Universal Vertex-IRF Transformation and Quantum Whittaker Vectors

Universal Vertex-IRF Transformation and Quantum Whittaker Vectors E.Buffenoir RAQIS - Annecy le Vieux - September 11-14, 2007 ENIGMA - Lalonde les Maures - October 14-19, 2007 Quantum Dynamical coboundary Equation for finite dimensional simple Lie algebras E.B., Ph.Roche, V.Terras, math.qa/0512500, Adv.Math.214 (2007) 181-229 Universal Vertex-IRF Transformation for Quantum Affine Algebras E.B., Ph.Roche, V.Terras, arxiv:0707.0955 submit. Adv.Math. Quantum Whittaker vectors and dynamical coboundary equation E.B.in preparation

Baxter s Vertex-IRF Transformation Vertex-IRF Transformation in Gervais-Neveu s work Notations on quantum affine envelopping algebras 8-vertex model: 2-d square lattice model link ε j = ± vertex Boltzmann weight z = spectral parameter, p = elliptic parameter R 8V (z 1/z 2) ε 1,ε 2 ε 1,ε 2 = z 2 ε 2 ε 1 ε 2 ε 1 z 1 R 8V (z) = 0 1 a(z; p) 0 0 d(z; p) B 0 b(z; p) c(z; p) 0 C @ 0 c(z; p) b(z; p) 0 A d(z; p) 0 0 a(z; p) a(z; p) = Θ p 4 (p 2 z)θ p 4 (p 2 q 2 ) Θ p 4 (p 2 )Θ p 4 (p 2 q 2 z) b(z; p) = q 1 Θ p 4 (z)θ p 4 (p 2 q 2 ) Θ p 4 (p 2 )Θ p 4 (q 2 z) c(z; p) = Θ p 4 (p 2 z)θ p 4 (q 2 ) Θ p 4 (p 2 )Θ p 4 (q 2 z) d(z; p) = pq 1 Θ p 4 (z)θ p 4 (q 2 ) Θ p 4 (p 2 )Θ p 2 (q 2 z) Θ p(x) := (x, px 1, p; p) satisfying the Quantum Yang-Baxter Equation (QYBE) with spectral param. No charge conservation through a vertex no direct Bethe Ansatz solution Baxter s solution (Ann.Phys.1973) map onto an IRF model (SOS model)

Baxter s Vertex-IRF Transformation Vertex-IRF Transformation in Gervais-Neveu s work Notations on quantum affine envelopping algebras SOS model (Interaction-Round-a-Face model): 2-d square lattice model vertex local height n j n j n k = ±1 (adjacent) face Boltzmann weight w dynamical parameter R SOS ( z 1 z 2 ; w 0q n ) ε 1,ε 2 ε 1,ε 2 = 0 1 1 0 0 0 R SOS (z; w) = B0 b(z; w, p) c(z; w, p) 0 @ 0 zc(z; p, p) b(z; p, p) 0 C A w w 0 0 0 1 z 2 n n + ε 1 n + ε 2 n + ε 1 + ε 2 z 1 = n + ε 1 + ε 2 b(z; w, p) = q 1 Θ p 2 (z)(q 2 p 2 w 2,q 2 p 2 w 2 ;p 2 ) Θ p 2 (q 2 z)(p 2 w 2 ;p 2 ) 2 c(z; w, p) = Θ p 2 (q 2 )Θ p 2 (w 2 z) Θ p 2 (w 2 )Θ p 2 (q 2 z) satisfying the Dynamical Quantum Yang-Baxter Equation (DQYBE): R SOS 12 (z 1/z 2; wq h 3 ) R SOS 13 (z 1; w) R SOS 23 (z 2; wq h 1 ) = R SOS 23 (z 2; w) R SOS 13 (z 1; wq h 2 ) R SOS 12 (z 1/z 2; w) with h = Charge conservation, solvable by Bethe Ansatz «1 0 0 1

