Tridiagonal pairs of q-racah type and the quantum enveloping algebra U q (sl 2 )

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1 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

2 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.

3 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 apple i apple d), where V 1 =0andV d+1 =0. 3. There exists an ordering {V i } i=0 of the eigenspaces of A such that AV i V i 1 + Vi where V 1 =0andV +1 =0. + V i+1 (0 apple i apple ), 4. There does not exist a subspace W of V such that AW W, A W W, W 6= 0,W 6= V.

4 Example: Q-polynomial distance-regular graph Let = (X, E) denote a Q-polynomial distance-regular graph. Let A denote the adjacency matrix of. Fix x 2 X.LetA = A (x) denote the dual adjacency matrix of respect to x. with Let W denote an irreducible (A, A )-module of C X. Then A, A form a TD pair on W.

5 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.

6 Standard ordering Definition Given a TD pair A, A,anordering{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 apple i apple d), where V 1 =0andV d+1 = 0. (A similar discussion applies to A.)

7 Standard ordering Definition Given a TD pair A, A,anordering{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 apple i apple d), where V 1 =0andV 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.

8 Tridiagonal system Definition By a tridiagonal system (or TD system) on V,wemeanasequence that satisfies (1) (3) below. 1. A, A is a tridiagonal pair on V. =(A; {V i } d i=0; A ; {V i } d i=0) 2. {V i } d i=0 is a standard ordering of the eigenspaces of A. 3. {Vi } d i=0 is a standard ordering of the eigenspaces of A.

9 Assumptions Until further notice, we fix a TD system =(A; {V i } d i=0; A ; {V i } d i=0). Let + =(A; {V d i } d i=0; A ; {V i } d i=0).

10 Notation Throughout this talk, we will focus on and its associated objects. Keep in mind that a similar discussion applies to + and its associated objects. For any object f associated with object associated with +., we let f + denote the corresponding

11 Some ratios For 0 apple i apple d, welet i (resp. i ) denote the eigenvalue of A (resp. A ) corresponding to the eigenspace V i (resp. V i ).

12 Some ratios For 0 apple i apple d, welet i (resp. i ) denote the eigenvalue of A (resp. A ) corresponding to the eigenspace V i (resp. V i ). Lemma (Ito, Tanabe, Terwilliger) The ratios i 2 i+1 i 1 i, i 2 i+1 i 1 i are equal and independent of i for 2 apple i apple 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.

13 q-racah case Definition We say that the TD system has q-racah type whenever there exist nonzero scalars q, a, b 2 K such that q 4 6=1and i = aq d 2i + a 1 q 2i d, i = bq d 2i + b 1 q 2i d for 0 apple i apple d. Assumption Throughout this talk, we assume that has q-racah type. For simplicity, we also assume that K is algebraically closed.

14 First split decomposition of V Definition For 0 apple i apple d, define U i =(V 0 + V V i ) \ (V i + V i V d ).

15 First split decomposition of V Definition For 0 apple i apple d, define U i =(V 0 + V V i ) \ (V i + V i V d ). Theorem (Ito, Tanabe, Terwilliger) V = U 0 + U U d (direct sum) We refer to {U i } d i=0 as the first split decomposition of V.

16 Second split decomposition of V Definition For 0 apple i apple d, define U + i =(V 0 + V V i ) \ (V 0 + V V d i ). Theorem (Ito, Tanabe, Terwilliger) V = U U U+ d (direct sum) We refer to {U + i } d i=0 as the second split decomposition of V.

17 The maps K, B Definition Let K : V! V denote the linear transformation such that for 0 apple i apple d, U i is an eigenspace of K with eigenvalue q d 2i.Thatis, for 0 apple i apple d. Definition (K q d 2i I )U i =0 Let B : V! V denote the linear transformation such that for 0 apple i apple d, U + i is an eigenspace of B with eigenvalue q d 2i.Thatis, for 0 apple i apple d. Note: B = K +. (B q d 2i I )U + i =0

18 Split decompositions of V Lemma (Ito, Tanabe, Terwilliger) Let 0 apple i apple d. A, A act on the first split decomposition in the following way: (A i I )U i U i+1, (A i I )U i U i 1. A, A act on the second split decomposition in the following way: (A d i I )U + i U + i+1, (A i I )U + i U + i 1.

19 The raising maps R, R + Definition Let R = A ak a 1 K 1. By construction, I R acts on U i as A i I for 0 apple i apple d, I RU i U i+1 (0 apple i < d), RU d = 0. Definition Let R + = A a 1 B ab 1. By construction, I R + acts on U + i as A d i I for 0 apple i apple d, I R + U + i U + i+1 (0 apple i < d), R + U + d = 0.

20 Relating R, R +, K, B Lemma (B. 2014) Both KR = q 2 RK, BR + = q 2 R + B.

21 The linear transformation In 2005, Nomura showed that d/2 X Xd i V = ij (A)K i, where i=0 j=i and K i = U i \ U + i ij (x) =(x i )(x i+1 ) (x j 1 ).

