Tetrahedron equation and quantum R matrices for q-oscillator representations

Transcription

1 Tetrahedron equation and quantum R matrices for q-oscillator representations Atsuo Kuniba University of Tokyo 17 July 2014, Group30@Ghent University Joint work with Masato Okado

2 Tetrahedron equation R : F F F F F F (3D R) R 1,2,4 R 1,3,5 R 2,3,6 R 4,5,6 = R 4,5,6 R 2,3,6 R 1,3,5 R 1,2,4 End(F 6 ) (AB Zamolodchikov 1980) = 1 6 6

3 3D R: q-oscillator solutions F = m 0 C m Fock space (q: generic) R( i j k ) = R a,b,c i,j,k a b c, R a,b,c i,j,k δ a j = = δa+b i+j δ b+c j+k { 1 a = j 0 a j, a,b,c 0 λ+µ=b ( 1) λ q i(c j)+(k+1)λ+µ(µ k) (q2 ) c+µ (q 2 ) c (q) m = m j=1 (1 qj ), Remark R a,b,c i,j,k is a polynomial in q For example, ( m j ) q = ( ) i µ q 2 (q) m (q) j (q) m j R = q 2 (1 q 4 )(1 q 6 )(1 q 8 ), R = (1 q 6 )(1 q 8 )(1 q 4 q 6 q 8 q 10 ), R = q 2 (1 + q 2 )(1 + q 4 )(1 q 6 )(1 q 6 q 10 ), etc ( ) j λ q 2

4 Origin of 3D R (1) Kapranov-Voevodsky (1994): Intertwiner of quantized coordinate ring A q (sl 3 ) (2) Bazhanov-Sergeev (2006): Quantization of Miquel s theorem (1838) (3) Sergeev (2008): (2)=Transition coefficients of PBW bases of U q + (sl 3 ) (4) K-Okado (2012): (1)=(2) solution of 3d reflection equation Remark: 3D R has Combinatorial and Birational counterparts 3D R q 0 Combinatorial 3D R ultradiscretization Birational 3D R

5 Reduction to 2D R 124 R 135 R 236 R 456 = R 456 R 236 R 135 R 124 2d reduction (eliminate spaces 4,5,6) S 12 (x)s 13 (xy)s 23 (y) = S 23 (y)s 13 (xy)s 12 (x) Yang-Baxter equation Prescription χ s (x, y) R 124 R 135 R 236 R 456 χ t (1, 1) = χ s (x, y) R 456 R 236 R 135 R 124 χ t (1, 1) by the boundary vectors χ s (x, y) = χ s (x) χ s (xy) χ s (y) F 4 F 5 F 6, χ t (x, y) = χ t (x) χ t (xy) χ t (y) 4 F 5 F 6 F satisfying χ s (x, y) R 456 = χ s (x, y), R 456 χ t (x, y) = χ t (x, y) Then S 12 (x) = χ s (x) R 123 χ t (1) End(F F ), etc

6 Boundary vectors There are 2 such boundary vectors [K-Sergeev (2013)]: χ 1 (z) = m 0 χ 1 (z) = m 0 z m (q) m m z m (q) m m, χ 2 (z) = m 0 χ 2 (z) = m 0 z m (q 4 ) m 2m, z m (q 4 ) m 2m So far: 1-layer version of reduction Possible to extend it to n-layer version

7 n-layer version of the tetrahedron equation 1 i n (R 1 i 2 i 4R 1i 3 i 5R 2i 3 i 6) R 456 = R i n (R 2 i 3 i 6R 1i 3 i 5R 1i 2 i 4) 1 1,, 1 n, 2 1,, 2 n, 3 1,, 3 n, 4, 5, 6: copies of the Fock space F The same reduction χ s (x, y) ( ) χ t (1, 1) works = Solution of the Yang-Baxter equation constructed as S s,t (z) = χ s (z) R R R 1n2 n3 χ t (1) End(F n F n ) (The evaluation is done in the space 3)

8 Matrix elements of S s,t (z) (s, t = 1, 2) Notations: a = a 1 a n F n for a = (a 1,, a n ) (Z 0 ) n e i = (0,, 0, i 1, 0,, 0), 0 = (0, 0,, 0) S s,t (z) ( i j ) = a,b S s,t (z) a,b i,j a b, S s,t (z) a,b i,j = c 0,,c n 0 z c 0 (q 2 ) sc0 R a 1,b 1,sc 0 (q s2 ) c0 (q t2 i ) 1,j 1,c 1 R a 2,b 2,c 1 i 2,j 2,c 2 R an,bn,c n 1 i n,j n,tc n cn b 2 b 1 i 2 c i 1 2 c 1 a 2 χ s (z) sc 0 a1 j 2 i n b n c n 1 j n tc n a n χ t(1) j 1

