Self duality of ASEP with two particle types via symmetry of quantum groups of rank two

Transcription

1 Self duality of ASEP with two particle types via symmetry of quantum groups of rank two 26 May 2015

2 Consider the asymmetric simple exclusion process (ASEP): Let q = β/α (or τ = β/α). Denote the occupation variables by η i {0, 1} for each lattice site i. Denote the particle variables by ξ (n1,...,nr).

3 ASEP has U q (sl 2 ) symmetry, which can be used to prove stochastic self duality. Recall that sl 2 is the Lie algebra of traceless 2 2 matrices, with basis e := ( ), f := ( ), h := ( The universal enveloping algebra U(sl 2 ) is the algebra generated by e, f, h with the relations ef fe = h, he eh = 2e, hf fh = 2f and co product : U(sl 2 ) U(sl 2 ) U(sl 2 ) defined on generators by (e) = 1 e + e 1, (f) = 1 f + f 1, (h) = 1 h + h 1. )

4 The quantum group U q (sl 2 ) is the algebra generated by e, f, k = q h, k 1 = q h with the q deformed relation and q deformed co product ef fe = qh q h q q 1 (e) = q h e + e 1, (f) = 1 f + f q h. In the q 1 limit, one recovers U(sl 2 ) from L Hôpital s rule.

5 Relationship to ASEP: sl 2 (and also U q (sl 2 )) has a natural two dimensional representation V with basis {v 1, v 2 }. Using the co product, V L is a 2 L dimensional representation. Each basis element of V L corresponds to a state of ASEP on {1,..., L}, with v 1 corresponding to a particle and v 2 corresponding to an empty site. The elements e and f act as creation and annihilation operators, respectively, and every element of U q (sl 2 ) is a symmetry (i.e. commutes with) of the Hamiltonian of ASEP.

6 This symmetry can be used to prove self duality. Recall that two Markov processes X(t), Y (t) are dual with respect to D(, ) if E x [D(X(t), y)] = E y [D(x, Y (t))]. Schutz 95 shows that ASEP is self dual with duality function D(η, ξ (n1,...,nr) ) = r s=1 2Nns (η)+2ns 1 {ηns =1}q where N i (η) is the number of particles to the right of site i.

7 Carinci Giardinà Redig Sasamoto (14) lay out a scheme for generalizing to other Lie algebras g and representations V Start from U q (g) with a co product : U q (g) U q (g) U q (g). Compute the action of (C) on V V for a central element C. Construct a quantum Hamiltonian by letting (C) act on neighboring sites in V L H := L 1 1 (C) 1 1 }{{}}{{} i=1 i 1 times L i 1 times By construction, H has U q (g) symmetry, i.e. [S, H] = 0 for every S U q (g).

8 If H is has non negative off diagonal entries, then apply an appropriate conjugation with a diagonal operator G to get a Markov generator L = G 1 HG. If H is symmetric, then self duality functions arise of the form D = G 1 SG 1 : LD = G 1 HSG 1 = (G 1 SG 1 )(GH G 1 ) = DL Carinci Giardinà Redig Sasamoto (14) do this for g = sl 2 and arbitrary finite dimensional representations V.

9 Consider gl 3, the Lie algebra of 3 3 matrices, with V the natural 3 dimensional representation. Its root system is two dimensional of type A 2 :

10 Gould Bracken Zhang (91) show that q 2 k (2,0,0) + k (0,2,0) + q 2 k (0,0,2) +(q q 1 ) 2( ) q 1 k (1,1,0) e 1 f 1 +qk (0,1,1) e 2 f 2 +qk (1,0,1) (e 1 e 2 q 1 e 2 e 1 )(f 2 f 1 q 1 f 1 f 2 ) is central in U q (gl 3 ). The proof applies Drinfeld s central element construction to Jimbo s quantum R(x) matrix.

11 Proposition (K.) The quantum Hamiltonian H is symmetric with non negative off diagonal entries, and there is a G such that G 1 HG is the generator of a particle system called spin 1/2 type A 2 ASEP. Red particles are type 2 (second class) particles, and type 1 (first class) particles can switch places with type 2 particles.

12 Two interesting projections: The projection onto the type 1 particles is usual ASEP: The projection onto the number of particles is also usual ASEP:

13 Theorem (K.) If ξ denotes the state with type 1 particles at n 1,..., n r, and type 2 particles at m 1,..., m r then D (η, ξ) = r s=1 r 1 {ηns =1}q 2Ñns (η)+2ns s =1 1 {ηms 0} q 2Nm s (η)+2m s is a self duality function, where Ñi(η) is the number of type 1 particles to the right of site i, and N i (η) is the total number of particles to the right of site i.

14 Two interesting cases: When ξ consists only of type 1 particles, D(η, ξ) is the Schütz duality function for the projection to type 1 particles: When ξ consists only of type 2 particles, D(η, ξ) is the Schütz duality function for the projection to the number of particles:

15 Belitsky Schütz proved similar results at around the same time, using the Perk Schultz quantum spin chain to prove U q (gl 3 ) symmetry.

16 Consider sp 4, the Lie algebra of 4 4 matrices of the form {( ) } A B : A = D T, B = B T, C = C T C D with V the natural four dimensional representation. Its root system is two dimensional of type C 2 :

17 Proposition (K.) The element q 4 k ( 2,0) + q 2 k (0, 2) + q 4 k (2,0) + q 2 k (0,2) + (q q 1 ) 2 (q 3 f 1 k ( 1, 1) e 1 + q 3 f 1 k (1,1) e 1 ) + (q 2 q 2 ) 2 f 2 e 2 + (q q 1 ) 2( q 1 (qf 1 f 2 q 1 f 2 f 1 )k ( 1,1) (qe 2 e 1 q 1 e 1 e 2 ) + q(qf 2 f 1 q 1 f 1 f 2 )k (1, 1) (qe 1 e 2 q 1 e 2 e 1 ) + ( f 1 f 1 f 2 (q 2 + q 2 ) ) )f 1 f 2 f 1 + f 2 f 1 f 1 (e1 e 1 e 2 (q 2 + q 2 )e 1 e 2 e 1 + e 2 e 1 e 1 ) is central in U q (sp 4 ). The proof uses an explicit construction of the quantum Harish Chandra isomorphism.

18 In this case, (C) does not have non negative off diagonal entries, which would correspond to a negative probability of having a site occupied by both a type 1 and a type 2 particle. However, we can take this probability to be ɛ 0. Proposition (K.) The quantum Hamiltonian H is symmetric, and there exist G ɛ such that lim ɛ 0 G 1 ɛ HG ɛ is the generator of spin 1/2 type C 2 ASEP. Here β 1 /α 1 = q 2 and α 1 = q 2 (q 2 + q 2 ) 2 (q 4 + q 6 ) 1 α. The projection to type 1 particles is not Markov.

19 Let ( Ñ L i ) N L i be the number of (type 1) particles to the left of site i. Proposition (K.) Spin 1/2 type C 2 ASEP is self dual with respect to the function D(η, ξ) = L i=1 ( 1 {ξi=η i=1}q 2(i 1) + 1 {ξi=η i=2}q 2(i 1+N L i (η)+n L i (ξ)) + 1 {ξi=1,η i=2}q 2(N L i (η)+i 1+2N L i (ξ) Ñ L i (ξ))) and is dual to usual ASEP with respect to the function D(η, ξ (n1,...,nr) ) = r s=1 2Nns (η)+2ns 1 {ηns 0}q

20 Thank you!