Quantum Symmetric Kac-Moody Pairs - Lancaster
Quantum symmetric Kac-Moody pairsStefan KolbSchool of Mathematics and StatisticsNewcastle UniversityLancasterSeptember 6, 2012based on arXiv:1207.6036
Symmetric pairs for quantum groupschar(k ) 0, k kg semisimple Lie algebra /k , θ : g g,θ2 Id,g k p.Generalizations:Uq (g) quantum envel. algebra.g symmetrizable Kac-Moody alg.P1. Construct θq : Uq (g) Uq (g)dim(θ(b ) b ) (1st kind)P2. Construct Uq0 (k) Uq (g)dim(θ(b ) b ) (2nd kind)Solved by M. Noumi, T. Sugitani,M. Dijkhuizen, G. Letzterclassified by V. Kac, S. Wang ’92.Main result of this talkLet g be a symmetrizable Kac-Moody algebra and θ an involutiveautomorphism of the second kind. Then both θq and Uq0 (k) can beconstructed. There exists a rich structure theory.
Symmetric pairs for quantum groupschar(k ) 0, k kg semisimple Lie algebra /k , θ : g g,θ2 Id,g k p.Generalizations:Uq (g) quantum envel. algebra.g symmetrizable Kac-Moody alg.P1. Construct θq : Uq (g) Uq (g)dim(θ(b ) b ) (1st kind)P2. Construct Uq0 (k) Uq (g)dim(θ(b ) b ) (2nd kind)Solved by M. Noumi, T. Sugitani,M. Dijkhuizen, G. Letzterclassified by V. Kac, S. Wang ’92.Main result of this talkLet g be a symmetrizable Kac-Moody algebra and θ an involutiveautomorphism of the second kind. Then both θq and Uq0 (k) can beconstructed. There exists a rich structure theory.
Symmetric pairs for quantum groupschar(k ) 0, k kg semisimple Lie algebra /k , θ : g g,θ2 Id,g k p.Generalizations:Uq (g) quantum envel. algebra.g symmetrizable Kac-Moody alg.P1. Construct θq : Uq (g) Uq (g)dim(θ(b ) b ) (1st kind)P2. Construct Uq0 (k) Uq (g)dim(θ(b ) b ) (2nd kind)Solved by M. Noumi, T. Sugitani,M. Dijkhuizen, G. Letzterclassified by V. Kac, S. Wang ’92.Main result of this talkLet g be a symmetrizable Kac-Moody algebra and θ an involutiveautomorphism of the second kind. Then both θq and Uq0 (k) can beconstructed. There exists a rich structure theory.
Symmetric pairs for quantum groupschar(k ) 0, k kg semisimple Lie algebra /k , θ : g g,θ2 Id,g k p.Generalizations:Uq (g) quantum envel. algebra.g symmetrizable Kac-Moody alg.P1. Construct θq : Uq (g) Uq (g)dim(θ(b ) b ) (1st kind)P2. Construct Uq0 (k) Uq (g)dim(θ(b ) b ) (2nd kind)Solved by M. Noumi, T. Sugitani,M. Dijkhuizen, G. Letzterclassified by V. Kac, S. Wang ’92.Main result of this talkLet g be a symmetrizable Kac-Moody algebra and θ an involutiveautomorphism of the second kind. Then both θq and Uq0 (k) can beconstructed. There exists a rich structure theory.
