Tetrahedron Equation and Quantum R Matrices for q-Oscillator Representations Mixing Particles and Holes

We construct 2 + 1 solutions to the Yang–Baxter equation associated with the quantum affine algebras Uq ( A (1) n−1 ) , Uq ( A (2) 2n ) , Uq ( C (1) n ) and Uq ( D (2) n+1 ) . They act on the Fock spaces of arbitrary mixture of particles and holes in general. Our method is based on new reductions of the tetrahedron equation and an embedding of the quantum affine algebras into n copies of the q-oscillator algebra which admits an automorphism interchanging particles and holes.


Introduction and main results
The principal structure in quantum integrable systems is the Yang-Baxter equation [1]: R 1,2 (x)R 1,3 (xy)R 2,3 (y) = R 2,3 (y)R 1,3 (xy)R 1,2 (x). (1.1) The tetrahedron equation [20] is a three dimensional (3d) analogue of it having the form where R lives in End(F ⊗F ⊗F ) for some vector space F .The equality holds in End The R i,j,k in (1.2) acts on the components F as R and as the identity elsewhere.Composing the above equation n times one gains a 'non-local' relation where the spaces 1, 2, 3 have been replaced by the copies 1 i , 2 i , 3 i with i = 1, . . ., n.It can be cast into the following form just by commuting R's without common indices: 4 ).(1.3)This is almost the Yang-Baxter equation except the conjugation by R 4,5,6 .In fact there are two ways to evaluate  F called boundary vectors will be explained in Section 2. We have regarded the F to be eliminated as an auxiliary space and labeled it with a.
The above reduction works for arbitrary n hence generates an infinite family of solutions to the Yang-Baxter equation from a single solution to the tetrahedron equation.This idea has a long history; see [3,9,17,18] and references therein.As for the input R, we will exclusively deal with the celebrated solution to the tetrahedron equation [8] discovered as the intertwiner of the quantized coordinate ring A q (sl 3 ).See (2.3) and (2.4) for an explicit formula and [2,10,11,16] for further aspects.The R is a linear operator on F ⊗3 with F being the q-oscillator Fock space F = m≥0 C(q 1 2 )|m .The reduction procedure based on this R has been studied extensively in recent years [3,12,13,14,17].By the construction (1.4) and (1.5), the resulting R matrices, i.e., solutions to the Yang-Baxter equation, become linear operators on the tensor product of the Fock spaces 1 F ⊗n ⊗ F ⊗n .
Having infinitely many R matrices at hand, a fundamental problem is to clarify their origin in the framework of quantum group theory.More specifically one should identify the quantum affine algebras U q , if any, which characterize the R matrices by the intertwining relation in an appropriate representation space.Here ∆ and ∆ op denote the coproduct and its opposite (cf.(3.2)).In short one should elucidate the quantum group symmetry of the R matrices [4,6].
Our aim in this paper is to generalize these results further by exploring new variants of the reduction method.Let us illustrate them along the trace reduction with n = 3.We write (1.4

