New Solutions to the Tetrahedron Equation Associated with Quantized Six-Vertex Models

We present a family of new solutions to the tetrahedron equation of the form RLLL=LLLR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$RLLL=LLLR$$\end{document}, where L operator may be regarded as a quantized six-vertex model whose Boltzmann weights are specific representations of the q-oscillator or q-Weyl algebras. When the three L’s are associated with the q-oscillator algebra, R coincides with the known intertwiner of the quantized coordinate ring Aq(sl3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_q(sl_3)$$\end{document}. On the other hand, L’s based on the q-Weyl algebra lead to new R’s whose elements are either factorized or expressed as a terminating q-hypergeometric type series.


Introduction
Tetrahedron equation [24] is a key to integrability for lattice models in statistical mechanics in three dimensions.Among its several versions and formulations, let us focus on the so-called RLLL relation: Indices here specify the tensor components on which the associated operators act non-trivially.When the spaces 4, 5, 6 are evaluated away appropriately, it reduces to the Yang-Baxter equation L 23 L 13 L 12 = L 12 L 13 L 23 [1].Thus (1) may be viewed as a quantization of the Yang-Baxter equation along the direction of the auxiliary spaces 4, 5 and 6.It has appeared in several guises and studied from various point of view.
In this paper we take the spaces 1, 2, 3 as V = C 2 and consider the three kinds of L operators: They all have the six-vertex model structure [1], i.e., weight conservation property, with respect to the component V ⊗ V .The last component is taken from specific representations π X , π Z of the q-Weyl algebra W q (6) on F = ⊕ m∈Z C|m or π O of the q-oscillator algebra O q (10) on F + = ⊕ m∈Z ≥0 C|m .In short, these L operators may be viewed as quantized six-vertex models whose Boltzmann weights are End(F ) or End(F + )-valued.They naturally lead to the generalizations of (1) to where A,B and C can be any one of Z, X and O. Let us temporarily call it the RLLL relation of type ABC.
The main result of this paper is the explicit solution R for types ZZZ, OZZ, ZZO, ZOZ, OOZ, ZOO, OZO, OOO, XXZ, ZXX and XZX.They turn out to be unique up to normalization in each sector specified by a parity condition in an appropriate sense.Elements of R are either factorized or expressed as a terminating q-hypergeometric type series.See Table 1 in Section 6 for a summary.They are new except for type OOO, where the RLLL relation [3] is equivalent (cf.Section 5.2 and [14, Lem 3.22]) with the intertwining relation of the quantized coordinate ring A q (sl 3 ), and the R coincides with the intertwiner obtained in [9].We will show a similar link to A q (sl 3 ) also for type ZZZ in Proposition 16.
The representations π Z and π X of the q-Weyl algebra XZ = qZX are natural ones in which Z and X become diagonal, respectively.See (8) and (9).They are q-analogue of the coordinate and the momentum representations of the canonical commutation relation, which are formally interchanged via a q-difference analogue of the Fourier transformation.The representation π O is a restriction of the special case of π X as explained around (12).One of our motivation is to investigate systematically how these L operators, including their mixtures, lead to a variety of solutions R for the associated RLLL relation.The new R's obtained in this paper will be important inputs to many interesting future problems which will be discussed in the last section.
The layout of the paper is as follows.In Section 2, the L operators L Z , L X associated with the q-Weyl algebra and L O for the q-oscillator algebra are introduced.L O is a restriction of L X , and appeared in the earlier works [3,5,18,17,23].The RLLL relation is formulated.In Section 3 and 4, the solutions R are presented for the choices L = L Z , L O and L = L Z , L X , respectively.Some results in the former case can be reproduced as a limit of the latter.In Section 5, a connection to the representation theory of A q (sl 3 ) is explained.A new result is Proposition 16.Section 6 contains a summary and discussion on the tetrahedron equation of the form RRRR = RRRR.Conjecture 17 is promising.Appendix A provides the list of explicit forms of the RLLL relation for type ZZZ.

Quantized six-vertex models
We assume that q is generic throughout the paper.
2.1.q-Weyl algebra W q and q-oscillator algebra O q .Let W q be the q-Weyl algebra, which is an associative algebra with generators X ±1 , Z ±1 obeying the relation and those following from the obvious ones XX −1 = X −1 X = ZZ −1 = Z −1 Z = 1.Introduce the infinite dimensional vector spaces 1 : C|m .
The algebra W q has irreducible representations π Z (resp.π X ) on F where Z (resp.X) is diagonal: They are q-analogue of the "coordinate" and the "momentum" representations of the canonical commutation relation.
