Universal K-matrices via quantum symmetric pairs
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1 Universal K-matrices via quantum symmetric pairs Martina Balagović (joint work with Stefan Kolb) School of Mathematics and Statistics Newcastle University Leicester, September 215
2 1. Introduction - Universal R-matrices and quantum groups QYBE Particle on a line Two particles Scattering: M M N c M,N : M N N M QYBE: (c N,L 1) (1 c M,L ) (c M,N 1) = = (1 c M,N ) (c M,L 1) (1 c N,L )
3 1. Introduction - Universal R-matrices and quantum groups Braided tensor categories Braided tensor category M in a tensor category commutativity constraint c M,N : M N N M QYBE = braid group action
4 1. Introduction - Universal R-matrices and quantum groups Braided tensor categories Braided tensor category M in a tensor category commutativity constraint c M,N : M N N M QYBE = braid group action hexagon axiom: c M N,L = (c M,L 1) (1 c N,L )
5 1. Introduction - Universal R-matrices and quantum groups Quasitriangular Hopf algebras Hopf algebra H, (some nice) category of representations quasitraingular = exists R H H Ř = R flip c M,N = Ř M N Ř (a) = (a)ř Consistency condition: ( 1)(R) = R 13 R 23 (1 )(R) = R 13 R 12 QYBE R 12 R 13 R 23 = R 23 R 13 R 12
6 1. Introduction - Universal R-matrices and quantum groups Quantum enveloping algebra U q g, category O int 1 Define the bar involution on U q g: E i E i, F i F i, K i K 1 i, q q 1 2 Find quasi R-matrix R U q n U q n + such that R (a) = (a)r 3 Set R = R q H H Ř = R q H H flip 4 Prove ( 1)(R) =... (1 )(R) =... 5 QYBE
7 1. Introduction - Universal R-matrices and quantum groups Reflection equation particle on a line + a wall: t M : M M? Reflection Equation: c N,M (t N 1) c M,N (t M 1) = (t M 1) c N,M (t N 1) c M,N
8 1. Introduction - Universal R-matrices and quantum groups Braided tensor categories with a cylinder twist [T. tom Dieck, R. Häring-Oldenburg, Quantum groups and cylinder braiding, 1998.] cylinder twist t N : N N? RE = action of the braid group of type B c N,M (t N 1) c M,N (t M 1) = (t M 1) c N,M (t N 1) c M,N Consistency condition: t M N = (t M 1) c N,M (t N 1) c M,N
9 1. Introduction - Universal R-matrices and quantum groups Cylinder braided coideal subalgebras of quasi triangular Hopf algebras H quasitriangular Hopf algebra, B subalgebra (B) B H Cylinder braided = exists K H such that t M = K M Kb = bk (K) = (K 1) Ř (K 1) Ř RE (K 1) Ř (K 1) Ř = Ř (K 1) Ř (K 1)
10 1. Introduction - Universal R-matrices and quantum groups Strategy U q g 1 bar involution 2 quasi R-matrix R 3 universal R-matrix 4 (1 )(R) 5 prove R sats QYBE
11 1. Introduction - Universal R-matrices and quantum groups Strategy U q g 1 bar involution 2 quasi R-matrix R 3 universal R-matrix 4 (1 )(R) 5 prove R sats QYBE B c,s U q g quantum symmetric pair 1 bar involution 2 quasi K-matrix 3 universal K-matrix K 4 (K) 5 prove K sats RE
12 2. Quantum symmetric pair coideal subalgebras U qg Dynkin diagram, vertices I, Kac-Moody type Quantized enveloping algebra U q g field: C(q) generators: E i, F i, K ±1 i, i I relations: Serre(E i, E j ) =... Hopf algebra: (E i ) = E i K i,...
13 2. Quantum symmetric pair coideal subalgebras U qg Dynkin diagram, vertices I, Kac-Moody type Quantized enveloping algebra U q g field: C(q) generators: E i, F i, K ±1 i, i I relations: Serre(E i, E j ) =... Hopf algebra: (E i ) = E i K i,... Involutions of U q g τ : I I involution of Dynkin diagram I of finite type + conditions [S. Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, 1962.] [V.G. Kac, S.P. Wang, On automorphisms of Kac-Moody algebras and groups, 1992.]