)| n=3 simply as Tr
paying attention only to which component is adopted as the auxiliary space a = •.We will show that the reduction to the Yang-Baxter equation works equally well and produces different R matrices for the following 23 + 1 arrangements: Tetrahedron Equation and Quantum R Matrices 3 ently understood as the composition [14]: where z is a spectral parameter and g n = A (1) 2n , C n , D n+1 are affine Lie algebras [7] which are already mentioned after (1.6).The homomorphism π z is specified in (3.6) and (3.7) depending on g n .The B q denotes the q-oscillator algebra generated by b + , b − , t ±1 obeying the relations Finally the map ρ : B q → End(F ) is the representation (3.3) sending b + to the creation operator, b − to the annihilation operator and t ±1 to the (exponentiated) number operators which are concretely realized on the Fock space as (2.9).A key observation at this point is that B q admits the automorphisms (u ∈ C × is a parameter) In particular the first one interchanges the creation and the annihilation operators.Thus B q is endowed with two types of representations defined by ρ u for ε = 1, 3. Now we are ready to digest our result on the quantum group symmetry of the R matrices obtained by the new 2 n reductions.The associated quantum affine algebra U q (g n ) remains unchanged from the previous result [3,12,14].Namely g n = A (1) n−1 for the trace reduction and n+1 for the boundary vector reduction depending on the boundary vectors.On the other hand the relevant representations (1.8) are generalized to Here the essential data is the array ε = (ε 1 , . . ., ε n ) ∈ {1, 3} n which is determined as ε i = 1 or The parameters u = (u 1 , . . ., u n ) ∈ (C × ) n do not play a significant role.The R matrices enjoy the U q symmetry (1.6) in the representation π x,u (ε) ⊗ π y,u (ε).These results are summarized in Theorem 3.1 and Theorem 3.2, respectively.They include the previous ones [12,14] as the special case ε = (3, . . ., 3).The representation n−1 is a direct sum of finite dimensional ones if and only if ε = (1, . . ., 1) or (3, . . ., 3).The irreducible components contained in π tr z,u (1, . . ., 1) are the symmetric tensor representations corresponding to the Young diagrams that have a single row.Similarly the irreducible components contained in π tr z,u (3, . . ., 3) are their duals corresponding to the Young diagrams of rectangular shape with depth n − 1.In the language of the q-oscillators, they correspond to a system of particles or holes only.In this sense π z,u (ε) in (1.9) with general ε ∈ {1, 3} n is viewed as a q-oscillator representation mixing particles and holes.These degrees of freedom live on the vertices of the Dynkin diagram of g n and hop to the neighboring 'sites' according to the rules (3.11)-(3.22)via pair creation and annihilation.The algebra A n+1 describe the systems with various injection/ejection at the boundaries.Let us turn to the exceptional reduction involving only R ••• like the bottom right case of (1.7).We find that the trace reduction produces the R matrix with the U q A (1) n−1 symmetry (1.6) on A. Kuniba the representation π tr x,u (3, . . ., 3) ⊗ π tr y,u (1, . . ., 1).See Theorem 3.3 for the precise statement.This R matrix is known to be a basic ingredient in the box-ball system with reflecting end via its geometric and combinatorial counterparts [15, equation (2.3), Appendix A.3].Our result here establishes a matrix product formula of it for the first time.It will also be an essential input to the project [16,Section 6(iii)] on the recently proposed quantized reflection equation.As for the boundary vector reduction involving only R ••• , the corresponding solution to the Yang-Baxter equation is not locally finite (see the end of Section 2.4) and we have not found a quantum group symmetry.
The variants of the reductions introduced in this paper have essentially emerged from the three local forms of the tetrahedron equation (2.2), (2.22) and (2.23).Similar possibilities have been pursued extensively in [19] including fermionic degrees of freedom.The importance of the automorphism of the q-oscillator algebra and the appearance of infinite dimensional representations mixing particles and holes were recognized there.
There is also another generalization of the reduction method [13] to include the 3d L operator obeying the RLLL = LLLR relation [3].Section 2.8 of [13] provides a concise survey of the status.Combining these degrees of freedom in full generality is beyond the scope of this work.We believe nonetheless that the treatise in this paper will serve as a basic step toward a thorough understanding of the subject.
In Section 2 we recall the solution R to the tetrahedron equation and demonstrate the reduction procedures generalizing the previous ones.They lead to the solutions to the Yang-Baxter equation listed in the above table.Their basic properties are described.In particular the subspaces of F ⊗n ⊗ F ⊗n that are kept invariant under these solutions are extracted in (2.56)-(2.60)and the corresponding decompositions are listed in (2.61)-(2.66).
In Section 3 we recall the q-oscillator algebra B q , its automorphisms and the homomorphism from U q to B ⊗n q [5,14].They are combined to define the representations π tr z,u (ε) of U q A (1) n+1 , where the superscripts s, t of π s,t z,u (ε) correspond to the choices of the boundary vectors in (1.5).We describe the actions of the generators in these representations explicitly.Our main results in this section are Theorems 3.1, 3.2 and 3.3 which clarify the U q symmetry of the solutions to the Yang-Baxter equations constructed in Section 2 (except S s,t (2, . . ., 2|z)).The tensor product representation of U q corresponding to each summand in the decompositions (2.61)-(2.66) is irreducible for (2.61) with ε = (1, . . ., 1), (3, . . ., 3) and (2.64).In the other cases the irreducibility is yet to be investigated.
Throughout the paper we assume that q is generic and use the following notations: 2 Solutions of the Yang-Baxter equation