Examples of quantum symmetric Kac-Moody pairsb 2 with Chevalley involutionq-Onsager algebra: slBaseilhac, Belliard: factorisable scattering on the half line.Terwilliger: polynomial association schemes.twisted q-Yangians: dim(g) .Consider bg0 g k [t, t 1 ] kc with involutionθb : bg0 bg0 ,b t n ) θ(x) t n ,θ(xb c.θ(c)bg0 k0 p0 . Construction of Uq0 (k0 ) Uq (bg0 )A. Molev, E. Ragoucy, P. Sorba ’03: θ of type AI, AIIN. Guay, X. Ma ’11: θ of type AIIIquantized generalized intersection matrix algebras(Y. Tan, R. Lv ’05, ’12)
Examples of quantum symmetric Kac-Moody pairsb 2 with Chevalley involutionq-Onsager algebra: slBaseilhac, Belliard: factorisable scattering on the half line.Terwilliger: polynomial association schemes.twisted q-Yangians: dim(g) .Consider bg0 g k [t, t 1 ] kc with involutionθb : bg0 bg0 ,b t n ) θ(x) t n ,θ(xb c.θ(c)bg0 k0 p0 . Construction of Uq0 (k0 ) Uq (bg0 )A. Molev, E. Ragoucy, P. Sorba ’03: θ of type AI, AIIN. Guay, X. Ma ’11: θ of type AIIIquantized generalized intersection matrix algebras(Y. Tan, R. Lv ’05, ’12)
Examples of quantum symmetric Kac-Moody pairsb 2 with Chevalley involutionq-Onsager algebra: slBaseilhac, Belliard: factorisable scattering on the half line.Terwilliger: polynomial association schemes.twisted q-Yangians: dim(g) .Consider bg0 g k [t, t 1 ] kc with involutionθb : bg0 bg0 ,b t n ) θ(x) t n ,θ(xb c.θ(c)bg0 k0 p0 . Construction of Uq0 (k0 ) Uq (bg0 )A. Molev, E. Ragoucy, P. Sorba ’03: θ of type AI, AIIN. Guay, X. Ma ’11: θ of type AIIIquantized generalized intersection matrix algebras(Y. Tan, R. Lv ’05, ’12)
Uq (g0 )A (aij )i,j I symmetric, g g(A), g0 [g, g] Uq (g0 ) k (q)hEi , Fi , Ki , Ki 1 i Ii relationswith the following relations (for all i, j I):(1) Ki Ki 1 1 Ki 1 Ki ,(2)Ki Ej Ki 1aij q Ej ,Ki Kj Kj KiKi Fj Ki 1 q aij FjKi Ki 1q q 1(4) Fij (Ei , Ej ) 0 Fij (Fi , Fj )(3) Ei Fj Fj Ei δijwhere Fij (x, y ) if i 6 jP1 aij 1 a nij( 1)n 1 ax ij y x nnq [N] [N 1] .[N n 1]with Nn q q [1]q [2]q q .[n]q q ,n 1[m]q Hopf algebra with coproduct: (Ei ) Ei 1 Ki Ei (Fi ) Fi Ki 1 1 Fi (Ki ) Ki Kiq m q m.q q 1
Uq (g0 )A (aij )i,j I symmetric, g g(A), g0 [g, g] Uq (g0 ) k (q)hEi , Fi , Ki , Ki 1 i Ii relationswith the following relations (for all i, j I):(1) Ki Ki 1 1 Ki 1 Ki ,(2)Ki Ej Ki 1aij q Ej ,Ki Kj Kj KiKi Fj Ki 1 q aij FjKi Ki 1q q 1(4) Fij (Ei , Ej ) 0 Fij (Fi , Fj )(3) Ei Fj Fj Ei δijwhere Fij (x, y ) if i 6 jP1 aij 1 a nij( 1)n 1 ax ij y x nnq [N] [N 1] .[N n 1]with Nn q q [1]q [2]q q .[n]q q ,n 1[m]q Hopf algebra with coproduct: (Ei ) Ei 1 Ki Ei (Fi ) Fi Ki 1 1 Fi (Ki ) Ki Kiq m q m.q q 1
Uq (g0 )A (aij )i,j I symmetric, g g(A), g0 [g, g] Uq (g0 ) k (q)hEi , Fi , Ki , Ki 1 i Ii relationswith the following relations (for all i, j I):(1) Ki Ki 1 1 Ki 1 Ki ,(2)Ki Ej Ki 1aij q Ej ,Ki Kj Kj KiKi Fj Ki 1 q aij FjKi Ki 1q q 1(4) Fij (Ei , Ej ) 0 Fij (Fi , Fj )(3) Ei Fj Fj Ei δijwhere Fij (x, y ) if i 6 jP1 aij 1 a nij( 1)n 1 ax ij y x nnq [N] [N 1] .[N n 1]with Nn q q [1]q [2]q q .[n]q q ,n 1[m]q Hopf algebra with coproduct: (Ei ) Ei 1 Ki Ei (Fi ) Fi Ki 1 1 Fi (Ki ) Ki Kiq m q m.q q 1
Right coideal subalgebrasObserve: Let x g0 .kx g0 (commutative) Lie subalgebra of g0 .k hxi Hopf subalgebra of U(g0 ).But k hFi i is not a Hopf subalgebra of Uq (g0 ).DefinitionA subalgebra C Uq (g0 ) is called a right coideal subalgebra if (C) C Uq (g0 ).General principle: Quantum group analogs of Lie subalgebras of g0appear as right (or left) coideal subalgebras of Uq (g0 ).Example: k hFi i is RCSA of Uq (g0 ).