22 The linear transformation d/2 X Xd i V = ij (A)K i, i=0 j=i Definition Let : V! V denote the linear transformation such that ij (A)v = ij i,j 1 (A)v for 0 apple i apple d/2, i apple j apple d i, andv 2 K i.here ij = [j i] q [d i j + 1] q [d] q and q n [n] q = qn q q 1.

23 The linear transformation Lemma (B. 2012) For 0 apple i apple d, both U i U i 1, U + i U + i 1. Moreover, d+1 =0. In light of the above result, we refer to as the double lowering operator.

24 The linear transformation Lemma (B. 2014) Both K = q 2 K, B = q 2 B.

25 Relating, R, R +, K, B Lemma (B. 2014) Both R R = q q 1 K K 1, R + R + = q q 1 B B 1.

26 Relation summary Our relations K = q 2 K, KR = q 2 RK, R R = q q 1 K K 1, + -analogues B = q 2 B, BR + = q 2 R + B, R + R + = q q 1 B B 1,

27 The quantum universal enveloping algebra U q (sl 2 ) Definition The quantum universal enveloping algebra U q (sl 2 )isdefinedtobe the unital associative K-algebra with generators and relations e, f, k, k 1 kk 1 =1=k 1 k, ke = q 2 ek, kf = q 2 fk, ef fe = k k 1 q q 1. The generators e, f, k, k 1 are known as the Chevalley generators for U q (sl 2 ).

28 Two U q (sl 2 )-module structures on V Theorem (B. 2014) There exists a U q (sl 2 )-module structure on V for which the Chevalley generators act as follows: element of U q (sl 2 ) e f k k 1 action on V (q q 1 ) 1 (q q 1 ) 1 R K K 1

29 Two U q (sl 2 )-module structures on V Theorem (B. 2014) There exists a U q (sl 2 )-module structure on V for which the Chevalley generators act as follows: element of U q (sl 2 ) e f k k 1 action on V (q q 1 ) 1 (q q 1 ) 1 R K K 1 Theorem (B. 2014) There exists a U q (sl 2 )-module structure on V for which the Chevalley generators act as follows: element of U q (sl 2 ) e f k k 1 action on V (q q 1 ) 1 (q q 1 ) 1 R + B B 1

30 Irreducible U q (sl 2 )-submodules of V Let denote M the subalgebra of End(V ) generated by A. For 0 apple i apple d/2 andv 2 K i = U i \ U + i,letmv denote the M-submodule of V generated by v.

31 Irreducible U q (sl 2 )-submodules of V Let denote M the subalgebra of End(V ) generated by A. For 0 apple i apple d/2 andv 2 K i = U i \ U + i,letmv denote the M-submodule of V generated by v. Lemma (B. 2014) Let 0 apple i apple d/2 and v 2 K i. For either of the U q (sl 2 )-actions on V from the previous slide, Mv is an irreducible U q (sl 2 )-submodule of V with dimension d 2i +1.

32 Irreducible U q (sl 2 )-submodules of V Let denote M the subalgebra of End(V ) generated by A. For 0 apple i apple d/2 andv 2 K i = U i \ U + i,letmv denote the M-submodule of V generated by v. Lemma (B. 2014) Let 0 apple i apple d/2 and v 2 K i. For either of the U q (sl 2 )-actions on V from the previous slide, Mv is an irreducible U q (sl 2 )-submodule of V with dimension d 2i +1. Lemma (B. 2014) For either of the U q (sl 2 )-actions on V from the previous slide, V can be written as a direct sum of its irreducible U q (sl 2 )-submodules.

33 The Casimir element of U q (sl 2 ) Definition Define the normalized Casimir element of U q (sl 2 )by = (q q 1 ) 2 ef + q 1 k + qk 1, =(q q 1 ) 2 fe + qk + q 1 k 1. I is in the center of U q (sl 2 ). I generates the center of U q (sl 2 )wheneverq is not a root of unity. I acts as a scalar multiple of the identity on irreducible U q (sl 2 )-modules.

34 The action of the Casimir element on V Lemma (B. 2014) With respect to the first U q (sl 2 )-module structure on V, the action of the on V is equal to both R + q 1 K + qk 1, R + qk + q 1 K 1.

35 The action of the Casimir element on V Lemma (B. 2014) With respect to the first U q (sl 2 )-module structure on V, the action of the on V is equal to both R + q 1 K + qk 1, R + qk + q 1 K 1. Lemma (B. 2014) With respect to the second U q (sl 2 )-module structure on V, the action of the on V is equal to both R + + q 1 B + qb 1, R + + qb + q 1 B 1.