Tetrahedron equation and 3d
m| be a Fock space 4 and its dual equipped with the bilinear pairing where δ m,m = θ(m = m ).In this paper we will study the tetrahedron equation of the form where R lives in End F ⊗3 .The equality (2.2) holds in End F ⊗6 , where R 1,2,4 for example means the operator acting on the 1st, the 2nd and the 4th component in F ⊗6 from the left as R and identity elsewhere.
This solution was originally obtained5 as the intertwiner of the quantum coordinate ring A q (sl 3 ) [8].Later it also emerged from a quantum geometry consideration [3], and the two R's in these literatures were identified in [11, equation (2.29)].Here we simply call it the 3d R. It satisfies the following: The second property is refered to as the conservation law.The third one is due to [11, Proposition 2.4].We let R act also on For later use, we introduce the creation, annihilation and number operators on F , F * by ) where | − 1 = −1| = 0. Due to (2.1) they satisfy ( m|X)|m = m|(X|m ).By definition, the identity k = q 1 2 +h holds.The extra 1 2 here is the celebrated zero point energy, which makes the coefficients in (2.13)-(2.16)free from q totally6 .It is easy to check the q-oscillator relations: It is known that the 3d R is uniquely characterized (up to sign) as the involutive operator on F ⊗3 satisfying the following relations (cf.[3,8,17]): ) Here for example Thus operators with different indices are commutative.In this notation, (2.6) is rephrased as for generic parameters x and y.Introduce the vectors Up to normalization they are characterized by the relations The following equalities are known to hold for s = 1, 2 [17, Proposition 4.1]: (2.21)

2.4
Conservation laws of S tr (ε|z) and S s,t (ε|z) Let us investigate the consequence of the conservation law (2.6).For instance consider S tr (ε|z) a,b i,j with ε = (ε 1 , . . ., ε n ) ∈ {1, 3} n .From (2.6), (2.41) and (2.42) we have where k ∈ Z n .They are equivalent to The former means a + b = i + j ∈ Z n whereas the latter leads, by elimination of c 0 , . . ., c n−1 , to |b| ε = |j| ε in terms of the symbol defined by We say that S tr (ε|z) and S s,t (ε|z) are locally finite if the summands in r.h.s. in (2.39) and (2.40) are nonzero only for finitely many (a, b)'s for any given (i, j).The result (2.51)-(2.54)tells that they are locally finite except S s,t (2, . . ., 2|z).In any case, the matrix elements of the Yang-Baxter equations (2.35) and (2.36) for the prescribed transition |i ⊗|j ⊗|k → |a ⊗|b ⊗|c in F ⊗n ⊗ F ⊗n ⊗ F ⊗n consist of finitely many summands.

Matrix product operators
In order to calculate the matrix elements (2.42) and (2.43), it is useful to reformulate the 3d R (2.41) as a family of operators on the auxiliary Fock space.Here we provide such operators. 12 (2.68) The sum (2.67) is taken in the same manner as (2.4), and the sum (2.68) ranges over µ, ν ∈ Z ≥0 satisfying µ − ν = i − a and µ ≤ i.The operators R a,b i,j , Q a,b i,j have been designed so that the action of the 3d R (2.3) is expressed as These relations can be checked by using, for example, (a i,j was first introduced in [10, equation ( 8)].Now the elements of (2.32) and (2.33) are expressed as where the family of matrix product operators R (ε) a,b i,j ∈ End(F ) are specified by and χ s (z)|(a ± ) j k m (−1) h |χ t .These quantities are evaluated explicitly as follows (m ≥ 0): q 2j+2m+2 z 2 w 2 ; q 4 ∞ q 2m z 2 w 2 ; q 4 ∞ . (2.74) These formulas are easily derived by only using the elementary identity j≥0 (ξ; q) j (q; q) j η j = (ξη; q) ∞ (η; q) ∞ .
An essential consequence of these formulas are that the matrix elements S tr (ε|z) a,b i,j and S s,t (ε|z) a,b i,j become rational functions of z and q via appropriate choice of tr (ε|z) and s,t (ε|z).We will specify them explicitly in the next subsection.
According to (2.71) and (2.73) they are derived from 020,130  101,211 = tr (z) Tr where tr (z) = 1 + q 9 z, which is tr (2, 2, 2|z) (2.75) with n = 3, l = 2, m = 4. Let us illustrate the calculation of the top left example.In terms of the number operator without zero point energy k := q − 1 2 k, the relevant Q a,b i,j (2.68) are given by Tetrahedron Equation and Quantum R Matrices 15 Thus Tr Upon multiplication of tr (z) = 1 + q 9 z, this agrees with S(z) 002,112 101,211 .
3 Quantum R matrices

