Tetrahedron equation and quantum R matrices for q-oscillator representations

We review and supplement the recent result by the authors on the reduction of the three dimensional R (3d R) satisfying the tetrahedron equation to the quantum R matrices for the q-oscillator representations of Uq(D(2)n+1), Uq(A(2)2n) and Uq(C(1)n). A new formula for the 3d R and a quantum R matrix for n = 1 are presented and a proof of the irreducibility of the tensor product of the q-oscillator representations is detailed.


Introduction
This paper is a summary and supplement of the recent result [9] by the authors, which is motivated by the earlier works [13,2,11]. The tetrahedron equation (1) [14] is a three dimensional generalization of the Yang-Baxter equation [1]. In [11] a new prescription was proposed to reduce it to the Yang-Baxter equation R 1,2 R 1,3 R 2,3 = R 2,3 R 1,3 R 1,2 by using the special boundary vectors defined by (3) and (10). Applied to a particular solution of the tetrahedron equation (3d L operator [2]), the reduction was shown [11] to give the quantum R matrices for the spin representations [12].
In [9] a similar reduction was studied for the distinguished solution of the tetrahedron equation which we call 3d R. The 3d R was obtained as the intertwiner of the quantum coordinate ring A q (sl 3 ) [6], (The original formula on p194 therein contains a misprint.) and was found later also in a different setting [2]. They were shown to coincide and to constitute the solution of the 3d reflection equation in [7]. See [9, App. A] for more detail. The main result of [9] was the identification of the reduction of the 3d R with the quantum R matrices for the quantum affine algebras U q = U q (D (2) n+1 ), U q (A (2) 2n ) and U q (C (1) n ). Their relevant representations turned out to be new infinite dimensional ones which we called the q-oscillator representations. There are two kinds of boundary vectors, which curiously correspond to the choices of the above three algebras. See Remark 5. This paper contains a summary of these results and a few supplements. The formula (9) for the 3d R and (19) for the quantum R matrix for n = s = t = 1 case are new. Section 4 recollects a proof of the irreducibility of the tensor product of the q-oscillator representations whose detail was omitted in [9]. The result for n = 1 was reported earlier in [8]. More recently it has been shown that the q-oscillator representations [9] quoted in Prop. 1-3 here actually factor through a homomorphism from U q to the n fold tensor product of the q-oscillator algebra [10].
Throughout the paper we assume that q is generic and use the following notations: where the q-binomial is to be understood as zero unless 0 ≤ k ≤ m.
[m] q t with t = 1 will simply be denoted by [m].

Reducing the tetrahedron equation to the Yang-Baxter equation 2.1. General scheme using boundary vectors
Let F be a vector space and R ∈ End(F ⊗3 ). Consider the tetrahedron equation: where R i,j,k acts as R on the i, j, k th components from the left in F ⊗6 . We recall the prescription which produces an infinite family of solutions to the Yang-Baxter equation from a solution to the tetrahedron equation based on special boundary vectors [11].
The index s is put to distinguish possibly more than one such vectors. Suppose there exist vectors in the dual space Then evaluating (2) between χ s (x, y)| and |χ t (1, 1) , one obtains where ̺ s,t (z) is inserted to control the normalization. The composition of R and matrix elements are taken for the space signified by 3. One may simply write it as S(z) ∈ End(F ⊗n ⊗ F ⊗n ) dropping the dummy labels. The S(z) depends on s and t although they have been temporarily suppressed. It follows from (2), (4) and (5) that S(z) satisfies the Yang-Baxter equation:

A realization of the scheme
We focus on the solution R of the tetrahedron equation mentioned in the introduction. Take F to be an infinite dimensional space F = m≥0 Q(q)|m with the dual F * = m≥0 Q(q) m| having the bilinear pairing l|m = (q 2 ) m δ l,m . Then the 3d R is given by where (9) is simpler than [9, eq.(2.10)]. Its derivation will be given elsewhere.
The two boundary vectors satisfying (4) and (5) are known [11] and given by Given two boundary vectors, one can construct four families of solutions to the Yang-Baxter equation S(z) = S s,t (z) = S s,t (z, q) (s, t = 1, 2) by (6) by substituting (9) and (10). Each family consists of the solutions labeled with n ∈ Z ≥1 . They are the matrices acting on F ⊗n ⊗ F ⊗n whose elements read where |a = |a 1 ⊗ · · · ⊗ |a n ∈ F ⊗n for a = (a 1 , . . . , a n ) ∈ (Z ≥0 ) n , etc. By Applying [9, eq.(A.1)] to (12) it is straightforward to show where a = (a n , . . . , a 1 ) is the reverse array of a = (a 1 , . . . , a n ) and similarly for b, i and j.
Henceforth we shall only consider S 1,1 (z), S 1,2 (z) and S 2,2 (z) in the rest of the paper. The matrix elements R a,b,c i,j,k (9) and S s,t (z) a,b i,j (12) are depicted as follows: Due to δ factors in (9), S s,t (z) obeys the conservation law  (12) is constrained by the n conditions b 1 + sc 0 = j 1 + c 1 , . . . , b n + c n−1 = j n + tc n leaving effectively a single sum. For (s, t) = (2, 2), they further enforce a parity constraint where |a| = a 1 + · · · + a n , etc. Thus we have a direct sum decomposition We dare allow the coexistence of somewhat confusing notations S s,t (z) and S ǫ 1 ,ǫ 2 (z) expecting that they can be properly distinguished from the context. (A similar warning applies to ̺ s,t (z) in the sequel.) We choose the normalization factors as Then the matrix elements of S 1,1 (z), S 1,2 (z) and S ǫ 1 ,ǫ 2 (z) are rational functions of q and z.