14 2. Quantum symmetric pair coideal subalgebras Satake diagrams
15 2. Quantum symmetric pair coideal subalgebras Involutions q = 1: involution θ T w ω τ symmetric pair (k, g), k = g θ
16 2. Quantum symmetric pair coideal subalgebras Involutions q = 1: involution θ T w ω τ symmetric pair (k, g), k = g θ q 1: quantum involution θ q T w ω τ quantum symmetric pair B c,s U q g
17 2. Quantum symmetric pair coideal subalgebras Involutions q = 1: involution θ T w ω τ symmetric pair (k, g), k = g θ q 1: quantum involution θ q T w ω τ quantum symmetric pair B c,s U q g generators: 1 U q(h) θ 2 U q(g ) 3 B i = F i + c i θ q(f i K i )K 1 i + s i K i parameters c i, s i coideal: (B c,s ) B c,s U q g Example: 1 K 1 K 1 3, K 2 2 E 2, F 2, K ±1 2 3 B 1 = F 1 c 1 (E 2 E 3 q 1 E 3 E 2 ) B 3 = F 3 c 3 (E 2 E 1 q 1 E 1 E 2 )
18 2. Quantum symmetric pair coideal subalgebras Involutions q = 1: involution θ T w ω τ symmetric pair (k, g), k = g θ q 1: quantum involution θ q T w ω τ quantum symmetric pair B c,s U q g generators: 1 U q(h) θ 2 U q(g ) 3 B i = F i + c i θ q(f i K i )K 1 i + s i K i parameters c i, s i coideal: (B c,s ) B c,s U q g [G. Letzter, Symmetric pairs for quantized enveloping algebras, 1999.] [G. Letzter, Coideal subalgebras and quantum symmetric pairs, 22.] [G. Letzter, Quantum symmetric pairs and their zonal spherical functions, 23.] [S. Kolb, Quantum symmetric Kac-Moody pairs, 212.]
19 2. Quantum symmetric pair coideal subalgebras The presentation Theorem (Letzter; Kolb; B-Kolb) B c,s is generated over (U q h θ ) (U q g ) with generators B i, relations: K β B i K 1 β = q (β,α i ) B i K [E i, B j ] = δ i K 1 i ij q i q 1 i Serre(B i, B j ) = C ij (c) = lower order terms in B k Eg. C 13 (c) = C 12 (c) = 1 ( q 1 (q q 1 ) 2 (1 q 2 )c 1 Z 1 + q(1 q 2 ) )c 3 Z 3
20 3. Bar involution Bar involution U q g U q g does not preserve B c,s Eg. B 1 = F 1 c 1 (E 2 E 3 q 1 E 3 E 2 ) B 1 / B c,s [H. Bao, W. Wang, A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs, 213.] [M. Ehrig, C. Stroppel, Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality, 213.]
21 3. Bar involution Want the internal bar involution B c,s B c,s such that: q B = q 1 E i B = Ei B i B = Bi K β B = K 1 β F i B = Fi
22 3. Bar involution Want the internal bar involution B c,s B c,s such that: q B = q 1 E i B = Ei B i B = Bi K β B = K 1 β F i B = Fi Relations must be bar invariant C 13 (c) = C 12 (c) = 1 ( q 1 (q q 1 ) 2 (1 q 2 )c 1 Z 1 + q(1 q 2 ) )c 3 Z 3 c 1 Z 1 = c 3 Z 3 c 1 = q 2 c 3
23 3. Bar involution Theorem (B-Kolb) The following are equivalent: 1 The map b b B given on generators by q B = q 1 E i B = Ei B i B = Bi K β B = K 1 β F i B = Fi extends to an algebra homomorphism B c,s B c,s ; 2 The parameters satisfy c τ(i) = q (α i,θ(α i ) 2ρ ) c i, s i = s i Corollary For every U q g,, τ, there exists a good choice of parameters.
