Tridiagonal pairs of q -Racah type and the quantum enveloping algebra U q ( sl 2 ) Sarah Bockting-Conrad University of Wisconsin-Madison June 5, 2014
Introduction This talk is about tridiagonal pairs and tridiagonal systems . We will focus on a special class of tridiagonal systems known as the q -Racah class. We introduce some linear transformations which act on the on the underlying vector space in an attractive manner. We give two actions of the quantum group U q ( sl 2 ) on the underlying vector space. We describe these actions in detail, and show how they are related. Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Definition of a tridiagonal pair Let K denote a field. Let V denote a vector space over K of finite positive dimension. Definition By a tridiagonal pair (or TD pair) on V we mean an ordered pair of linear transformations A : V ! V and A ∗ : V ! V satisfying: 1. Each of A , A ∗ is diagonalizable. 2. There exists an ordering { V i } d i =0 of the eigenspaces of A such that A ∗ V i ✓ V i − 1 + V i + V i +1 (0 i d ) , where V − 1 = 0 and V d +1 = 0. i =0 of the eigenspaces of A ∗ such that i } δ 3. There exists an ordering { V ∗ AV ∗ i ✓ V ∗ i − 1 + V ∗ i + V ∗ (0 i δ ) , i +1 where V ∗ − 1 = 0 and V ∗ δ +1 = 0. 4. There does not exist a subspace W of V such that AW ✓ W , A ∗ W ✓ W , W 6 = 0, W 6 = V . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Example: Q -polynomial distance-regular graph Let Γ = Γ ( X , E ) denote a Q -polynomial distance-regular graph. Let A denote the adjacency matrix of Γ . Fix x 2 X . Let A ∗ = A ∗ ( x ) denote the dual adjacency matrix of Γ with respect to x . Let W denote an irreducible ( A , A ∗ )-module of C | X | . Then A , A ∗ form a TD pair on W . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Connections There are connections between TD pairs and I Q -polynomial distance-regular graphs I representation theory I orthogonal polynomials I partially ordered sets I statistical mechanical models I other areas of physics For further examples, see “Some algebra related to P - and Q -polynomial association schemes” by Ito, Tanabe, and Terwilliger or the survey “An algebraic approach to the Askey scheme of orthogonal polynomials” by Terwilliger. Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Standard ordering Definition Given a TD pair A , A ∗ , an ordering { V i } d i =0 of the eigenspaces of A is called standard whenever A ∗ V i ✓ V i − 1 + V i + V i +1 (0 i d ) , where V − 1 = 0 and V d +1 = 0. (A similar discussion applies to A ∗ .) Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Standard ordering Definition Given a TD pair A , A ∗ , an ordering { V i } d i =0 of the eigenspaces of A is called standard whenever A ∗ V i ✓ V i − 1 + V i + V i +1 (0 i d ) , where V − 1 = 0 and V d +1 = 0. (A similar discussion applies to A ∗ .) Lemma (Ito, Terwilliger) If { V i } d i =0 is a standard ordering of the eigenspaces of A, then { V d − i } d i =0 is standard and no other ordering is standard. Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Tridiagonal system Definition By a tridiagonal system (or TD system) on V , we mean a sequence Φ = ( A ; { V i } d i =0 ; A ∗ ; { V ∗ i } d i =0 ) that satisfies (1)–(3) below. 1. A , A ∗ is a tridiagonal pair on V . 2. { V i } d i =0 is a standard ordering of the eigenspaces of A . 3. { V ∗ i } d i =0 is a standard ordering of the eigenspaces of A ∗ . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Assumptions Until further notice, we fix a TD system Φ = ( A ; { V i } d i =0 ; A ∗ ; { V ∗ i } d i =0 ) . Let Φ ⇓ = ( A ; { V d − i } d i =0 ; A ∗ ; { V ∗ i } d i =0 ) . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Notation Throughout this talk, we will focus on Φ and its associated objects. Keep in mind that a similar discussion applies to Φ ⇓ and its associated objects. For any object f associated with Φ , we let f ⇓ denote the corresponding object associated with Φ ⇓ . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Some ratios For 0 i d , we let θ i (resp. θ ∗ i ) denote the eigenvalue of A (resp. A ∗ ) corresponding to the eigenspace V i (resp. V ∗ i ). Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Some ratios For 0 i d , we let θ i (resp. θ ∗ i ) denote the eigenvalue of A (resp. A ∗ ) corresponding to the eigenspace V i (resp. V ∗ i ). Lemma (Ito, Tanabe, Terwilliger) The ratios θ ∗ i − 2 � θ ∗ θ i − 2 � θ i +1 i +1 , θ ∗ i − 1 � θ ∗ θ i − 1 � θ i i are equal and independent of i for 2 i d � 1 . This gives two recurrence relations, whose solutions can be written in closed form. The most general case is known as the q -Racah case and is described below. Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
q -Racah case Definition We say that the TD system Φ has q -Racah type whenever there exist nonzero scalars q , a , b 2 K such that q 4 6 = 1 and θ i = aq d − 2 i + a − 1 q 2 i − d , i = bq d − 2 i + b − 1 q 2 i − d θ ∗ for 0 i d . Assumption Throughout this talk, we assume that Φ has q -Racah type. For simplicity, we also assume that K is algebraically closed. Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
First split decomposition of V Definition For 0 i d , define U i = ( V ∗ 0 + V ∗ 1 + · · · + V ∗ i ) \ ( V i + V i +1 + · · · + V d ) . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
First split decomposition of V Definition For 0 i d , define U i = ( V ∗ 0 + V ∗ 1 + · · · + V ∗ i ) \ ( V i + V i +1 + · · · + V d ) . Theorem (Ito, Tanabe, Terwilliger) V = U 0 + U 1 + · · · + U d (direct sum) We refer to { U i } d i =0 as the first split decomposition of V . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Second split decomposition of V Definition For 0 i d , define U ⇓ i = ( V ∗ 0 + V ∗ 1 + · · · + V ∗ i ) \ ( V 0 + V 1 + · · · + V d − i ) . Theorem (Ito, Tanabe, Terwilliger) V = U ⇓ 0 + U ⇓ 1 + · · · + U ⇓ (direct sum) d We refer to { U ⇓ i } d i =0 as the second split decomposition of V . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
The maps K , B Definition Let K : V ! V denote the linear transformation such that for 0 i d , U i is an eigenspace of K with eigenvalue q d − 2 i . That is, ( K � q d − 2 i I ) U i = 0 for 0 i d . Definition Let B : V ! V denote the linear transformation such that for 0 i d , U ⇓ i is an eigenspace of B with eigenvalue q d − 2 i . That is, ( B � q d − 2 i I ) U ⇓ i = 0 for 0 i d . Note: B = K ⇓ . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Split decompositions of V Lemma (Ito, Tanabe, Terwilliger) Let 0 i d. A , A ∗ act on the first split decomposition in the following way: ( A ∗ � θ ∗ ( A � θ i I ) U i ✓ U i +1 , i I ) U i ✓ U i − 1 . A , A ∗ act on the second split decomposition in the following way: ( A ∗ � θ ∗ ( A � θ d − i I ) U ⇓ i ✓ U ⇓ i I ) U ⇓ i ✓ U ⇓ i +1 , i − 1 . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
The raising maps R , R + Definition Let R = A � aK � a − 1 K − 1 . By construction, I R acts on U i as A � θ i I for 0 i d , I RU i ✓ U i +1 (0 i < d ) , RU d = 0. Definition Let R ⇓ = A � a − 1 B � aB − 1 . By construction, I R ⇓ acts on U ⇓ i as A � θ d − i I for 0 i d , I R ⇓ U ⇓ i ✓ U ⇓ R ⇓ U ⇓ (0 i < d ) , d = 0. i +1 Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
Relating R , R + , K , B Lemma (B. 2014) Both KR = q − 2 RK , BR ⇓ = q − 2 R ⇓ B . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
The linear transformation ψ In 2005, Nomura showed that d / 2 d − i X X V = τ ij ( A ) K i , i =0 j = i where K i = U i \ U ⇓ i and τ ij ( x ) = ( x � θ i )( x � θ i +1 ) · · · ( x � θ j − 1 ) . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
The linear transformation ψ d / 2 d − i X X V = τ ij ( A ) K i , i =0 j = i Definition Let ψ : V ! V denote the linear transformation such that ψτ ij ( A ) v = γ ij τ i , j − 1 ( A ) v for 0 i d / 2, i j d � i , and v 2 K i . Here [ j � i ] q [ d � i � j + 1] q γ ij = [ d ] q and [ n ] q = q n � q − n q � q − 1 . Sarah Bockting-Conrad Tridiagonal pairs of q -Racah type and the quantum enveloping algebra Uq (
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