Example
Let us present an explicit form of the matrix element (12) for n = 1. It was worked out earlier in [8, Prop.2] by using a formula for R a,b,c i,j,k different from (9). For simplicity we concentrate on the case s = t = 1 and write S s,t (z) a,b i,j as S(z) a,b i,j with a, b, i, j ∈ Z ≥0 . A direct calculation using (9) and (18) leads to The last sum is over λ, µ ∈ Z ≥0 such that λ + µ = j and λ + i ≥ b. Thus it is actually a single sum over max(0, b − i) ≤ λ ≤ j. The formula (19) is simpler than [8, eq.(2.19)]. From our main Theorem 4 it follows that S a,b i,j (z = 1) = δ a j δ b i , which is consistent with the above result.

Quantum R matrices for q-oscillator representations
The Drinfeld-Jimbo quantum affine algebras without derivation U q = U q (D 2n ) and U q (C (1) n ) are the Hopf algebras generated by e i , f i , k ±1 i (0 ≤ i ≤ n) satisfying the relations [3,4]: The Cartan matrix (a ij ) 0≤i,j≤n [5] is given by a i,j = 2δ i,j − max((log q j )/(log q i ), 1)δ |i−j|,1 . The data q i is specified We employ the coproduct ∆ of the form ∆(k ±1

q-oscillator representations
We introduce representations of U q on the tensor product of the Fock spaceF ⊗n or F ⊗n , wherê F = m≥0 C(q Proposition 2. The following defines an irreducible U q (A (2) 2n ) module structure onF ⊗n .
We call these irreducible representations the q-oscillator representations of U q . For the twisted case U q (D (2) n+1 ) and U q (A (2) 2n ), they are singular at q = 1 because of the factor κ.

Quantum R matrices
be the representation space of U q in Propositions 1 and 2. By the existence of the universal R matrix [3] there exists an element R ∈ End(V x ⊗ V y ) such that up to an overall scalar. Here ∆ ′ is the opposite coproduct defined by ∆ ′ = P • ∆, where P (u ⊗ v) = v ⊗ u is the exchange of the components. A little inspection of our representations shows that R depends on x and y only through the ratio z = x/y. Moreover V x ⊗ V y is irreducible ([9, Prop. 12] and Sec. 4 of this paper) hence R is determined only by postulating (20) for g = k r , e r and f r with 0 ≤ r ≤ n. Thus denoting the R by R(z), we may claim [4] that it is determined by the conditions for 0 ≤ r ≤ n up to an overall scalar. We fix the normalization of R(z) by where |0 ∈F ⊗n is defined in the beginning of Section 3.1 with 0 = (0, . . . , 0). We call the intertwiner R(z) the quantum R matrix for q-oscillator representation. It satisfies the Yang-Baxter equation  (17) for the definition of (F ⊗n ) ± . We define the quantum R matrix R(z) to be the direct sum where each R ǫ 1 ,ǫ 2 (z) ∈ End(V ǫ 1 x ⊗V ǫ 2 y ) is the quantum R matrix with the normalization condition The R matrix R(z) satisfies the Yang-Baxter equation (25). In fact it is decomposed into the finer equalities (ǫ 1 , ǫ 2 , ǫ 3 = ±)

Main theorem
Define the operator K acting onF ⊗n by K|m = (−iq 1 2 ) m 1 +···+mn |m . Introduce the gauge transformed quantum R matrix bỹ It is easy to see thatR(z) also satisfies the Yang-Baxter equation (25). In Section 2.2 the solutions S s,t (z) of the Yang-Baxter equation have been constructed from the 3d R in (11), (12) and (18). In Section 3.2 the quantum R matrices for q-oscillator representations of U q (D (2) n+1 ), U q (A (2) 2n ) and U q (C (1) n ) have been defined. The next theorem, which is the main result of [9], states the precise relation between them. (See (13) for S 2,1 (z).) Theorem 4. Denote byR g (z) the gauge transformed quantum R matrix (28) for U q (g). Then the following equalities hold: where the last one means S ǫ 1 ,ǫ 2 (z) =R ǫ 1 ,ǫ 2 (z) between (16) and (26) with the gauge transformation (28).
Remark 5. Theorem 4 suggests the following correspondence between the boundary vectors (10) with the end shape of the Dynkin diagrams: Consistently with Remark 5, S 2,1 (z), which is reducible to S 1,2 (z 1/2 ) by (13), is identified [10] with the quantum R matrix for q-oscillator representation of another U q (A (2) 2n ) realized as the affinization of the classical part U q (B n ). (Proposition 2 corresponds to taking the classical part to be U q (C n ).) As far as χ 1 (z)| and |χ 1 (1) are concerned, the above correspondence agrees with the observation made in [11,Remark 7.2] on the similar result concerning a 3d L operator. With regard to χ 2 (z)| and |χ 2 (1) , the relevant affine Lie algebras A (2) 2n and C (1) n in this paper are the subalgebras of B

Proof of the irreducibility of the tensor product
In [9] we gave a proof of the following proposition.
Proposition 7. For g = D and for g = A 2n , C