Admissible pairsA generalized Cartan matrix,g g(A)DefinitionLet X I be of finite type and τ Aut(A, X ).The pair (X , τ ) is called admissible if1τ 2 Id2τ X wX3If j I \ X and τ (j) j then αj (ρ X ) Z.Notation: Aut(A, X ): diagram automorphisms which fix X .wX : longest element in the parabolic subgroup WX W .ρ X : half sum of positive coroots for mX .(mX : semisimple Lie subalgebra corresponding to X .)
S. Araki: J. Math. Osaka City Univ. 13 (1962), 1 – 34.
Involutive automorphisms of the second kindBr (W ): Braid group associated to W .Action: Ad : Br (W ) Aut(g)si 7 Ad(si ) exp(ad(ei )) exp(ad( fi )) exp(ad(ei ))Theorem (Kac, Wang ’92)The following map is a bijection: , ,admissibleinvolutive automorphismsAut(A) Aut(g)pairsof g of the second kind(X , τ ) 7 θ(X , τ ) : Ad(s(X , τ )) Ad(wX ) τ ω.Notation:ω Chevalley involution:ω(ei ) fi , ω(fi ) ei , ω(h) h i I, h h.s(X , τ ) Hom(Q, k ) if j X or τ (j) j 1 s(X , τ )(αj ) i αj (2ρX )if j / X and τ (j) j αj (2ρ )X( i)if j / X and τ (j) j
Lusztig’s braid group actionBr (W ) Aut(Uq (g0 )),si 7 Ti ,w 7 TwwhereTi (Fi ) Ki 1 Ei ,Ti (Ei ) Fi Ki , aijTi (Kj ) Kj Kiand for j 6 i one has aTi (Ej ) ad(Ei ij )(Ej ),[ aij ]q [ aij 1]q . . . [1]qTi (Fj ) q aij ω(Ti (Ej )).Notation:ω Chevalley involution for Uq (g0 ):ω(Ei ) Fi , ω(Fi ) Ei , ω(Ki ) Ki 1ad(Ei )(x) Ei x Ki xKi 1 Ei i I.(adjoint action).
Quantum involution(X , τ ) admissible pair, θ θ(X , τ )Notation: MX k (q)hEi , Fi , Ki 1 i X iPQKh i I Kimi for h i I mi hi Q Goal: Define algebra automorphismθq θq (X , τ ) : Uq (g0 ) Uq (g0 )such that (1) θq (Kh ) Kθ(h)(2) θq MX IdMX(3) θqq 1 h Q θThings we do not ask for:θq Hopf algebra automorphism(θq )2 IdUq (g0 )The ansatz θq Ad(s(X , τ )) TwX τ ω does not satisfy (2).
