute the irreducible representations, in terms of path algebras, of the centralizer algebras corresponding to the quantum groups ${\U0001d518}_{h}\left(\U0001d530\U0001d529(r+1)\right)$ and the fundamental representation.

In Section 5 we apply the results of the first two sections to compute the irreducible representations, in terms of path algebras, of the centralizer algebras corresponding to the quantum groups corresponding to complex simple Lie algebras of Types B, C, D, and the fundamental representation. This derivation is only slightly more complex than that for the Type A case given in Section 4.

We finish in Section 6 by deriving, explicitly, irreducible representations of the Iwahori-Hecke algebras, the Birman-Wenzl-Murakami algebras, and the Brauer algebras.

Some further remarks on the results in this paper:

(1) All of the representations obtained in this paper are, in some sense, analogues of Young's orthogonal representations for the symmetric group. This is due to the way that we inductively identify the centralizer algebras ${\mathcal{Z}}_{m}$ with path algebras.

(2) Hoefsmit determined explicit irreducible representations of the Iwahori-Hecke algebras of Type A in [Hoe1974]. One of the consequences of our approach is that the mysterious axial distances which have appeared in the work of Hoefsmit are completely explained in terms of the values of the Casimir element of the complex simple Lie algebras of type A acting on irreducible representations. Similarly, some of the constants appearing in the formulas for the irreducible representations of the Birman-Wenzl-Murakami algebras are obtained from the values of the Casimir elements of the complex simple Lie algebras of type B or C acting on irreducible representations. In fact, the only other values that are needed in order to give closed form formulas for the irreducible representations are the "quantum dimensions" of the irreducible representations of the corresponding Lie algebra. These are determined by the Weyl character formula.

(3) Although our formulas for the irreducible representations of the Birman-Wenzl-Murakami algebra are equivalent to those in [Mur1990] we have found ours to be more tractable, in particular, it is a trivial matter to specialize appropriately to give formulas, to our knowledge new ones, for the irreducible representations of the Brauer algebras [Bra1937, Wen1988-2].

(4) We have found that it is quite easy to derive the formulas for the basic construction element (which was obtained by various authors [RWe1992, Theorem 1.4; GHJ1989, (2.6.5.4); Sun1987]) by simple path algebra (matrix algebra) computations and thus we give an alternate and elementary proof of some of the results in [Wen1988-2, Section 1]. This result appears in our Theorem (3.12).

(5) In Sections 4 and 5 we give formulas for matrix units in the centralizer algebras corresponding to quantum groups of types A, B, C, and D. Similar formulas have been given in [RWe1992]. The formulas we give here, in the cases of types B, C, and D, are new formulas for the same matrix units that were given in [RWe1992].

This is a typed version of the paper *A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras* by Robert ${\text{Leduc}}^{*}$ and Arun ${\text{Ram}}^{\u2020}\text{.}$

The paper was received June 24, 1994; accepted September 12, 1994.

${}^{*}$Supported in part by National Science Foundation Grant DMS-9300523 to the University of Wisconsin.

${}^{\u2020}$Supported in part by National Science Foundation Postdoctoral Fellowship DMS-9107863.