arXiv:0707.4029v1 [math.QA] 27 Jul 2007 - CiteSeerX

TETRAHEDRON EQUATIONS AND NILPOTENT SUBALGEBRAS OF Uq (sln ).

arXiv:0707.4029v1 [math.QA] 27 Jul 2007

S. M. SERGEEV

Abstract. A relation between q-oscillator R-matrix of the tetrahedron equation and decompositions of Poinkar´ e-Birkhoff-Witt type bases for nilpotent subalgebras Uq (n+ ) ⊂ Uq (sln ) is observed.

The method of q-oscillator three-dimensional auxiliary problem, derivation of three-dimensional Rmatrix for the Fock space representation of the algebra of observables, various q-oscillator tetrahedron equations and details of 3D → 2D dimension-rank transmutation may be found in [1]. In the first part of this letter I remind the definition of q-oscillator R-matrix of the tetrahedron equation. Surprisingly, this R-matrix may be derived in a completely different framework. In the second part of this letter it is shown that matrix elements of q-oscillator R-matrix relate two certain Poinkar´e-Birkhoff-Witt type bases of the nilpotent (positive roots) subalgebra Uq (n+ ) ⊂ Uq (sl3 ). The tetrahedron equation in this framework follows from the completeness and irreducibility of Poinkar´e-Birkhoff-Witt type bases of the nilpotent subalgebra of Uq (sl4 ). Q-oscillator R-matrix. Let elements elements a+ , a− and q N generate the q-oscillator algebra A : (1)

a+ a− = 1 − q 2N ,

a− a+ = 1 − q 2N +2 ,

q N a± = a± q N ±1 .

−1 There is a special automorphism x → r123 · x · r123 of A ⊗3 defined by

(2)

 ∓ N3 ±   r123 · q N 2 a± a1 + q N 1 a±  1 = (q 2 a3 ) · r123 ,        ± ± 1+N 1 +N 3 ± r123 · a± a2 ) · r123 , 2 = (a1 a3 − q           r · q N 2 a± = (q N 1 a± + q N 3 a∓ a± ) · r . 123 123 3 3 1 2

Here the indices of q-oscillator generators stand for the components of the tensor product A ⊗3 . In the Fock space (non-hermitian) representation (3)

a− |0i = 0 ,

|ni = a+n |0i ,

q N |ni = |niq n ,

1991 Mathematics Subject Classification. 17B37,81R50. 1

hm|ni = δn,m

2

S. M. SERGEEV

the matrix elements of Eqs. (2) provide a set of recursion equations for the matrix elements of r123 with the unique solution def

, n2 , n3 = hm1 , m2 , m3 |r|n1 , n2 , n3 i = rmn11 ,m 2 ,m3

(4) δm1 +m2 ,n1 +n2 δm2 +m3 ,n2 +n3

q (m1 −n2 )(m3 −n2 ) Pm2 (q 2n1 , q 2n2 , q 2n3 ) , (q 2 ; q 2 )m2

where Pm (x, y, z) is defined recursively by (5)

Pm+1 (x, y, z) = (1 − x)(1 − z)Pm (

x z xz y , y, 2 ) − 2m (1 − y)Pm (x, 2 , z) , 2 q q q q

P0 (x, y, z) = 1 .

Pohgammer’s symbol is defined as usual, (q 2 ; q 2 )n = (1 − q 2 )(1 − q 4 ) · · · (1 − q 2n ). Pm (q

2n1

,q

2n2

,q

2n3

Coefficients

) may be expressed in terms of q-hypergeometric function 2 ϕ1 . Matrix r123 is a

square root of unity, r2123 = 1. Note the symmetry property, (6)

Pm2 (q 2n1 , q 2n2 , q 2n3 ) =

(q 2 ; q 2 )n1 (q 2 ; q 2 )n3 Pn (q 2m1 , q 2m2 , q 2m3 ) , (q 2 ; q 2 )m1 (q 2 ; q 2 )m3 2

where m1 + m2 = n1 + n2 and m2 + m3 = n2 + n3 . Spectral parameters may be introduced as follows: (7)

R123 =

 N 1  N 3 N 2  λ3 λ1 µ3 µ1 r123 − , q λ2 µ2

where C-valued parameters λj , µj are associated with jth component of a tensor power of A . Matrices Rijk satisfy the tetrahedron equation in A ⊗6 , (8)

R123 R145 R246 R356 = R356 R246 R145 R123 .

Due to the δ-symbols structure of (4), the spectral parameters may be removed from the tetrahedron equation, Eq. (8) is equivalent to the constant tetrahedron equation. However, the constant r-matrix provides spectral parameter dependent solutions for the Yang-Baxter equation (9)

Rs1 ,s2 (u)Rs1 ,s3 (uv)Rs2 ,s3 (v) = Rs2 ,s3 (v)Rs1 ,s3 (uv)Rs1 ,s2 (u) .

b 2 ), matrix elements of two-dimensional R-matrix For instance, in the lowest rank case of affine Uq (sl

of the Yang-Baxter equation in spin s1 × spin s2 evaluation representations are given by (10)

hm1 , m2 |Rs1 ,s2 (u)|n1 , n2 i =

∞ X

+n1 , s2 +n2 , n3 s1 −n1 , s2 −n2 ,m3 , r um3 rss11+m 1 ,s2 +m2 ,m3 s1 −m1 ,s2 −m2 , n3

n3 ,m3 =0

where mj , nj ∈ {−sj , −sj + 1, · · · , sj − 1, sj }, and the sum converges if |q| < 1 and |u| < 1. In particular, formula (10) gives the six-vertex R-matrix for s1 = s2 = 1/2.