Let O q be the q-oscillator algebra, which is an associative algebra with generators a + , a − , k obeying the relation There is an embedding ι : O q ֒→ W q given by ι The composition O q ι ֒→ W q πX −→ End(F ) yields the representation: Due to a − |0 = 0, the subspace F + ⊂ F becomes invariant and irreducible.We let π O : O q → End(F + ) denote the resulting irreducible representation obtained by restricting (12) to m ≥ 0.
1 The actual coefficient field will contain many parameters introduced subsequently including q.
2.2.3D L operator.Let V = Cv 0 ⊕ Cv 1 be the two dimensional vector space.We consider q-Weyl algebravalued L operator L 00 00 = r, L 11 11 = s, L 10 10 = twX, Here r, s, t, w are parameters whose dependence has been suppressed in the notation L ab ij .They are assumed to be generic throughout.The symbol E ij denotes the matrix unit on V acting on the basis as The L operator L may be viewed as a quantized six-vertex model where the Boltzmann weights are W q -valued.See Figure 1 for a graphical representation.
Figure 1.L = L r,s,t,w as a W q -valued six-vertex model.Assigning another perpendicular arrow corresponding to the W q -modules leads to a unit of the three dimensional (3D) lattice.
In this context, L will also be called the 3D L operator.
Note that L does not contain X −1 , which will be a key in Remark 1 below.Although t can be absorbed into the normalization of X, we keep it for convenience.It is easy to see For the special choice of the parameters (r, s, t, w) = (1, 1, µ −1 , µ 2 ), L only contains the combinations appearing in the RHS of (11) which can be pulled back to the q-oscillator algebra.Therefore we regard it as O q -valued, i.e., Its elements are given by See Figure 2. Now we introduce the three types of (represented) L operators: From ( 16) and (17) we have Remark 1.The operator L Z in (19) keeps the subspace V ⊗ V ⊗ m≤n C|m ⊂ F invariant for any n ∈ Z.
We also call it RLLL relation.The indices denote the tensor components on which the respective operators act non-trivially.The operator L will be taken as L Z , L X or L O in ( 19)- (21).The conjugation operator R, which we call 3D R in this paper, will be the main object of study in what follows.In terms of the components of L, (23) reads as for arbitrary a, b, c, i, j, k ∈ {0, 1}.See Figure 3. From the conservation condition (14), the equation ( 24) becomes 0 = 0 unless a+b+c = i+j +k.There are 20 choices of (a, b, c, i, j, k) ∈ {0, 1} 6 satisfying it.Among them, the cases (0, 0, 0, 0, 0, 0) and (1, 1, 1, 1, 1, 1) yield the trivial relation they are translated into linear recursion relations on the matrix elements R a,b,c i,j,k .We say that R is locally finite if the sum (25) consists of finitely many terms, i.e., R a,b,c i,j,k = 0 for all but finitely many (a, b, c) for any given (i, j, k).

Solutions of RLLL relation for L = L Z and L O
In this section we treat the cases in which L 124 , L 135 and L 236 are chosen as L Z or L O independently.It turns out that they always admit a unique R up to normalization in a sector specified by appropriate parity conditions.Their explicit forms will be presented case by case.We write the characteristic function as θ(true) = 1, θ(false) = 0, δ a b = θ(a = b) and use the following notation: The above convention for (z; q) m valid for any m ∈ Z is standard and essential in the working below.In particular 1/(q; q) a = 0 for a ∈ Z <0 , and we will freely use (z; q) m = 1/(zq m ; q) −m and (z; q) m /(z; q) n = (zq n ; q) m−n , etc.The q-binomial n m q is zero unless 0 ≤ m ≤ n.The q-hypergeometric series will always appear in the terminating situation, i.e., α or β ∈ q Z ≤0 .
