## Affine and degenerate affine BMW algebras: Actions on tensor space

Last update: 13 October 2013

## Introduction

This paper is a continuation of our study of the affine Birman-Murakami-Wenzl (BMW) algebras ${W}_{k}$ and their degenerate versions ${𝒲}_{k}\text{.}$ In [DRV1105.4207] we defined the algebras ${W}_{k}$ and ${𝒲}_{k}$ and determined their centers. Each of these algebras contains a commuting family of “Jucys-Murphy elements”; following Nazarov and Beliakova-Blanchet [Naz1996, BBl1866492] we derived generating function formulas for specific central elements ${z}_{k}^{\left(\ell \right)}\in {𝒲}_{k}$ and ${Z}_{k}^{\left(\ell \right)}\in {W}_{k}$ in terms of the Jucys-Murphy elements.

In this paper we show that the algebras ${W}_{k}$ and ${𝒲}_{k}$ have a natural action on tensor space which provides a Schur-Weyl duality with the quantum group and enveloping algebras of classical type Lie algebras. In particular, the algebras ${W}_{k}$ and ${𝒲}_{k}$ arise in orthogonal and symplectic type, though we treat all the classical type cases uniformly. In complete analogy with the fact that affine BMW algebra is a quotient of the group algebra $C{B}_{k}$ of the affine braid group of type A, the degenerate affine BMW algebra is a quotient of the degenerate affine braid algebra ${ℬ}_{k}$ which we first defined in [DRV1105.4207]. This analogy extends to the Schur-Weyl duality. In Theorem 1.2, we show that there is a natural action of ${ℬ}_{k}$ on tensor space which commutes with an arbitrary finite-dimensional complex reductive Lie algebra $𝔤\text{.}$ In Theorem 3.3 we show that, when $𝔤$ is of classical type and the tensor space is constructed from the $n\text{-dimensional}$ defining representation, the action of ${ℬ}_{k}$ becomes an action of familiar algebras : the degenerate affine BMW algebra arises when $𝔤={𝔰𝔬}_{n}$ or ${𝔰𝔭}_{n},$ and the degenerate affine Hecke algebra arises when $𝔤={𝔤𝔩}_{n}$ or ${𝔰𝔩}_{n}\text{.}$

The affine and degenerate affine BMW algebras depend on the choice of an infinite number of parameters. This is analogous to the way that the Iwahori-Hecke algebra depends on one parameter, often called $q\text{.}$ Unfortunately, the infinite collection of parameters for the BMW algebras is not free; significant work has been done on when a collection is admissible [AMR0506467, WYu0611518, WYu0911.5284, Goo0905.4258, Goo0905.4253, GHa0411155, GMo0612064, GMo0612065, Yu0810.0069]. In this work, we take a different point of view and produce universal parameters for the affine and degenerate affine BMW algebras. These universal parameters are symmetric functions which satisfy the admissibility conditions. In future work, we hope to show via representation theory that every choice of admissible complex parameters is a specialization of our universal parameters.

To compute the symmetric functions which arise as universal parameters, we use the Schur-Weyl duality to naturally identify them as elements of the center of the corresponding symplectic or orthogonal enveloping algebra or quantum group (which, by the Harish-Chandra isomorphism, is isomorphic to a ring of symmetric functions). Specifically, in Theorem 3.3 and Theorem 3.5 we execute computations which push the recursive formulas of Nazarov [Naz1996] and Beliakova-Blanchet [BBl1866492] to the other side of the Schur-Weyl duality. This produces explicit formulas for the Harish-Chandra images of the corresponding central elements ${z}_{V}^{\left(\ell \right)}$ and ${Z}_{V}^{\left(\ell \right)}$ of the orthogonal and symplectic enveloping algebras and quantum groups. These computations are related to the calculations in, for example, [Naz1996], [Mol2355506, Ch. 7] and [MRo1112.0620v1].

In Section 4 we show that the central elements ${z}_{V}^{\left(\ell \right)}$ and ${Z}_{V}^{\left(\ell \right)}$ are the natural higher Casimir elements for orthogonal and symplectic enveloping algebras and quantum groups. In fact, we are able to show that the universal parameters ${z}_{V}^{\left(\ell \right)}$ of the degenerate affine BMW algebras coincide exactly with the higher Casimirs for orthogonal and symplectic Lie algebras given by Perelomov-Popov [PPo0214703, PPo0205621]. Expositions of the Perelomov-Popov results are also found in [Zeh1973, §127] and [Mol2355506, §7.1]. Another computation shows that the universal parameters ${Z}_{V}^{\left(\ell \right)}$ of the affine BMW algebras coincide with the central elements in quantum groups defined by Reshetikhin-Takhtajan-Faddeev central elements defined in [RTF1015339, Theorem 14]. To execute our computation we have relied on a remarkable identity of Baumann [Bau1620662, Theorem 1].

Acknowledgements: Significant work on this paper was done while the authors were in residence at the Mathematical Sciences Research Institute (MSRI) in 2008. We thank MSRI for hospitality, support, and a wonderful working environment. We thank F. Goodman and A. Molev for their helpful comments and references. This research has been partially supported by the National Science Foundation DMS-0353038 and the Australian Research Council DP0986774 and DP120101942.

## Notes and references

This is a typed exert of the paper Affine and degenerate affine BMW algebras: Actions on tensor space by Zajj Daugherty, Arun Ram and Rahbar Virk.