9 Examples Substitute the matrix elements of 3D R R a,b,c i,0,k = qac (q 2 ) i (q 2 ) k (q 2 ) a (q 2 ) b (q 2 δ a+b i ) c δ b+c k, R 0,b,c i,j,k = ( 1)j q j(c+1) (q2 ) k (q 2 δ ) i+jδ b b+c j+k c Up to an overall factor, the following formulas are valid (t = 1, 2): S 1,t (z) a,0 a,0 = ( q) a S 1,t (z) 0,a 0,a = (zt ; q t ) a ( z t q; q t ) a ( a = a a n ), S 1,1 (z) 2e 1,0 e 1,e 1 = ( q) 1 S 1,1 (z) 0,2e 1 e 1,e 1 = where (z; q) m = m j=1 (1 zqj 1 ) (1 + q)(1 z) (1 + zq)(1 + zq 2 ),

10 Proposition (summary so far) S(z) = S s,t (z) End(F n F n ) satisfies the Yang-Baxter equation S 12 (x)s 13 (xy)s 23 (y) = S 23 (y)s 13 (xy)s 12 (x) Problem: Find a characterization of S 1,1 (z), S 1,2 (z), S 2,2 (z) in the framework of the quantum group theory (S 2,1 (z) is simply related to S 1,2 (z) ) Result They are quantum R-matrices intertwining the q-oscillator representations of U q (D (2) n+1 ), U q(a (2) 2n ), U q(c n (1) )

11 Affine Lie algebras D (2) n+1, A(2) 2n, C n (1) Dynkin diagrams D (2) n+1 < > n 1 n A (2) 2n < < n 1 n C (1) n > < n 1 n

12 q-oscillator representations V x := F n [x, x 1 ] (x : spectral parameter) Let e j, f j, k ±1 j 0 j n act on V x by ([m] = (q m q m )/(q q 1 )) e 0 m = x m + e 1 f 0 m = 1κ[m 1 ]x 1 m e 1 κ = (q + 1)/(q 1) k 0 m = 1q m m e j m = [m j ] m e j + e j+1 (0 < j < n) f j m = [m j+1 ] m + e j e j+1 (0 < j < n) k j m = q m j +m j+1 m (0 < j < n) e n m = 1κ[m n ] m e n f n m = m + e n k n m = 1q m n 1 2 m e j = (0,, j 1,, 0), m = n j=1 m je j Z n, m = m 1 m n F n

13 Proposition V x is an irreducible representation (q-oscillator representation) of the Drinfeld-Jimbo quantum affine algebra U q (D (2) n+1 ) = e j, f j, k ±1 j 0 j n U q (A (2) 2n ) and U q(c n (1) ) also have similar q-oscillator representations (U q (D (2) 2 ) and U q(c (1) 1 ) are regarded as U q(a (1) 1 )) For U q (D (2) n+1 ) and U q(a (2) 2n ), the q-oscillator representation is singular at q = 1 due to κ= (q + 1)/(q 1) The q-oscillator representations for U q (A (1) n ), U q (C n ) were known by Hayashi (1990)

14 Quantum R matrix for q-oscillator representation For simplicity, consider U q = U q (D (2) n+1 ) for the time being R(z) End(V x V y ) (z = x/y) is characterized by (i) Commutativity: [PR(z), (g)] = 0 ( : coproduct of U q P(u v) = v u) g U q (ii) Normalization: R(z)( 0 0 ) = ( zq;q) (z;q) 0 0 Introduce a gauge transformed R(z) R(z) := (K 1 1)R(z)(1 K) K m = ( 1q 1 2 ) m 1+ +m n m

15 Properties of the R-matrix The both R(z) and R(z) satisfy the Yang-Baxter equation The following spectral decomposition holds: (z;q) ( zq;q) PR(z) = ( k ) z+q j k=0 j=1 P 1+zq j k P k = projector onto the irreducible component with respect to the classical subalgebra U q (B n ) U q (D (2) n+1 ) labeled by k For U q (C n (1) ), V x V y consists of 4 irreducible components and a slightly finer characterization is necessary

16 Main result R g (z) := R(z) of the q-oscillator representation of U q (g) Theorem (K-Okado, arxiv: ) S 1,1 (z) = R D (2) n+1 (z), S 1,2 (z) = R A (2) 2n (z), S 2,2 (z) = R (1) C (z) n Remark: Boundary vector End shape of the Dynkin diagram of g 0 n 0 n 0 n χ 1 (z) χ 1 (1) χ 1 (z) χ 2 (1) χ 2 (z) χ 2 (1) Proof: Can check the commutativity of S s,t (z) with U q

17 Related results and outlook Bazhanov-Sergeev (JPA 2006) (L = 3D L-operator satisfying RLLL = LLLR) Tr(R R), Tr(L L) = (R for type A sym or anti-sym tensor rep) K-Sergeev (CMP 2013) χ s (z) L L χ t (1) = R-matrix for spin representation K-Okado (to appear in CMP, this talk) χ s (z) R R χ t (1) = R-matrix for q-oscillator representation K-Okado (in preparation) Mixture of 3D R and L like Tr(RLLRL), χ s (z) LRLLR χ t (1), etc Commutant examples of generalized quantum group (Lusztig, Heckenberger, Batra-Yamane, etc)