Quantum involution(X , τ ) admissible pair, θ θ(X , τ )Notation: MX k (q)hEi , Fi , Ki 1 i X iPQKh i I Kimi for h i I mi hi Q Goal: Define algebra automorphismθq θq (X , τ ) : Uq (g0 ) Uq (g0 )such that (1) θq (Kh ) Kθ(h)(2) θq MX IdMX(3) θqq 1 h Q θThings we do not ask for:θq Hopf algebra automorphism(θq )2 IdUq (g0 )The ansatz θq Ad(s(X , τ )) TwX τ ω does not satisfy (2).
Quantum involution(X , τ ) admissible pair, θ θ(X , τ )Notation: MX k (q)hEi , Fi , Ki 1 i X iPQKh i I Kimi for h i I mi hi Q Goal: Define algebra automorphismθq θq (X , τ ) : Uq (g0 ) Uq (g0 )such that (1) θq (Kh ) Kθ(h)(2) θq MX IdMX(3) θqq 1 h Q θThings we do not ask for:θq Hopf algebra automorphism(θq )2 IdUq (g0 )The ansatz θq Ad(s(X , τ )) TwX τ ω does not satisfy (2).
Quantum involution(X , τ ) admissible pair, θ θ(X , τ )Notation: MX k (q)hEi , Fi , Ki 1 i X iPQKh i I Kimi for h i I mi hi Q Goal: Define algebra automorphismθq θq (X , τ ) : Uq (g0 ) Uq (g0 )such that (1) θq (Kh ) Kθ(h)(2) θq MX IdMX(3) θqq 1 h Q θThings we do not ask for:θq Hopf algebra automorphism(θq )2 IdUq (g0 )The ansatz θq Ad(s(X , τ )) TwX τ ω does not satisfy (2).
Quantum involution IIDefine algebra homomorphismΨ : Uq (g0 ) Uq (g0 )Ψ(Ei ) Ei Ki ,ThenΨ(Fi ) Ki 1 Fi ,θq Ad(s(X , τ )) TX τ ωΨ(Ki ) Ki .with TX TwX Ψsatisfies (1), (2), and (3). Moreover, TX commutes with τ .
Quantum symmetric pair coideal subalgebras(X , τ ) admissible pair, θ θ(X , τ ), g0 k0 p0The Lie algebra k0 is generated byhfor all h h0 with θ(h) hmX(semisimple Lie subalgebra corresp. to X )fi θ(fi )for all i I \ X .Notation: Kh Qi IKini if h Pi Ini hi Q .Define Uq0 (k0 ) to be the subalgebra of Uq (g0 ) generated byKhfor all h Q with θ(h) hMXBi : Fi θq (Fi Ki )Ki 1for all i I \ X .Why θq (Fi Ki )Ki 1 ?Because there exist Zτ (i) Ei1 . . . Eir and ai , ai0 k (q) s.t.θq (Fi Ki ) ai TwX (Eτ (i) ) ai0 ad(Zτ (i) )(Eτ (i) ).Consequence: Uq0 (k0 ) is a right coideal subalgebra of Uq (g0 ), i.e. (Uq0 (k0 )) Uq0 (k0 ) Uq (g0 ).
Quantum symmetric pair coideal subalgebras(X , τ ) admissible pair, θ θ(X , τ ), g0 k0 p0The Lie algebra k0 is generated byhfor all h h0 with θ(h) hmX(semisimple Lie subalgebra corresp. to X )fi θ(fi )for all i I \ X .Notation: Kh Qi IKini if h Pi Ini hi Q .Define Uq0 (k0 ) to be the subalgebra of Uq (g0 ) generated byKhfor all h Q with θ(h) hMXBi : Fi θq (Fi Ki )Ki 1for all i I \ X .Why θq (Fi Ki )Ki 1 ?Because there exist Zτ (i) Ei1 . . . Eir and ai , ai0 k (q) s.t.θq (Fi Ki ) ai TwX (Eτ (i) ) ai0 ad(Zτ (i) )(Eτ (i) ).Consequence: Uq0 (k0 ) is a right coideal subalgebra of Uq (g0 ), i.e. (Uq0 (k0 )) Uq0 (k0 ) Uq (g0 ).