Baxter s Vertex-IRF Transformation Vertex-IRF Transformation in Gervais-Neveu s work Notations on quantum affine envelopping algebras Baxter s transformation: It is equivalent to the following Dynamical Gauge Equivalence: R SOS 12 (z 1/z 2; w) = M 1(z 1; wq h 2 )M 2(z 2; w) R 8V 12 (z 1/z 2) M 1(z 1; w) 1 M 2(z 2; wq h 1 ) 1 with M(z, w, p) 1 = «ϑp 4( p 3 w 2 z) pz 1 ϑ p 4( p 1 w 2 z) w ϑ p 4( pw 2 z) wϑ p 4( pw 2 Λ(z; w, p) z) where Λ(z; w, p) = diagonal matrix. Eigenvalues and eigenvectors of the 8-Vertex Transfer Matrix (Baxter, 1973) Correlation Functions of the 8-Vertex Model from the SOS ones (Lashkevich and Pugai, 1997)

Baxter s Vertex-IRF Transformation Vertex-IRF Transformation in Gervais-Neveu s work Notations on quantum affine envelopping algebras Gervais-Neveu-Felder R matrix : In their study of sl r+1 Toda Field theory, Gervais-Neveu (r = 1 case, 1984), and then Cremmer-Gervais (1989), introduced the Face-type standard trigonometric solution j R GN (x) = q r+1 1 q X i ( Q r+1 i=1 ν i = 1 and x 2 i = ν i ν i+1 E i,i E i,i + X q q 1 ν i 1 ν 1 i Ei,i E j,j ν j ν j i j + (q q 1 ) X 1 ν ff 1 i Ei,j E j,i ν j i j ) of the QDYBE: R GN 12 (x)r GN 13 (xq h 2 )R GN 23 (x) = R GN 23 (xq h 1 )R GN 13 (x)r GN 12 (xq h 3 ) where xq h = (x 1q hα 1,, xr q hαr ).

Cremmer-Gervais-Bilal R matrix : It exists, in the fundamental representation of U q(sl r+1), an invertible element M(x) Baxter s Vertex-IRF Transformation Vertex-IRF Transformation in Gervais-Neveu s work Notations on quantum affine envelopping algebras a certain non-standard solution R J of the (non-dynamical) QYBE (Cremmer-Gervais s R-matrix) such that R GN (x)m 1(xq h 2 )M 2(x) = M 2(xq h 1 )M 1(x)R CG Explicitly, j R CG = q r+1 1 q X t,s q 2(s t) r+1 E tt E s,s + (q q 1 ) X i,j,k η(i, j, k) q 2(i k) r+1 E i,j+i k E j,k ff, (with η(i, j, k) = 1 if i k < j, 1 if j k < i, 0 otherwise) and M(x) is given by a Van der Monde matrix (up to a diagonal matrix U(x)): X M(x) 1 = ν i 1 j E i,j U(x). i

Baxter s Vertex-IRF Transformation Vertex-IRF Transformation in Gervais-Neveu s work Notations on quantum affine envelopping algebras g fin. dim. or affine Lie algebra, h its Cartan subalg., let I = {α 1,, α r }, and if g = sl r+1, I := I, if g = A (1) r, I = {α 0} I (ζ i, h αj ) = δ i,j, i, j I, (ζ d, h αj ) = 0, j I, (ζ d, d) = 1. U q(g) is the algebra generated by e ±αi, i I and q h, h h with relations: q h e ±αi q h = q ±α i (h) e ±αi, q h q hα q h = q h+h i q hα i, [e αi, e αj ] = δ ij, q q 1 1 a X ij» ( 1) k 1 aij e 1 a ij k ±α k i e ±αj e±α k i = 0, i j q k=0 U q(b ± ) (resp. U q(n ± )): sub Hopf algebra generated by q h, e ±αi (resp. e ±αi, i I ). U ± q (g) = Ker(ι ± ) with ι ± : U q(b ± ) U q(h) projections U q(g) is a quasitriangular Hopf algebra with (e αi ) = e αi q hα i i + 1 e αi, (e αi ) = e αi 1 + q hα i i e αi, and (h) = h 1 + 1 h with universal R-matrix: R = K b R with K = q P l h l ζ l U q(h) 2, b R 1 2 U + q (g) U q (g)