24 4. Quasi K-matrix fix a good choice of c i, s i two bar involutions: a a on U q g and b b B on B c,s B i B i B
25 4. Quasi K-matrix fix a good choice of c i, s i two bar involutions: a a on U q g and b b B on B c,s B B i B i Theorem (B-Kolb) There exists a unique Ûqn + such that for all b B c,s b = b B
26 4. Quasi K-matrix fix a good choice of c i, s i two bar involutions: a a on U q g and b b B on B c,s B B i B i Theorem (B-Kolb) There exists a unique Ûqn + such that for all b B c,s b = b B = µ µ, µ U + µ, = 1 [H. Bao, W. Wang, A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs, 213.]
27 4. Quasi K-matrix 1 Rewrite b = b B as 2 Proposition r i ( µ ) = some expression in lower ν i r ( µ ) = some expression in lower ν For given A i, i A, i I, the following are equivalent: 1 The following system has a unique solution: 3 Lemma r i ( ) = A i i r ( ) = i A. 2 A i, i A satisfy: i) r i ( j A) = j r (A i ) ii) Some analogue of Serre relations. Some expression in lower ν satisfy Conditions i), ii).
28 5. Universal K-matrix From now on: g finite type w longest element of the Weyl group of g, w (α i ) = α τ (i) w longest element of the Weyl group of g ξ a certain character of weight lattice M a finite dimensional U q g module Theorem (B-Kolb) Let K = ξ T 1 w T 1 w. The action of K is a B c,s -isomorphism M M ττ.
29 5. Universal K-matrix From now on: g finite type w longest element of the Weyl group of g, w (α i ) = α τ (i) w longest element of the Weyl group of g ξ a certain character of weight lattice M a finite dimensional U q g module Theorem (B-Kolb) Let K = ξ T 1 w T 1 w. The action of K is a B c,s -isomorphism M M ττ. K = the universal K-matrix. [H. Bao, W. Wang, A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs, 213.] [T. tom Dieck, R. Häring-Oldenburg, Quantum groups and cylinder braiding, 1998.]
30 6. The coproduct of K Theorem (B-Kolb) (K) = (K 1) Ř ττ (K 1) Ř
31 6. The coproduct of K Theorem (B-Kolb) (K) = (K 1) Ř ττ (K 1) Ř Proof (ignoring ττ ): K = ξ Tw 1 Tw 1 R = R q H H flip
32 6. The coproduct of K Theorem (B-Kolb) (K) = (K 1) Ř ττ (K 1) Ř = ( ξ Tw 1 Tw 1 1) R q H H (1 ξ Tw 1 Tw 1 ) R 21 q H H Proof (ignoring ττ ): K = ξ Tw 1 Tw 1 R = R q H H flip
33 6. The coproduct of K Theorem (B-Kolb) (K) = (K 1) Ř ττ (K 1) Ř = ( ξ Tw 1 Tw 1 1) R q H H (1 ξ Tw 1 Tw 1 ) R 21 q H H (Tw 1 ) = R Tw 1 Tw 1 (Tw 1 ) = R Tw 1 Tw 1 (ξ) = ξ ξ q H (H+Θ(H)) () = ( 1) (R R 1 ) (1 )