Quantum symmetric pair coideal subalgebras(X , τ ) admissible pair, θ θ(X , τ ), g0 k0 p0The Lie algebra k0 is generated byhfor all h h0 with θ(h) hmX(semisimple Lie subalgebra corresp. to X )fi θ(fi )for all i I \ X .Notation: Kh Qi IKini if h Pi Ini hi Q .Define Uq0 (k0 ) to be the subalgebra of Uq (g0 ) generated byKhfor all h Q with θ(h) hMXBi : Fi θq (Fi Ki )Ki 1for all i I \ X .Why θq (Fi Ki )Ki 1 ?Because there exist Zτ (i) Ei1 . . . Eir and ai , ai0 k (q) s.t.θq (Fi Ki ) ai TwX (Eτ (i) ) ai0 ad(Zτ (i) )(Eτ (i) ).Consequence: Uq0 (k0 ) is a right coideal subalgebra of Uq (g0 ), i.e. (Uq0 (k0 )) Uq0 (k0 ) Uq (g0 ).
Quantum symmetric pair coideal subalgebras(X , τ ) admissible pair, θ θ(X , τ ), g0 k0 p0The Lie algebra k0 is generated byhfor all h h0 with θ(h) hmX(semisimple Lie subalgebra corresp. to X )fi θ(fi )for all i I \ X .Notation: Kh Qi IKini if h Pi Ini hi Q .Define Uq0 (k0 ) to be the subalgebra of Uq (g0 ) generated byKhfor all h Q with θ(h) hMXBi : Fi θq (Fi Ki )Ki 1for all i I \ X .Why θq (Fi Ki )Ki 1 ?Because there exist Zτ (i) Ei1 . . . Eir and ai , ai0 k (q) s.t.θq (Fi Ki ) ai TwX (Eτ (i) ) ai0 ad(Zτ (i) )(Eτ (i) ).Consequence: Uq0 (k0 ) is a right coideal subalgebra of Uq (g0 ), i.e. (Uq0 (k0 )) Uq0 (k0 ) Uq (g0 ).
SpecializationKi 1Define A k [q, q 1 ](q 1) and (Ki ; 0)q .q 1DE0UA: A Ei , Fi , Ki 1 , (Ki ; 0)q i I’A-form’0U10 : k A UAProposition (‘well known’)There exists an isomorphism of algebras U10 U(g0 ) such that1 Ei 7 ei ,1 Fi 7 fi ,1 (Ki ; 0)q 7 hi00φ : Uq (g0 ) Uq (g0 ) linear map s.t. φ(UA) UA.φ Id φ UA0 : U10 U10(’specialization of φ’)00V Uq (g ) subspace. Define VA V UAandV k A VA U1(’specialization of V ’)Propositionθq (X , τ ) θ(X , τ ) and Uq0 (k0 ) U(k0 ).
SpecializationKi 1Define A k [q, q 1 ](q 1) and (Ki ; 0)q .q 1DE0UA: A Ei , Fi , Ki 1 , (Ki ; 0)q i I’A-form’0U10 : k A UAProposition (‘well known’)There exists an isomorphism of algebras U10 U(g0 ) such that1 Ei 7 ei ,1 Fi 7 fi ,1 (Ki ; 0)q 7 hi00φ : Uq (g0 ) Uq (g0 ) linear map s.t. φ(UA) UA.φ Id φ UA0 : U10 U10(’specialization of φ’)00V Uq (g ) subspace. Define VA V UAandV k A VA U1(’specialization of V ’)Propositionθq (X , τ ) θ(X , τ ) and Uq0 (k0 ) U(k0 ).
SpecializationKi 1Define A k [q, q 1 ](q 1) and (Ki ; 0)q .q 1DE0UA: A Ei , Fi , Ki 1 , (Ki ; 0)q i I’A-form’0U10 : k A UAProposition (‘well known’)There exists an isomorphism of algebras U10 U(g0 ) such that1 Ei 7 ei ,1 Fi 7 fi ,1 (Ki ; 0)q 7 hi00φ : Uq (g0 ) Uq (g0 ) linear map s.t. φ(UA) UA.φ Id φ UA0 : U10 U10(’specialization of φ’)00V Uq (g ) subspace. Define VA V UAandV k A VA U1(’specialization of V ’)Propositionθq (X , τ ) θ(X , τ ) and Uq0 (k0 ) U(k0 ).