Quantum affine algebras
Let n be affine Kac-Moody algebras [7].The Ã(2) 2n is isomorphic to A 2n and their difference is only the enumeration of vertices.We keep it for uniformity of the description.The Drinfeld-Jimbo quantum affine algebras (without derivation operator) U q = U q (g tr n ), U q (g s,t n ) are the Hopf algebras generated by e i , f i , k ±1 i , 0 ≤ i ≤ n, satisfying the relations [4,6] where e The data (a ij ) 0≤i,j≤n is the Cartan matrix in the convention of [7].It is given by a i,j = 2δ i,j − max((log q j )/(log q i ), 1)(δ i,j+1 + δ i,j−1 ), where δ i,j = θ(i − j ∈ (n + 1)Z) for g tr n and δ i,j = θ(i = j) for g s,t n .The data q i in (3.1) are specified above the associated vertex i, 0 ≤ i ≤ n, in the Dynkin diagrams (see Fig. 1).For U q (g s,t ), we have q 0 = q s 2 /2 , q n = q t 2 /2 and q i = q, 0 < i < n.The coproduct ∆ has the form The opposite coproduct is denoted by ∆ op = P • ∆, where P (u ⊗ v) = v ⊗ u is the exchange of the components.
A. Kuniba a i,j = 2δ i,j − max((log q j )/(log q i ), 1)(δ i,j+1 + δ i,j−1 ), where δ i,j = θ(i − j ∈ (n + 1)Z) for g tr n and δ i,j = θ(i = j) for g s,t n .The data q i in (3.1) are specified above the associated vertex i (0 ≤ i ≤ n) in the Dynkin diagrams: For U q (g s,t ), we have q 0 = q s 2 /2 , q n = q t 2 /2 and q i = q (0 < i < n).The coproduct ∆ has the form The opposite coproduct is denoted by ∆ op = P • ∆, where P (u ⊗ v) = v ⊗ u is the exchange of the components.

q-oscillator algebra
Let B q be the algebra over C(q 2 ) generated by b + , b − , t and t −1 obeying the relations We call B q the q-oscillator algebra.Comparing it with (2.12) we see that the map provides a representation of B q on the Fock space F .