3

Nilpotent subalgebras. Let Bn = Uq (n+ ) be the nilpotent subalgebra of Uq (sln+1 ) = Uq (n− ) ⊗ Uq (h) ⊗ Uq (n+ ). It is generated by elements e1 , e2 , . . . , en satisfying q-Serre relations: e2i ei±1 + ei±1 e2i = (q −1 + q)ei ei±1 ei

(11)

and ei ej = ej ei if |i − j| > 1 .

Consider for a moment B2 and define (12)

e12 =

e1 e2 − qe2 e1 q −1 − q

and e e12 =

e2 e1 − qe1 e2 . q −1 − q

It is well known [2], there are many ways to define a Poinar´e-Birkhoff-Witt type basis in Bn . For instance, two following bases of B2 ,     1 m2 m3 e (13) en2 3 en122 en1 1 , nj ≥ 0 e and em e , m ≥ 0 , j 12 2 1

are complete, irreducible and therefore equivalent. The equivalence of the bases is given by the decomposition (14)

n3 m3 2 1 X e em en122 en1 1 em 12 e2 1 , n2 , n3 e2 rmn11 ,m = ,m 2 3 [m1 ]! [m2 ]! [m3 ]! [n3 ]! [n2 ]! [n1 ]! {n}

where we use for shortness (15)

[n] = q −n − q n ,

[n]! = [1][2] · · · [n] = q −n(n+1)/2 (q 2 , q 2 )n .

, n2 , n3 in (14) are given exactly by the formula (4). Theorem 1. The coefficients rmn11 ,m 2 ,m3 , n2 , n3 Sketch proof is the following. Coefficients rmn11 ,m in (14) satisfy a set of recursion relations following 2 ,m3

from the structure of bases. For instance, a simple identity following from (11) (16) provides (17)

1 +1 m2 m3 1 m2 m3 −1 1 m2 m3 e em e12 e2 = q m2 [m3 ]em e12 e2 e12 e2 e1 e12 + q m2 +m3 em 1 1 e 1 e

, n2 −1, n3 1 , n2 , n3 1 −1, n2 , n3 n2 q (1 − q 2n1 ) . (1 − q 2m1 +2 )rmn1 +1,m = rmn11 ,m (1 − q 2n2 ) + q m3 rnm 1 ,m2 ,m3 2 ,m3 2 ,m3 −1

In terms of q-oscillators in the basis (3) it corresponds to (18)

+ − N3 r123 q N 2 a− a− 1 , 2 r123 = a3 r123 a2 + q

what is simple consequence of (2). The complete collection of all such recursion relations for (14) is equivalent to the system (2). Consider next B3 and equivalence of the complete irreducible bases  n6 n5 n4 n3 n2 n1  e3 e23 e123 e2 e12 e1 (19) , nj ≥ 0 , [n6 ]![n5 ]! · · · [n1 ]!



4

S. M. SERGEEV

where (20)

e12 =

e1 e2 − qe2 e1 , [1]

e23 =

e2 e3 − qe3 e2 , [1]

e123 =

e1 e23 − qe23 e1 , [1]

and 

(21)

1 m2 m4 3 m5 m6 e23 e3 e12 e em e123 em 1 e 2 e , mj ≥ 0 [m1 ]![m2 ]! · · · [m6 ]!



,

where (22)

e e12 =

e2 e1 − qe1 e2 , [1]

e e23 =

e3 e2 − qe2 e3 , [1]

e e123 =

e e23 e1 − qe1 e e23 . [1]

,..., n6 for these bases may be constructed in two alternative ways, The decomposition matrix Tmn11,...,m 6 (1)

(23)

T12...6 = r123 r145 r246 r356

(2)

and T12...6 = r356 r246 r145 r123 . (1)

(2)

Due to the uniqueness of the decomposition, T12...6 = T12...6 , what is the tetrahedron equation.

“Coherent” states. The result of equivalence of (2) and (14) may be written in a remarkable simple form in terms of “coherent states”. Let ∞ ∞ X X un un (24) ψ(u) = = q n(n+1)/2 2 2 , [n]! (q ; q )n n=0 n=0

ψ(u/q) − ψ(qu) = uψ(u) .

Then (2) and (14) may be combined together into the basis-invariant form in A ⊗3 ⊗ B2 : (25)

+ + + + r123 · ψ(e2 a+ e12 a+ 3 )ψ(e12 a2 )ψ(e1 a1 )|0i = ψ(e1 a1 )ψ(e 2 )ψ(e2 a3 )|0i ,

where |0i is the total Fock vacuum for A ⊗3 , r123 is the q-oscillator R-matrix of (2), and the auxiliary coefficients e1 , e2 ∈ B2 . Presumably, relation (25) is the master equation for three-dimensional variant of the vertex-IRF duality. References [1] V. V. Bazhanov and S. M. Sergeev, “Zamolodchikov’s tetrahedron equation and hidden structure of quantum groups”, J. Phys. A 39 (2006) 3295–3310 [2] G. Lusztig, “Introduction to quantum groups”, volume 110 of Progress in Mathematics. Birkh¨ auser Boston Inc., Boston, MA, 1993

Department of Theoretical Physics & Department of Mathematics, Australian National University, Canberra ACT 0200, Australia E-mail address: [email protected]