Generalization and classificationLet c (ci )i I\X , s (si )i I\X AI\X such that ci (1) 1.Define Uq0 (k0 )c,s to be the subalgebra of Uq (g0 ) generated byKhfor all h Q with θ(h) hMXBi : Fi ci θq (Fi Ki )Ki 1 si Ki 1Properties of C Uq0 (k0 )c,sfor all i I \ X .(for ‘good’ c, s):(1) C is a right coideal subalgebra of Uq (g0 ).(2) C specializes to U(k0 ).(3) Maximality condition: Let V Uq (g0 ) be a subspace such thatV specializes to U(k0 )C Vthen V C.Theorem (Letzter ’02)Let g be of finite type. If C Uq (g) is a subalgebra satisfying (1), (2),and (3) above. Then C Uq0 (k)c,s for some c, s as above.
Generalization and classificationLet c (ci )i I\X , s (si )i I\X AI\X such that ci (1) 1.Define Uq0 (k0 )c,s to be the subalgebra of Uq (g0 ) generated byKhfor all h Q with θ(h) hMXBi : Fi ci θq (Fi Ki )Ki 1 si Ki 1Properties of C Uq0 (k0 )c,sfor all i I \ X .(for ‘good’ c, s):(1) C is a right coideal subalgebra of Uq (g0 ).(2) C specializes to U(k0 ).(3) Maximality condition: Let V Uq (g0 ) be a subspace such thatV specializes to U(k0 )C Vthen V C.Theorem (Letzter ’02)Let g be of finite type. If C Uq (g) is a subalgebra satisfying (1), (2),and (3) above. Then C Uq0 (k)c,s for some c, s as above.
Generalization and classificationLet c (ci )i I\X , s (si )i I\X AI\X such that ci (1) 1.Define Uq0 (k0 )c,s to be the subalgebra of Uq (g0 ) generated byKhfor all h Q with θ(h) hMXBi : Fi ci θq (Fi Ki )Ki 1 si Ki 1Properties of C Uq0 (k0 )c,sfor all i I \ X .(for ‘good’ c, s):(1) C is a right coideal subalgebra of Uq (g0 ).(2) C specializes to U(k0 ).(3) Maximality condition: Let V Uq (g0 ) be a subspace such thatV specializes to U(k0 )C Vthen V C.Theorem (Letzter ’02)Let g be of finite type. If C Uq (g) is a subalgebra satisfying (1), (2),and (3) above. Then C Uq0 (k)c,s for some c, s as above.
Generators and relationsLet Uθ0 k (q)hKh h Q , θ(h) hi and M X k (q)hEi i X i.Theorem0The algebra Uq0 (k0 )c is generated over M X Uθ by elements Bi for alli I and relationsKh Bi q αi (h) Bi KhEi Bj Bj Ei Ki Ki 1δijq q 1for all Kh Uθ0 , i I(1)for all i X , j I(2)Fij (Bi , Bj ) Cij (c)for some Cij (c) P J 1 aij(3)0M X Uθ BJ .Here, for any multi-index J (j1 , j2 , . . . , jn ) I n we defineBJ Bj1 Bj2 . . . Bjnwith Bi Fi if i X .and J n
Problems and outlookComplete the classification of subalgebras C satisfying (1), (2),and (3) up to Hopf algebra automorphism of Uq (g0 ) in theKac-Moody case.Relate the result to Poisson geometry for the correspondingsymmetric space in the finite case.Perform the program of harmonic analysis/special functions inthe non-finite case.Dream something up !
Quantum symmetric Kac-Moody pairs Stefan Kolb School of Mathematics and Statistics Newcastle University Lancaster September
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