Universal solutions of QDYBE and Quantum Dynamical Cocycles IRF Dynamical Cocycles and Fusion operators of Verma Modules General Solution of quantum dynamical cocycle equation Vertex-IRF problem and Dynamical Coboundary Equation Universal solutions R(x) of the QDYBE can be build in terms of the standard solution R of the QYBE and Quantum Dynamical Cocycles as R(x) = J 21(x) 1 R 12 J 12(x), Definition : Quantum Dynamical Cocycle J : (C ) dim l U q(g) 2 where J(x) is invertible, is of zero l -weight, and satisfies the Quantum Dynamical Cocycle Equation (QDCE): ( id)(j(x)) J 12(xq h 3 ) = (id )(J(x)) J 23(x). Quasi-Hopf interpretation, Dynamical Vertex-Operator approaches of 8 vertex correlation functions. Explicit construction by means of an auxiliary linear equation: Standard solution R(x) in the finite dimensional case (Arnaudon, Buffenoir, Ragoucy, Roche, 1998) Standard IRF solution and Belavin-Baxter s Vertex solution in the A (1) r case (Jimbo, Konno, Odake, Shiraishi, 1999) Extension to other solutions (in the formalism of generalized Belavin-Drinfeld Triple) (Etingof, Schedler, Schiffmann, 2000)

Universal solutions of QDYBE and Quantum Dynamical Cocycles IRF Dynamical Cocycles and Fusion operators of Verma Modules General Solution of quantum dynamical cocycle equation Vertex-IRF problem and Dynamical Coboundary Equation Verma modules V η = U q(g). > η u. > η= Υ η(u) > η, u U q(h ), x. > η= 0, x U + q (g). Shapovalov s form unique hermitian form such that S η : V η V η C S η( > η, > η) = 1 S η(v, x.v ) = S η(x.v, v ) with q = q e α i = e αi q hα i e α i = q hα i e αi h α i = h αi non-degenerate if and only if V η is irreducible.

Universal solutions of QDYBE and Quantum Dynamical Cocycles IRF Dynamical Cocycles and Fusion operators of Verma Modules General Solution of quantum dynamical cocycle equation Vertex-IRF problem and Dynamical Coboundary Equation Fusion Operators of Verma Modules Φ element of Hom Uq(g )(V η, V V η) (where V is a finite dimensional U q(g ) module). We define < Φ > by < Φ > V[η η] / Φ( > η ) =< Φ > > η + X a (γ) f (γ) > η. (γ) If V η is irreducible then <>: Hom Uq(g )(V η, V V η) V[η η] is an isomorphism. For any homogeneous element v µ of weight µ, we denote Φ vµ η Hom Uq(g )(V η+µ, V V η) the unique element such that < Φ vµ η >= v µ. Fusion Operators and Face Type Quantum Dynamical Cocycles For any homogeneous vector v µi V of weight µ i and any η h, we have Φ vµ η > η+µ= ( V π id)`j F (bx). (v µ > η) bx(h αi ) = q, i v µ2 µ1 `1 Φ η Φv η+µ 2 > η+µ1 +µ 2 =.Φ J F (bxq η )(v µ1 v µ2 ) η > η+µ1 +µ 2 J F verifies quantum Dynamical Cocycle equation. Relation of matrix elements of J F and Shapovalov s coefficients...