34 6. The coproduct of K Theorem (B-Kolb) (K) = (K 1) Ř ττ (K 1) Ř = ( ξ Tw 1 Tw 1 1) R q H H (1 ξ Tw 1 Tw 1 ) R 21 q H H (Tw 1 ) = R Tw 1 Tw 1 (Tw 1 ) = R Tw 1 Tw 1 (ξ) = ξ ξ q H (H+Θ(H)) () = ( 1) (R R 1 ) (1 )
35 6. The coproduct of K Theorem (B-Kolb) (K) = (K 1) Ř ττ (K 1) Ř = ( ξ Tw 1 Tw 1 1) R q H H (1 ξ Tw 1 Tw 1 ) R 21 q H H (Tw 1 ) = R Tw 1 Tw 1 (Tw 1 ) = R Tw 1 Tw 1 (ξ) = ξ ξ q H (H+Θ(H)) () = ( 1) (R R 1 ) (1 )
36 6. The coproduct of K Theorem (B-Kolb) (K) = (K 1) Ř ττ (K 1) Ř = ( ξ Tw 1 Tw 1 1) R q H H (1 ξ Tw 1 Tw 1 ) R 21 q H H (Tw 1 ) = R Tw 1 Tw 1 (Tw 1 ) = R Tw 1 Tw 1 (ξ) = ξ ξ q H (H+Θ(H)) () = ( 1) (R R 1 ) (1 )
37 6. The coproduct of K Theorem (B-Kolb) (K) = (K 1) Ř ττ (K 1) Ř = ( ξ Tw 1 Tw 1 1) R q H H (1 ξ Tw 1 Tw 1 ) R 21 q H H (Tw 1 ) = R Tw 1 Tw 1 (Tw 1 ) = R Tw 1 Tw 1 (ξ) = ξ ξ q H (H+Θ(H)) () = ( 1) (R R 1 ) (1 )
38 6. The coproduct of K Theorem (B-Kolb) (K) = (K 1) Ř ττ (K 1) Ř = ( ξ Tw 1 Tw 1 1) R q H H (1 ξ Tw 1 Tw 1 ) R 21 q H H (Tw 1 ) = R Tw 1 Tw 1 (Tw 1 ) = R Tw 1 Tw 1 (ξ) = ξ ξ q H (H+Θ(H)) () = ( 1) (R R 1 ) (1 )
39 6. The coproduct of K Theorem (B-Kolb) (K) = (K 1) Ř ττ (K 1) Ř = ( ξ Tw 1 Tw 1 1) R q H H (1 ξ Tw 1 Tw 1 ) R 21 q H H (Tw 1 ) = R Tw 1 Tw 1 (Tw 1 ) = R Tw 1 Tw 1 (ξ) = ξ ξ q H (H+Θ(H)) () = ( 1) (R R 1 ) (1 )
40 6. The coproduct of K Theorem (B-Kolb) (K) = (K 1) Ř ττ (K 1) Ř = ( ξ Tw 1 Tw 1 1) R q H H (1 ξ Tw 1 Tw 1 ) R 21 q H H (Tw 1 ) = R Tw 1 Tw 1 (Tw 1 ) = R Tw 1 Tw 1 (ξ) = ξ ξ q H (H+Θ(H)) () = ( 1) (R R 1 ) (1 )
41 6. The coproduct of K Theorem (B-Kolb) (K) = (K 1) Ř ττ (K 1) Ř = ( ξ Tw 1 Tw 1 1) R q H H (1 ξ Tw 1 Tw 1 ) R 21 q H H (Tw 1 ) = R Tw 1 Tw 1 (Tw 1 ) = R Tw 1 Tw 1 (ξ) = ξ ξ q H (H+Θ(H)) () = ( 1) (R R 1 ) (1 )
42 7.The reflection equation Theorem (B-Kolb) K satisfies the reflection equation, (K 1) Ř ττ (K 1) Ř = Ř (K 1) Ř ττ (K 1)
43 7.The reflection equation Theorem (B-Kolb) K satisfies the reflection equation, (K 1) Ř ττ (K 1) Ř = Ř (K 1) Ř ττ (K 1) Proof: (K) = (K 1) Ř ττ (K 1) Ř
44 7.The reflection equation Theorem (B-Kolb) K satisfies the reflection equation, (K 1) Ř ττ (K 1) Ř = Ř (K 1) Ř ττ (K 1) Proof: (K) = (K 1) Ř ττ (K 1) Ř (K) = Ř (K) Ř 1
45 7.The reflection equation Theorem (B-Kolb) K satisfies the reflection equation, (K 1) Ř ττ (K 1) Ř = Ř (K 1) Ř ττ (K 1) Proof: (K) = (K 1) Ř ττ (K 1) Ř (K) = Ř (K) Ř 1 (K 1) Ř ττ (K 1) Ř = Ř (K 1) Ř ττ (K 1) Ř Ř 1
46 THANK YOU! [B., Kolb The bar involution for quantum symmetric pairs, ariv: ] [B., Kolb Universal K-matrix for quantum symmetric pairs, ariv: ]
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