Tetrahedron equation and generalized quantum groups

Transcription

1 Atsuo Kuniba University of Tokyo 25 June June / 1

2 Key to integrability in 2D Yang-Baxter equation Reflection equation R 12 R 13 R 23 = R 23 R 13 R 12 R 21 K 2 R 12 K 1 = K 1 R 21 K 2 R = 2 = R : 2 particle scattering K : Reflection at boundary PMNP2015@Gallipoli, 25 June / 1

3 What about 3D? Tetrahedron equation (AB Zamolodchikov, 1980) R : F F F F F F (3D R) 3 5 R 123 R 145 R 246 R 356 = R 356 R 246 R 145 R = 6 6 { 3 string scattering amplitude in (2+1)D R = local Boltzmann weight of the vertex in 3D PMNP2015@Gallipoli, 25 June / 1

4 Status of finding solutions and relevant maths 2D 3D Infinitely many solutions constructed systematically by representation theory of the Drinfeld-Jimbo quantum affine algebra U q (ĝ) (ĝ = affine Kac-Moody algebra) A few classes of solutions are known Systematic framework yet to be developed One such approach is by quantized algebra of functions A q (g) which is the quantum group corresponding to the dual of U q (g) (g = finite dimensional simple Lie algebra) PMNP2015@Gallipoli, 25 June / 1

5 Simplest example: {( ) t11 t SL 2 = 12 t 21 t 22 } [t ij, t kl ] = 0, t 11 t 22 t 12 t 21 = 1 A q (sl 2 ) is generated by t 11, t 12, t 21, t 22 with the relations t 11 t 21 = qt 21 t 11, t 12 t 22 = qt 22 t 12, t 11 t 12 = qt 12 t 11, t 21 t 22 = qt 22 t 21, [t 12, t 21 ] = 0, [t 11, t 22 ] = (q q 1 )t 21 t 12, t 11 t 22 qt 12 t 21 = 1 Hopf algebra with coproduct t ij = k t ik t kj Fock representation π 1 : A q (sl 2 ) End(F q ) F q = m 0 C m : q-oscillator Fock space ( ) t11 t π 1 : 12 t 21 t 22 ( a ) k qk a + k m = q m m, a + m = m + 1, a m = (1 q 2m ) m 1 PMNP2015@Gallipoli, 25 June / 1

6 A q (g) for general g = finite dim simple Lie algebra i : a vertex of the Dynkin diagram of g π i : = irreducible rep of A q (g) factoring through A q (g) A qi (sl 2,i ) s i : = simple reflection in the Weyl group W (g) Theorem (Soibelman 1991) 1 If s ii s ir W (g) is a reduced word, π i1 π ir is irreducible 2 If s i1 s ir = s j1 s jr are two reduced words, the associated irreps are equivalent: π i1 π ir π j1 π jr Corollary: Exists unique (up to overall) intertwiner Φ: (π i1 π ir ) Φ = Φ (π j1 π jr ) PMNP2015@Gallipoli, 25 June / 1

7 Example π 1 π 2 A q (sl 3 ) = t ij 3 i,j=1 Fock representations π 1 π 2 t 11 t 12 t 13 a k t 21 t 22 t 23 qk a + 0, 0 a k t 31 t 32 t qk a + W (sl 3 ) = s 1, s 2 s 2 s 1 s 2 = s 1 s 2 s 1 (Coxeter relation) = π 2 π 1 π 2 π 1 π 2 π 1 as representations on (F q ) 3 Exists the intertwiner Φ : (F q ) 3 (F q ) 3 (π 2 π 1 π 2 ) Φ = Φ (π 1 π 2 π 1 ) such that PMNP2015@Gallipoli, 25 June / 1

8 Explicit form R := ΦP 13, P 13 (x y z) = z y x, R( i j k ) = abc Rijk abc a b c R abc ijk = δ i+j,a+b δ j+k,b+c λ,µ 0,λ+µ=b [ ( 1) λ q i(c j)+(k+1)λ+µ(µ k) i, j, c + µ µ, λ, i µ, j λ, c ] m (q) m = (1 q j ), j=1 [ ] i1,, i r = j 1,, j s r m=1 (q2 ) im s m=1 (q2 ) jm PMNP2015@Gallipoli, 25 June / 1