36 The action of the Casimir element on V Lemma (B. 2014) Let 0 apple i apple d/2 and v 2 K i. With respect to either of the U q (sl 2 )-module structures on V, acts on Mv as q d 2i+1 + q d+2i 1 times the identity.

37 The action of the Casimir element on V Corollary (B. 2014) Let 0 apple i apple d/2. With respect to either of the U q (sl 2 )-module structures on V, acts on MK i as q d 2i+1 + q d+2i 1 times the identity.

38 The action of the Casimir element on V Corollary (B. 2014) Let 0 apple i apple d/2. With respect to either of the U q (sl 2 )-module structures on V, acts on MK i as q d 2i+1 + q d+2i 1 times the identity. Lemma (B. 2012) The following sum is direct. V = rx MK i, i=0 where r = bd/2c.

39 Comparing the actions of on V Lemma (B. 2014) The following coincide: (i) the action of on V for the first U q (sl 2 )-module structure on V, (ii) the action of on V for the second U q (sl 2 )-module structure on V.

40 Comparing the actions of on V Lemma (B. 2014) The following coincide: (i) the action of on V for the first U q (sl 2 )-module structure on V, (ii) the action of on V for the second U q (sl 2 )-module structure on V. Corollary (B. 2014) The following four expressions are equal: R + q 1 K + qk 1, R + qk + q 1 K 1, R + + q 1 B + qb 1, R + + qb + q 1 B 1.

41 Relating K, B, andtheirinverses Theorem (B. 2014) Each of the following holds: KB 1 = I a 1 q I aq, BK 1 = I aq I a 1 q, K 1 B = I a 1 q 1 I aq 1, B 1 K = I aq 1 I a 1 q 1. Note that since is nilpotent, each of the above expressions is equal to a polynomial in.

42 How R, K ±1 and R +, B ±1 are related Lemma (B. 2014) The following equations hold: dx B = a 2 K +(1 a 2 )K a i q i i, i=0 dx B 1 = a 2 K 1 +(1 a 2 )K 1 a i q i i, i=0 dx R + = R +(a a 1 ) (a i q i K a i q i K 1 ) i. i=0

43 How R, K ±1 and R +, B ±1 are related Lemma (B. 2014) The following equations hold: dx K = a 2 B +(1 a 2 )B a i q i i, i=0 dx K 1 = a 2 B 1 +(1 a 2 )B 1 a i q i i, i=0 dx R = R + +(a a 1 ) (a i q i B 1 a i q i B) i. i=0

44 Four equations for Theorem (B. 2014) Each of the following expressions is equal to : I BK 1 q(ai a 1 BK 1 ), I KB 1 q(a 1 I akb 1 ), q(i K 1 B) ai a 1 K 1 B, q(i B 1 K) a 1 I ab 1 K.

45 Relating K, B, andtheirinverses Theorem (B. 2014) We have ak 2 + aq 1 a 1 q q q 1 KB + a 1 q 1 aq q q 1 BK + a 1 B 2 =0. Theorem (B. 2014) We have ab 2 + aq 1 a 1 q q q 1 K 1 B 1 + a 1 q 1 aq q q 1 B 1 K 1 + a 1 K 2 =0.

46 The subalgebra U _ q of U q (sl 2 ) There exists a second presentation of U q (sl 2 )knownastheequitable presentation. The generators are x, y ±1, z and are subject to the relations yy 1 = y 1 y = 1, qxy q 1 yx q q 1 =1, qyz q 1 zy q q 1 =1, qzx q 1 xz q q 1 =1. Let U _ q denote the subalgebra of U q (sl 2 ) generated by x, y 1, z. The above relations involving K ±1, B ±1 helped lead to the discovery of two new presentations for U _ q.

47 The subalgebra U _ q of U q (sl 2 ) Theorem (B.,Terwilliger 2014) Let U denote the K-algebra with generators K, B, A and relations qka q 1 AK q q 1 = ak 2 + a 1 I, qba q 1 AB q q 1 = a 1 B 2 + ai, ak 2 + aq 1 a 1 q q q 1 KB + a 1 q 1 aq q q 1 BK + a 1 B 2 =0. Then U is isomorphic to U q _. The second presentation of U q _ is obtained by inverting q, a, K, B.

48 Summary Given a TD system on V of q-racah type, we defined several linear transformations which act on V in an attractive way. We used these linear transformations to obtain two U q (sl 2 )-module structures for V and discussed how these actions are related. Using information about U q (sl 2 ), we derived several new equations relating our linear transformations. These equations helped lead to the discovery of new information about U q (sl 2 ).

49 The End Thank you for your attention!

Bidiagonal pairs, Tridiagonal pairs, Lie algebras, and Quantum Groups

Bidiagonal pairs, Tridiagonal pairs, Lie algebras, and Quantum Groups Darren Funk-Neubauer Department of Mathematics and Physics Colorado State University - Pueblo Pueblo, Colorado, USA darren.funkneubauer@colostate-pueblo.edu