Last update: 13 October 2013
Our goal in Section 2 is to provide a tensor space action of the affine Birman-Murakami-Wenzl (BMW) algebra and its degenerate version by way of the group algebra of the affine braid group and the degenerate affine braid algebra respectively. The definition of the degenerate affine braid algebra makes the Schur-Weyl duality framework completely analogous in both the affine and degenerate affine cases.
In this section, we define and and show that both act on tensor space of the form In the degenerate affine case this action commutes with complex reductive Lie algebras in the affine case this action commutes with the Drinfeld-Jimbo quantum group As we will see in Section 2, the affine and degenerate affine BMW algebras arise when is orthogonal or symplectic and is the defining representation; similarly, the degenerate affine Hecke algebras arise when is of type or and is the defining representation. In the case when is the trivial representation and is the elements in become the Jucys-Murphy elements for the Brauer algebras used in [Naz1996]; in the case that these become the classical Jucys-Murphy elements in the group algebra of the symmetric group.
The action of the degenerate affine braid algebra and the action of the affine braid group on each provide Schur functors where in each case is the vector space of highest weight vectors of weight in These ubiquitous functors transfer representation theoretic information back and forth either between and or between and
Let be a commutative ring and let denote the symmetric group on For write for the transposition in that switches and The degenerate affine braid algebra is the algebra over generated by with relations and, for This presentation highlights the “Jucys-Murphy” elements for the degenerate BMW algebra as in [Naz1996]. However, the algebra also admits the following presentation, which highlights its natural action on tensor space (as we will see in Theorem 1.2).
[DRV1105.4207, Theorem 2.1] The degenerate affine braid algebra has another presentation by generators and relations for and and and and and
The conversion between the two presentations is given by the formulas and the formulas in (1.8). Set for Then pairwise commute,
Let be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form i.e., and for Let be a basis for and let be the dual basis with respect to The Casimir is where is the center of the enveloping algebra (see [Bou1990, I §3 Prop. 11]). Since the coproduct on is defined by for
Let be a finite-dimensional complex Lie algebra with a symmetric nondegenerate
be the universal enveloping algebra. Let
be the center of
be the Casimir in and
as in (1.15). Let and be
The degenerate affine braid algebra acts on via defined by so that by permuting tensor factors of
is acting in the factor of in and is acting on
is acting in the and factors of in
is acting on and the factor of in
is acting on and the first factors of
for This action of commutes with the on
Since the operators are in From the relation it also follows that
All of the relations in Theorem 1.1 except the last relations in (1.10) follow from consideration of the tensor factors being acted upon. The last relations in (1.10) are established by the computation for Recursively applying the coproduct formula from (1.15) connects the action of with the action of and as in the second formula in (1.13).
For a let as in (1.14). If is a generated by a highest weight vector of weight then is the half-sum of the positive roots (see [Bou1990, VIII §2 no. 3 Prop. 6 and VIII §6 no. 4 Cor. to Prop. 7]). By equation (1.15), if and are finite-dimensional irreducible of highest weights and respectively, then acts on the component of the decomposition by the constant Pictorially, By (1.17) and (1.12), the eigenvalues of are functions of the eigenvalues of the Casimir.
The affine braid group is the group given by generators and with relations The generators and can be identified with the diagrams For define The pictorial computation shows that the pairwise commute.
Let be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form, and let be the Drinfel’d-Jimbo quantum group corresponding to The quantum group is a ribbon Hopf algebra with invertible where (see [LRa1977, Corollary (2.15)]). For and the map is a isomorphism. The quasitriangularity of a ribbon Hopf algebra provides the braid relation (see, for example, [ORa0401317, (2.12)]),
Let be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form, let be the corresponding Drinfeld-Jimbo quantum group and let be the center of Let and be Then is a with action given by where with as in (1.24). The action commutes with the on
The relations (1.18) and (1.21) are consequences of the definition of the action of and The relations (1.19) and (1.20) follow from the following computations: and
Let be the ribbon element in For a define (see [Dri1990, Prop. 3.2]). If is a generated by a highest weight vector of weight then (see [LRa1977, Prop. 2.14] or [Dri1990, Prop. 5.1]). From (1.27) and the relation (1.26) it follows that if and are finite-dimensional irreducible of highest weights and respectively, then acts on the component of the decomposition By the definition of in (1.23), so that, by (1.26), the eigenvalues of are functions of the eigenvalues of the Casimir.
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.