9 Theorem (Kapranov-Voevodsky 1994) Ṛ satisfies the tetrahedron eq R 123 R 145 R 246 R 356 = R 356 R 246 R 145 R 123 Essence of proof Consider A q (sl 4 ) and W (sl 4 ) = s 1, s 2, s 3 s 2 s 1 s 2 = s 1 s 2 s 1, s 3 s 2 s 3 = s 2 s 3 s 2, s 1 s 3 = s 3 s 1, s 1 s 2 s 3 s 1 s 2 s 1 = s 3 s 2 s 3 s 1 s 2 s 3 (longest element) The intertwiner for the last one is constructed in 2 different ways as Φ P Φ Φ P 12 P Φ Φ P 23 P Φ Φ P Φ Equate the 2 sides, substitute Φ ijk = R ijk P ik and cancel P ij s PMNP2015@Gallipoli, 25 June / 1

10 Summary so far (type SL case) Remark Weyl group elements Multi-string states Cubic Coxeter relation 3D R matrix Reduced words for longest element Tetrahedron equation 3D R = Quantization of Miquel s theorem (1838) (Bazhanov-Sergeev-Mangazeev 2008) q = 0: set-theoretical sol to tropical (ultradiscrete) tetrahedron eq Recent developments Type SO, Sp, F 4 cases: 3D analogue of reflection equation Reduction to 2D: Quantum R s for generalized quantum groups Connection to Poincaré-Birkhoff-Witt basis of U + q (g) Application to multispecies TASEP: Hidden 3D structure PMNP2015@Gallipoli, 25 June

11 A q (sp 6 ) = t ij 6 i,j=1 : (Reshetikhin-Takhtajan-Faddeev 1990) π 1 π 2 π 3 F q F q F q 2 π k (t ij ) are given as follows a k qk a a k π 1 : , π 2 : 0 qk a a k a k qk a qk a π 3 : 0 0 A K q 2 K A + 0 0, A ±, K = a ±, k q q PMNP2015@Gallipoli, 25 June

12 W (sp 6 ) = s 1, s 2, s 3 s 1 s 3 = s 3 s 1, s 1 s 2 s 1 = s 2 s 1 s 2, s 2 s 3 s 2 s 3 = s 3 s 2 s 3 s 2 Write simply as π i1,,i r := π i1 π ir Then, Equivalence Intertwiner π 13 π 31, P 12 (x y) = y x, (trivial) π 121 π 212, Φ = RP 13 (same as SL case), π 2323 π 3232, Ψ = KP 14 P 23 (new) K End(F q 2 F q F q 2 F q ), R End((F q ) 3 ) PMNP2015@Gallipoli, 25 June

13 Explicit form K( a i b j ) = c,m,d,n K c,m,0,n a, i, 0, j = [ ( 1) m+λ (q4 ) c+λ (q 4 q ϕ 2 ) c λ 0 ϕ 2 = (a + c +1)(m+j 2λ)+m j K cmdn a i b j = (q4 ) a (q 4 ) c [ α,β,γ 0 Ka cmdn i b j c m d n i, j λ, j λ, m λ, i m + λ ( 1) α+γ (q 4 ) d β q ϕ 1 K a,i+b α β γ,0,j+b α β γ c,m+d α β γ,0,n+d α β γ b, d β, i + b α β, j + b α β α, β, γ, m α, n α, b α β, d β γ ϕ 1 = α(α+2d 2β 1)+(2β d)(m+n+d)+γ(γ 1) b(i +j +b) ], ], PMNP2015@Gallipoli, 25 June