Universal solutions of QDYBE and Quantum Dynamical Cocycles IRF Dynamical Cocycles and Fusion operators of Verma Modules General Solution of quantum dynamical cocycle equation Vertex-IRF problem and Dynamical Coboundary Equation Definition : Generalized Translation Quadruple (θ + [x], θ [x], ϕ, S) θ ± [x] End(Uq(b± )), ϕ and S are invertible elements of U q(h) 2 such that θ ± [xq h 2 ]1 = Ad 1 ϕ θ ± [x]1 θ + [x]1 (b R) = (Ad ϕ θ [x]2 )(b R) θ ± [x]1 (ϕ12) = θ± [x]2(ϕ12) = ϕ12 θ+ [x]1 (S12) = θ [x]2 (S12) v U q(b ˆ ), ϕ12k 12S 1 21 S 12 θ 1 [x]1(k12s21s12 ), θ [x]1 (v) = 0. Theorem : Auxilliary Linear Problem (ABRR equation) Let (θ + [x], θ [x], ϕ, S) be a generalized translation quadruple. The linear equation b J(x) = θ [x]2 Ad S 1( R) b b J(x) admits a unique solution b J(x) 1 1 + U + q (g) U q (g) and J(x) = S b J(x) satisfies the Quantum Dynamical Cocycle Equation. explicit formula for quantum dynamical cocycles as infinite product: Q J 12(x) = S + 12 k=1 (θ [x]2 )k S 1 12 br S 12

Standard IRF solutions Universal solutions of QDYBE and Quantum Dynamical Cocycles IRF Dynamical Cocycles and Fusion operators of Verma Modules General Solution of quantum dynamical cocycle equation Vertex-IRF problem and Dynamical Coboundary Equation Standard IRF Solution g = sl r+1 or g = A (1) r, l = h θ ± [x] =Ad B(x) ±1, ϕ = K 2, S = 1, with B(x) U q(h) such that (B(x)) = B 1(x) B 2(x) K 2, B 1(xq h 2 ) = B 1(x) K 2. This solution leads to the fusion matrix J F. Arnaudon, Buffenoir, Ragoucy, Roche, 1998 for the finite case, Jimbo, Konno, Odake, Shiraishi, 1999, for the affine case Face-type solution R GN (x) in the g = A n case in the fundamental evaluation representation Face-type solution R SOS (z; w, p) in the g = A (1) 1 case (and higher rank generalizations) in the fundamental evaluation representation

Extremal Vertex Solutions Universal solutions of QDYBE and Quantum Dynamical Cocycles IRF Dynamical Cocycles and Fusion operators of Verma Modules General Solution of quantum dynamical cocycle equation Vertex-IRF problem and Dynamical Coboundary Equation Cremmer-Gervais-Bilal s Vertex Solutions (g = A r, l = 0) Obtained with the following choice of Generalized Translation Quadruple: θ ± [x] (e±α i ) = e±α i 1, θ+ [x] (eα 1) = 0, θ [x] (e αr ) = 0, ϕ = 1, θ ± [x] (ζα i ) = ζ α i 1, θ + [x] (ζα 1 ) = 0, θ [x] (ζαr ) = 0, S = q P r 1 i=1 ζα i ζ α i 1. Cremmer-Gervais s vertex solution R CG (and higher rank generalizations) in the fundamental representation Belavin-Baxter Elliptic Vertex solutions (g = A (1) r, l = cc) θ ± [x] = Ad D ± (p) σ ±, D + (p) = p 2 r+1 ϖ, D (p) = p 2 r+1 ϖ q 2 r+1 cϖ, σ ± (e ±αi ) = e ±α[i 1], σ ± (h αi ) = h α[i 1], σ ± (ϖ) = ϖ, where ϖ = P r i=0 ζα i, [i] = i mod r + 1, and p is the component of x along q c. eight-vertex solution R 8V (z; p) (and higher rank generalizations)