Given Proof.Claim (i) can be checked directly.Let us prove Claim (ii).First, we reduce b, c and k to 0 by using (34) and (36).The result reads Applying this to (37) and ( 38) with b = c = k = 0 we get Eliminating R a,0,0 i+1,j,0 here leads to the recursion relation We remark that combination of (39) and (42) allows one to express R a,b,c i,j,k in terms of R 0,0,0 Next, consider (35) and (37) again with a = b = c = k = 0. Reducing them to the relations among R 0,0,0 •,•,0 by the above remark, and taking a suitable combination, we get Thus we find any R a,b,c i,j,k is uniquely expressed as R 0,0,0 p2,p1,0 times known factors, where p 1 , p 2 ∈ {0, 1} are determined by where d 1 and d 2 are the same as those in Proposition 2. It is easy to see ϕ ∈ Z+(d 1 −1)d 2 /2.The dependence on t 1 , t 2 , t 3 is actually by the combination t −a+i

3
, which corresponds to a similarity transformation.By Proposition 2, we know that the solution R of (28), if exists, is unique up to normalization in each sector specified by (d 1 , d 2 ) mod 2 .The following result establishes the existence together with an explicit form.Proof.From Proposition 2 and changes the individual recursion relations only by an overall scalar.The results become the relations among finitely many rational functions.To check them is straightforward.
Similar decompositions according to a parity condition also take place in the forthcoming Theorems 9, 10, 11, 13, 14 and 15.
3.2.OZZ type.We consider the RLLL relation where L Z 135 and L Z 236 are given by ( 19) with (r, s, t, w) = (r 2 , s 2 , t 2 , w 2 ) and (r 3 , s 3 , t 3 , w 3 ), respectively.In this case, R ∈ End(F + ⊗ F ⊗ F ) and the sum (25) extends over a ∈ Z ≥0 and b, c ∈ Z.The equality (57) Here are some examples of the RLLL relation (57): The boundary condition has to be taken into account.Thus for example when a = 0, (60) is to be understood as For the convenience of the proof of Theorem 5, we have enlarged the range of the indices a and i from Z ≥0 to Z.The property (62) is satisfied thanks to the factor i β q 2 /(q 2 ; q 2 ) a .The formula (63) is also presented as a terminating q-hypergeometric series: Theorem 5.The RLLL relation (57) has a unique solution R up to normalization.It is given by ( 63)-(65).
Proof.The first claim, i.e., uniqueness, can be shown by an argument similar to Proposition 2. To prove the second claim, let S a i,j−b,k−c (x, y) denote the second line of (63).One sees that S a i,j,k (x, y) , where S a i,j,k,β (x) = q β(β+2j−1) i β q 2 (xq 2k−2β+2 ; q 2 ) a is a polynomial in x and y.The equation ( 57) is reduced to the recursion relations among S a i,j,k (x, y) with coefficients including q, q a , q i , q j , q k , x, y only.By picking the coefficients of y β , they are reduced to the relations containing finitely many S a i,j,k,β (x)'s.To check them is straightforward.This proves the recursion relations for generic a and i.This fact together with (62) assure that they are also valid in the vicinity of a = 0 and i = 0.
As for the last point of the proof, a similar and more detailed explanation is available in the proof of Theorem 9.The R is not locally finite.
3.3.ZZO type.We consider the RLLL relation where L Z 124 and L Z 135 are given by ( 19) with (r, s, t, w) = (r 1 , s 1 , t 1 , w 1 ) and (r 2 , s 2 , t 2 , w 2 ), respectively.In this case, R ∈ End(F ⊗ F ⊗ F + ) and the sum (25) extends over a, b ∈ Z and c ∈ Z ≥0 .The equality (66) Here are some examples of the RLLL relation (66): One has the boundary condition analogous to (62): where we have redefined x, y, z changing (64).It is also presented as a terminating q-hypergeometric series: Theorem 6.The RLLL relation (66) has a unique solution R up to normalization.It is given by ( 72)-(74).
The proof is similar to Theorem 5.The R is not locally finite.
The proof is similar to Theorem 5.The R is not locally finite.
3.5.OOZ type.We consider the RLLL relation where L O 124 and L O 135 are given by ( 21) with µ = µ 1 and µ 2 , respectively, and L Z 236 is given by ( 19) with (r, s, t, w) = (r 3 , s 3 , t 3 , w 3 ).In this case, R ∈ End(F + ⊗ F + ⊗ F ) and the sum (25) extends over a, b ∈ Z ≥0 and c ∈ Z.The equality (84 Here are some examples of the RLLL relation (84): As these examples indicate, every recursion relation consists of those R a,b,c i,j,k having the same parity of a − c + j + k.