14 Theorem (K-Okado 2012) R and K satisfy the 3D reflection equation (Isaev-Kulish 1997): R 489 K 3579 R 269 R 258 K 1678 K 1234 R 654 = R 654 K 1234 K 1678 R 258 R 269 K 3579 R 489 Two sides come from the 2 ways of constructing the intertwiners for π π as A q (sp 6 ) modules which correspond to the two reduced words of the longest element s 1 s 2 s 3 s 2 s 1 s 2 s 3 s 2 s 3 = s 3 s 2 s 3 s 2 s 1 s 2 s 3 s 2 s 1 W (sp 6 ) PMNP2015@Gallipoli, 25 June

15 Physical and geometric interpretation of the 3D reflection eq R 489 K 3579 R 269 R 258 K 1678 K 1234 R 654 = R 654 K 1234 K 1678 R 258 R 269 K 3579 R 489 is a factorization of 3 string scattering with boundary reflections R : Scattering amplitude of 3 strings K: Reflection amplitude with boundary freedom signified by spaces 1, 3, = PMNP2015@Gallipoli, 25 June

16 F 4 case F q F q F q 2 F q 2 R : 121 = 212 K : 2323 = 3232 S = R q q 2 : 434 = 343 π π reverse order corresponding to the longest element of W (F 4 ) (length 24) leads to the F 4 -analogue of the tetrahedron equation: S 14,15,16 S 9,11,16 K 16,10,8,7 K 9,13,15,17 S 4,5,16 R 7,12,17 S 1,2,16 R 6,10,17 S 9,14,18 K 1,3,5,17 S 11,15,18 K 18,12,8,6 S 1,4,18 S 1,8,15 R 7,13,19 K 1,6,11,19 K 4,12,15,19 R 3,10,19 S 4,8,11 K 1,7,14,20 S 2,5,18 R 6,13,20 R 3,12,20 S 1,9,21 K 2,10,15,20 S 4,14,21 K 21,13,8,3 S 2,11,21 S 2,8,14 R 6,7,22 K 2,3,4,22 S 5,15,21 K 11,13,14,22 R 10,12,22 K 2,6,9,23 R 3,7,23 R 19,20,22 K 16,17,18,22 R 10,13,23 K 5,12,14,23 R 3,6,24 K 16,19,21,23 K 4,7,9,24 R 17,20,23 K 5,10,11,24 R 12,13,24 R 17,19,24 K 18,20,21,24 S 5,8,9 R 22,23,24 = product in reverse order PMNP2015@Gallipoli, 25 June

17 Now we proceed to the last topic: 2D Reduction Tetrahedron equation = Yang-Baxter equation R 124 R 135 R 236 R 456 = R 456 R 236 R 135 R 124 = R 12 R 13 R 23 = R 23 R 13 R 12 Contents 3D L-operator : RLLL = LLLR Mixed n-product of R and L = 2 n -solutions to YBE Generalized quantum groups U A (ϵ 1,, ϵ n ), U B (ϵ 1,, ϵ n ) (ϵ i = 0, 1) Main theorem PMNP2015@Gallipoli, 25 June

18 3D L-operator: q-oscillator valued 6-vertex model V = Cv 0 Cv 1, F = F q, L = (L γ,δ α,β ) End(V V F ) L(v α v β m ) = γ,δ v γ v δ L γ,δ α,β m, Lγ,δ α,β End(F ) L 0,0 0,0 = L1,1 1,1 = 1, L0,1 0,1 = qk, L1,0 1,0 = k, L0,1 1,0 = a, L 1,0 0,1 = a+ L 124 L 135 L 236 R 456 = R 456 L 236 L 135 L 124 (Bazhanov-Sergeev 2006) = PMNP2015@Gallipoli, 25 June