Universal solutions of QDYBE and Quantum Dynamical Cocycles IRF Dynamical Cocycles and Fusion operators of Verma Modules General Solution of quantum dynamical cocycle equation Vertex-IRF problem and Dynamical Coboundary Equation Vertex-IRF problem and Dynamical Coboundary Equation In terms of Quantum Dynamical Cocycles, the Vertex-IRF is reformulated as Generalized Quantum Dynamical Coboundary Problem: Let J F (x) be the standard Face-type solution of the Quantum Dynamical Cocycle Equation, and J V (x) a Vertex-type solution Does it exists an invertible element M(x) U q(g) such that J F (x) = (M(x)) J V (x) M 2(x) 1 M 1(xq h 2 ) 1? Such a generalized Quantum Dynamical Coboundary is a Vertex-IRF transformation: R F (x) = J F (x) 1 21 R J F (x) 12, and R V (x) = J V (x) 1 21 R J V (x) 12 satisfy R F (x) 12 = M(xq h 1 ) 2 M(x) 1 R V (x) 12M(x) 1 2 M(xq h 2 ) 1 1.

Gauss decomposition of Quantum Dynamical Coboundaries Explicit solutions of Auxiliary Linear System Theorem : Auxiliary Linear Problem (Buffenoir, Roche, Terras) If M(x) is expressed as M(x) = M (0) (x) M ( ) (x) 1 M (+) (x), M (±) (x) = + Y k=1 C [±k] (x) ±1 with C [+k] (x) = (θ + [x] )k 1`C [+] (x), C [ k] (x) = B(x) k C [ ] (x) B(x) k. and if M (0) (x) U q(h), Fusion C [±] (x) 1 U ± q (g) satisfy (M (0) (x)) = S 1 12 M (0) 1 (xqh 2 ) M (0) 2 (x), `C[±] (C [±] (x)) = K 12 Ad [S21 ] 1 (x) Ad [K 12 S ]`C[±] 2 (x) K 1 12 12 Shift C [±] 1 (xq h 2 ) = Ad [S 1 12 S 21 K ]`C[±] 1 (x), 12 Hexagonal C [ ] 2 (x) C [+] 1 (xqh 2 ) ˆAd B2 (x) θ 1 [x]2 `S b 12 R 12 S 12 1 = `S b 12 R 12 S 12 C[+] 1 (xqh 2 ) C [ ] 2 (x). then it satisfies quantum dynamical coboundary equation.

Explicit Universal Vertex-IRF transform Gauss decomposition of Quantum Dynamical Coboundaries Explicit solutions of Auxiliary Linear System trivial to find M (0) (x) solution of fusion + adequate shift in terms of S easy to build explicit solutions C [±] (x) of fusion and shift: χ they are generically associated to couple χ = (χ +, χ ) of some non singular characters χ of the subalgebras U q(n ) ω of U q(b ) generated respectively by ω e + αi = q (α i id)(ω +) ω e αi e αi = e αi q (α i id)(ω ) q ω+ := S 21, q ω := KS 21 ω C [±] [±] χ (x) = Ad M (0) (x) 2(C χ ) C [±] χ = (id χ )(S 21 R (±) 12 K 1 S 1 21 ) hexagonal relation is satisfied only for specific Lie algebras and only for Standard IRF-type dynamical cocycles and Extremal Vertex-type cocycles (verified for the Cremmer-Gervais and Belavin-Baxter case for any rank) as soon as q 1 (q q 1 ) 2 χ + ( ω+ e αi )χ ( ω e αi ) = 1.

Quantum Whittaker vectors and functions Whittaker vectors and Toda Quantum Mechanics Explicit realization of quantum Whittaker vectors Fusion of quantum Whittaker Vectors and Vertex-IRF transform Quantum Whittaker Vectors Let V be a U q(g ) module and let χ + be a non-singular character of U q(n + ) ω+, a vector W ω χ + V is said to be a χ + quantum Whittaker vector if x U q(n + ) ω+, x. W ω χ + = χ + (x) W ω χ +. Quantum Whittaker Functions A quantum Whittaker function denoted W ω ξ associated to a central ϱ,ϱ + character ξ and non-singular characters ϱ ± of U q(n ± ) ω±, is an element of (U q(g )) defined such that x U q(n ) ω, z U q(n + ) ω+, y U q(g ), a Z q(g ) we have ω W ξ (x.y.z) = ϱ (x)ϱ + (z) W ω ξ (y). ϱ,ϱ + ϱ,ϱ + ω W ξ (a.y) = ξ(a) W ω ξ (y). ϱ,ϱ + ϱ,ϱ +