q-oscillator algebra
Let B q the algebra over C(q 2 ) generated by b + , b − , and −1 obeying the relations We call B q the q-oscillator algebra.Comparing it with (2.12) we see that the map provides a representation of B q on the Fock space F .The q-oscillator algebra admits families of automorphisms as where u ∈ C × , ν ∈ Z.The first family is notable in that it interchanges the creation and the annihilation operators.In this paper we will be concerned with their special cases: The compositions ρ u , ε = 1, 3, define irreducible representations B q → End(F q ).Explicitly they read ρ (1)  u 3.3 Homomorphism from U q to q-oscillator algebra (ε j , ε j+1 ) = (3, 1) : As is clear from (3.11), (3.12) and (2.50), the representation n−1 on V = F ⊗n decomposes into those on V l (ε).Each V l (ε) is irreducible for any ε ∈ {1, 3} n .It is finite dimensional if and only if ε is uniform, i.e., ε = (1, 1, . . ., 1) or (3, 3, . . ., 3).As a module over the classical subalgebra U q (A n−1 ), the V −l (1, 1, . . ., 1) with l ∈ Z ≥0 is equivalent to the degree-l symmetric tensor representation with highest weight vector |le 1 .It corresponds to the Young diagram of 1 × l row shape.The V l (3, 3, . . ., 3) with l ∈ Z ≥0 is equivalent to its dual, i.e., the degree-l symmetric tensor of the anti-vector representation with highest weight vector |le n .It corresponds to the Young diagram of (n − 1) × l rectangular shape.In these two cases of the uniform ε, one may regard the base vector |m ∈ V l (ε) as specifying a configuration of particles or holes on a ring Z n in terms of their occupation number m j at site j.The generators e j , f j in (3.11) and (3.14) represent 'ordinary' nearest neighbor hopping.In general the representation V l (ε) with a non-uniform ε ∈ {1, 3} n corresponds to the mixture of particles and holes.A site j accommodates only particles if ε j = 1 and only holes if ε j = 3.The base vector |m signifies the configuration in which there are m j particles (resp.holes) at site j if ε j = 1 (resp.ε j = 3).Then e j in (3.12) and (3.13) for example is interpreted as a particle hopping from the site j + 1 to j via pair creation and pair annihilation, respectively.

Representation π
Let us write down π s,t z,u (ε) (3.9) concretely for u = (u 1 , . . ., u n ) chosen in the same manner as (3.10).Since (3.6) and (3.7) are the same for 0 < j < n, the corresponding 'generic' generators e j , f j , k j are again given by Table 1 and described concretely as (3.11)- (3.14).The other 'exceptional' generators depend on the parameters u, u in (3.10) not only via the ratio but individually.Below we present them with the choice u = −dq −1 and u = 1 keeping (3.10).
The representation of e 0 , f 0 , k 0 are determined according to s = 1, 2 and ε 1 = 1, 3 as   A. Kuniba is valid, where x i,j = x i /x j and R i,j (x i,j ) acts on the ith and the jth components (from the left) of W x 1 ⊗ W x 2 ⊗ W x 3 as R(x i,j ) and identity elsewhere.We call the elements R satisfying (3.23)-(3.27)quantum R matrices.In short the U q symmetry serves as a characterization of a quantum R matrix up to the irreducibility of the relevant representations [4,6].
3.6 U q symmetry of S tr (ε|z) and S s,t (ε|z) Let us state the U q symmetry for the locally finite solutions to the Yang-Baxter equation S tr (ε|z) and S s,t (ε|z) in (2.61)-(2.64).We will be concerned with the spaces (2.56)-(2.60).We also assume z = x/y throughout this subsection.(I) S tr (ε|z) with ε ∈ {1, 3} n in (2.61).To recall this, see (2.32) for the matrix product construction, (2.51) for the weight conservation and (2.35) for the Yang-Baxter equation.As for the relevant representations W z and W z , we take the both to be π tr z,u (ε) : n−1 → End(V ) defined in (3.8).The parameters u ∈ (C × ) n are arbitrary and not restricted to (3.10).

4 F , 5 F , 6 F 4 F , 5 F , 6 F 4 F ⊗ 5 F ⊗ 6 F
) for the array of labels α = (α 1 , . . ., α n ).The notations β F and γ F should be understood similarly.The argument so far is just a 3d analogue of the simple fact in 2d that a single RLL = LLR relation for a local L operator implies a similar relation for the n-site monodromy matrix in the quantum inverse scattering method.In this terminology play the role of auxiliary spaces.Now we are going to eliminate R 4,5,6 by evaluating the auxiliary spaces away.There are two ways to do this.The first one is to multiply x h 4 (xy) h 5 y h 6 R −1 4,5,6 to (2.29) from the left and take the trace over .From (2.17) the result becomes Tr 4,5,6 x h 4 (xy) h 5 y h 6 P (ε 1 ) 1 • • • P (εn) n = Tr 4,5,6 x h 4 (xy) h 5 y h 6 P (ε 1 ) 1 • • • P (εn) n .