19 n-layer tetrahedron equation Write RRRR = RRRR and LLLR = RLLL as ( (ϵ) ) ( M αβ4 M(ϵ) αγ5 M(ϵ) (ϵ) βγ6 R456 = R 456 M βγ6 M(ϵ) αγ5 αβ4) M(ϵ), M (0) = R, M (1) = L M (ϵ) αβ4 End( α W (ϵ) β W (ϵ) 4 F ), etc, ( (ϵ 1 i n M i ) α i β i 4 M(ϵ i ) α i γ i 5 M(ϵ i ) ) β i γ i 6 R456 = R β 1 β 2 α 1 γ 2 = α 2 γ n β n α n 1 i n β γ 2 1 β 2 α 2 γ 1 γ 1 α W (0) = F, W (1) = V ( (ϵ M i ) β i γ i 6 M(ϵ i ) α i γ i 5 M(ϵ i ) ) α i β i 4 PMNP2015@Gallipoli, 25 June γ n β n α n

20 2D reduction i ( M (ϵ i ) α i β i 4 M(ϵ i ) α i γ i 5 M(ϵ i ) β i γ i 6 ) R456 = R F 5 F 6 F can be eliminated in two ways: i ( (ϵ M i ) β i γ i 6 M(ϵ i ) α i γ i 5 M(ϵ i ) ) α i β i 4 ( ) (A) Tr 456 ( x h 4 (xy) h 5 y h 6 ( ) ), (B) 456 χ x h 4 (xy) h 5 y h 6 ( ) χ 456 where [x h 4 (xy) h 5 y h 6, R 456 ] = 0, 456 χ R 456 = 456 χ, R 456 χ 456 = χ 456 h m = m m, χ 456 = χ 4 χ 5 χ 6, χ = m (q) m m 0 Both lead to YBE: S α,β (x)s α,γ (xy)s β,γ (y) = S β,γ (y)s α,γ (xy)s α,β (x) for (A) S α,β (z) = Tr 3 ( z h 3 M (ϵ 1) α 1 β 1 3 M(ϵn) α n β n3), (B) S α,β (z) = 3 χ z h 3 M (ϵ 1) α 1 β 1 3 M(ϵ n) α nβ n3 χ 3 S α,β (z) End(W W ), W = W (ϵ 1) W (ϵ n) PMNP2015@Gallipoli, 25 June

21 Matrix elements of S(z) S(z)( i j ) = a,b S a,b i,j (z) a b, a = a 1,, a n W (ϵ 1) W (ϵn), etc Matrix element S a,b i,j (z) is depicted as i 1 b 1 i 2 j 1 a 1 b 2 i n a 2 j 2 (A) Trace reduction b n an j n i 1 χ b 1 i 2 j 1 a 1 b 2 a 2 j 2 (B) Boundary vector reduction i n b n χ an j n PMNP2015@Gallipoli, 25 June

22 Problem: Find a characterization of S(z) obtained by (A) and (B) in the framework of the quantum group theory for each choice of (ϵ 1,, ϵ n ) {0, 1} n Example: (ϵ 1,, ϵ 5 ) = (01101) (A) S(z) = Tr(RLLRL), S(z) End(W W ), (B) S(z) = χ RLLRL χ, W = F V V F V Result They are quantum R-matrices for some specific representations of generalized quantum groups U A = U A (ϵ 1,, ϵ n ) and U B = U B (ϵ 1,, ϵ n ) defined in the sequel PMNP2015@Gallipoli, 25 June

23 Def U A (ϵ 1,, ϵ n ), U B (ϵ 1,, ϵ n ) (ϵ i = 0, 1) U A and U B are Hopf algebras over C(q 1 2 ) with generators e i, f i, k ±1 i (0 i ñ) and relations (ñ = n 1 for U A, ñ = n for U B ) k i k 1 i = k 1 i k i = 1, [k i, k j ] = 0, k i e j = D i,j e j k i, k i f j = D 1 i,j f j k i, k i k 1 i [e i, f j ] = δ i,j r i r 1 i p = iq 1 2, qi = q ( ϵ i = 0), q i = q 1 ( ϵ i = 1), { p i = 0, n, r i = q for U A, r i = p 2 0 < i < n for U B, D i,j = { 2δ q i,j 1 {i, i + 1} for U A, k, i = {i, i + 1} [1, n] for U B k i j k ±1 i = k ±1 i k ±1 i, e i = 1 e i + e i k i, f i = f i 1 + k 1 i f i PMNP2015@Gallipoli, 25 June