Quantum Whittaker vectors and functions Whittaker vectors and Toda Quantum Mechanics Explicit realization of quantum Whittaker vectors Fusion of quantum Whittaker Vectors and Vertex-IRF transform The basic motivation is the Toda Quantum Mechanics (ex. q = 1, g = A 1) ξ(c)w ξ (e φh ) = W e ξ φh (e ϱ,ϱ + ϱ,ϱ e + + + 1 «2 h2 + h) = ϱ (e )ϱ + (e +)e 2φ + 12 «2φ + φ W ξ (e φh ) ϱ,ϱ + Then, f (φ) = e φ W ξ (e φh ) obeys ϱ,ϱ + φ 2 + 2ϱ (e )ϱ + (e +)e 2φ 2ξ(c) 1. f = 0 (φ i ) i ω W ξ ϱ,ϱ + (e φi h αi ) is a wave function of q deformed quantum Toda hamiltonian derived from the action of the casimir ( 1 := m(s)b(bx)) 0 = tr V q cq V := (tr V q id)`ad q ω (b R 1 12 )K 2 12 Ad q ω + (b R 1 Ad e φ i hα i (C [+] 1 ϱ )q 2ζi φi C [ ] ϱ + ξ`c V q 21 )) ω W ξ (e φi h αi ϱ,ϱ ) +

Quantum Whittaker vectors and functions Whittaker vectors and Toda Quantum Mechanics Explicit realization of quantum Whittaker vectors Fusion of quantum Whittaker Vectors and Vertex-IRF transform Quantum Whittaker vectors can be identified as elements of a certain completion of a Verma module: Explicit realization of quantum Whittaker vectors in Verma modules An explicit realization of quantum Whittaker vectors is given by ω W ξη χ + = M(bx) 1 > η The quantum Whittaker function is given by ω W ξη χ +,χ (u) = S η( eω W ξ eω ± 12 = (ω ± 21) η eχ +, u ω W ξη χ + ) eχ + (x) = χ (Ad (x ))

Quantum Whittaker vectors and functions Whittaker vectors and Toda Quantum Mechanics Explicit realization of quantum Whittaker vectors Fusion of quantum Whittaker Vectors and Vertex-IRF transform Using the dynamical Coboundary equation as well as expression of Fusion operators, we deduce Φ vµ η ω W ξη+µ = Φ vµ χ + η M(bx) 1 > η+µ = ( V π id)` (M(bx) 1 ) Φ vµ η > η+µ = ( V π id)` (M(bx) 1 ) ( V π id)`j F (bx). (v µ > η) = ( V π id)`j V (bx)m(bx) 1 2 M(bxq h 2 ) 1 1. (vµ > η) = ( V π id)`j V (bx)m(bxq η(h) ) 1 1. v µ W ω ξη. χ + «These fusion formulas implies the basic finite difference equations describing the shift on η for Whittaker functions in terms of finite difference operators acting on the φ i variables.

Explicit universal solutions of quantum dynamical cocycle and coboundary equation in the finite dimensional as well as affine Lie algebra case: universality is fundamental to build correlation functions of the model as matrix elements in highest weight reps. of quantum algebras Relation between Face type quantum dynamical cocycles and the Fusion theory of Verma modules Relation between quantum dynamical coboundaries and Whittaker vectors Relation between Extremal Vertex type quantum dynamical cocycles and the Fusion theory of Whittaker modules Main Open Problems The explicit expression of quantum dynamical coboundary (in the quantum affine case) in the highest weight modules gives the tail operator used in Lashkevich/Pugai and Konno/Kojima/Weston s works New approach to compute correlation functions of the 8 vertex model Analysis of the quantum dynamical coboundary equation new results on Whittaker functions