24 Special cases: (up to Serre-type relations) ϵ i = 0, 1 cases = quantized Kac-Moody algebras of affine type U A (0,, 0) = U q (A (1) n 1 ), U A(1,, 1) = U q 1(A (1) n 1 ), U B (0,, 0) = U q (D (2) n+1 ), U B(1,, 1) = U q 1(D (2) n+1 ) κ κ {}}{{}}{{}}{{}}{ U A ( 0,, 0, 1,, 1), U B ( 0,, 0, 1,, 1) = affinization of quantum superalgebras of type A and B In general, U A and U B are examples of generalized quantum groups introduced and being developed by Heckenberger (2010), Andruskiewitsch-Schneider (2010), Angiono-Yamane (2015), Azam-Yamane-Yousofzadeh, etc κ κ PMNP2015@Gallipoli, 25 June

25 Relevant irreducible representation π x (U B case) m = m 1,, m n W = W (ϵ 1) W (ϵ n) (W (0) = F, W (1) = V ) e i = (0,, i 1,, 0), [m] = (q m q m )/(q q 1 ) π x : U B (ϵ 1,, ϵ n ) End(W ) is defined by e 0 m = x m + e 1, e n m = [m n ] m e n, f 0 m = x 1 [m 1 ] m e 1, f n m = m + e n, k 0 m = p 1 (q 1 ) m 1 m, e i m = [m i ] m e i + e i+1 f i m = [m i+1 ] m + e i e i+1 k i m = (q i ) m i (q i+1 ) m i+1 m k n m = p(q n ) mn m, (0 < i < n), (0 < i < n), (0 < i < n) ϵ i = 0 case: q-oscillator representation of U q (D (2) n+1 ) ϵ i = 1 case: spin representation of U q 1(D (2) n+1 ) PMNP2015@Gallipoli, 25 June

26 Quantum R matrix R(z) End(W W ) is characterized up to an overall scalar by [PR(z), x,y (g)] = 0 g U B, where x,y := (π x π y ), z = x/y, P(u v) = v u Theorem (K-Okado-Sergeev 2015) S(z) s obtained by (A) trace reduction and (B) boundary vector reduction are the quantum R matrices of U A and U B, respectively PMNP2015@Gallipoli, 25 June

27 Ending remarks More is known for homogeneous case ϵ 1 = = ϵ n = 0, 1 m (q) m, χ 2 = m Two boundary vectors χ 1 = m End shape of relevant Dynkin diagrams 2m (q 4 ) m 0 n 0 n 0 n χ 1 R R χ 1 U q (D (2) n+1 ) χ 1 R R χ 2 U q (A (2) 2n ) χ 2 R R χ 2 U q (C (1) n ) Quantum R matrix for U q (A (1) n 1 ) = Tr(LL L) = Matrix product formula for steady state probability in 1D Totally Asymmetric Simple Exclusion Process (TASEP) (K-Maruyama-Okado, arxiv ) PMNP2015@Gallipoli, 25 June

28 AK and S Sergeev, Tetrahedron Equation and quantum R matrices for spin representations of B n (1), D n (1) and D (2) n+1 Commun Math Phys 324 (2013) AK and M Okado, Tetrahedron and 3D reflection equations from quantized algebra of functions J Phys A: 45 (2012) Tetrahedron equation and quantum R matrices for q-oscillator representations of U q (A (2) 2n ), U q(c n (1) ) and U q (D (2) n+1 ) Commun Math Phys 334 (2015) AK, M Okado and Y Yamada, A common structure in PBW bases of the nilpotent subalgebra of U q (g) and quantized algebra of functions SIGMA 9 (2013), 049, 23 AK, M Okado and S Sergeev, Tetrahedron equation and quantum R matrices for modular double of U q (D (2) n+1 ), U q(a (2) 2n ) and U q(c n (1) ) Lett Math Phys 105 (2015) , Tetrahedron equation and generalized quantum groups, J Phys A to appear PMNP2015@Gallipoli, 25 June