For a, b, c, i, j, k, d ∈ Z, set bk (q 2+2e−2j ; q 2 ) j (q 2a+2 ; q 2 ) i−a (q 2 ; q 2 ) f (q 2a−2e ; q 2 ) e−a , For the convenience of the proof of Theorem 9, we have defined R(d) a,b,c i,j,k enlarging the range of the indices a, b, i, j from Z ≥0 to Z.We note that R(d) a,b,c i,j,k = q bd R(0) a,b,c i,j,k+d and θ(e ∈ Z)δ a+b i+j = θ(f ∈ Z)δ a+b i+j since e + f = i + j ∈ Z holds when a + b = i + j.The combinations e and f can be either positive or negative.Proof.The assertion is obvious if min(i, j) < 0. Thus we are to show that min(a, b) < 0 leads to R(d) a,b,c i,j,k = 0 assuming that min(i, j) ≥ 0. Suppose a < 0. Then (90) indeed vanishes due to (q 2a+2 ; q 2 ) i−a = (q 2a+2 ; q 2 ) ∞ /(q 2i+2 ; q 2 ) ∞ = 0. Suppose b < 0. We may further concentrate on the non-trivial case e ≥ a since otherwise 1/(q 2a−2e ; q 2 ) e−a = 0. Then 1/(q 2 ; q 2 ) f = 0 because of f = i + j − e = (a − e) + b < 0. When a, b, i, j ≥ 0, R(d) a,b,c i,j,k is divergence-free and R(d) a,b,c i,j,k = 0 unless e ≥ max(a, j) and f ≥ 0. From these conditions it follows that Theorem 9.The RLLL relation (84) has a non-trivial solution if and only if d := log q µ1 µ2 ∈ Z. Up to overall normalization it is given by R a,b,c i,j,k = R(d) a,b,c i,j,k specified by (90) and (91).Proof.The only if part of the first claim can be shown by an argument similar to Proposition 2. To show the rest, one first checks that the formula (90) satisfies the recursion relation when a, b, c, i, j, k are generic, i.e., when θ(min(i, j) ≥ 0) = 1.This can be done easily since (90) is factorized.The remaining task is to verify the boundary condition (92) to assure that the contribution from the "unwanted terms" to the recursion relation is zero.This has been guaranteed by Lemma 8.For example in (88) at b = 0, i.e., the first term is unwanted.
3.6.ZOO type.We consider the RLLL relation where L O 135 and L O 236 are given by ( 21) with µ = µ 2 and µ 3 , respectively, and L Z 124 is given by ( 19) with (r, s, t, w) = (r 1 , s 1 , t 1 , w 1 ).In this case, R ∈ End(F ⊗ F + ⊗ F + ) and the sum (25) extends over a ∈ Z and b, c ∈ Z ≥0 .The equality (96) holds in End( Here are some examples of the RLLL relation ( 96): As these examples indicate, every recursion relation consists of those R a,b,c i,j,k having the same parity of −a + c + i + j.The boundary condition is given by bi (q 2+2e−2j ; q 2 ) j (q 2+2c ; q 2 ) k−c (q 2 ; q 2 ) f (q 2c−2e ; q 2 ) e−c , We note that R(d) a,b,c i,j,k = q −bd R(0) a,b,c i−d,j,k and θ(e ∈ Z)δ b+c j+k = θ(f ∈ Z)δ b+c j+k since e + f = j + k ∈ Z when b + c = j + k.The combinations e and f can be either positive or negative.From b, c, j, k ≥ 0 and the definition (26), R abc ijk (d) is divergence-free and R(d) a,b,c i,j,k = 0 unless e ≥ max(c, j) and f ≥ 0. From these conditions it follows that Theorem 10.The RLLL relation (96) has a non-trivial solution if and only if d := log q µ3 µ2 ∈ Z. Up to overall normalization it is given by R a,b,c i,j,k = R(d) a,b,c i,j,k specified by ( 103) and ( 104).The proof is similar to Theorem 9. From ( 103) and (105), R is locally finite.From (22), its inverse is given by where the normalization has been deduced from R a,b,c 0,0,0 3.7.OZO type.We consider the RLLL relation where L Z 135 is given by ( 19) with (r, s, t, w) = (r 2 , s 2 , t 2 , w 2 ), and L O 124 and L O 236 are given by ( 21) with µ = µ 1 and µ 3 , respectively.In this case, R ∈ End(F + ⊗ F ⊗ F + ) and the sum (25) extends over a, c ∈ Z ≥0 and b ∈ Z.The equality (107 Here are some examples of the RLLL relation (107): As these examples indicate, every recursion relation consists of those R a,b,c i,j,k having the same parity of a We note that R(d) a,b,c i,j,k = q −dk R(0) a,b+d,j i,j,k The combinations e and f can be either positive or negative.From a, c, i, k ≥ 0 and the definition (26), R(d) a,b,c i,j,k is divergence-free and R(d) a,b,c i,j,k = 0 unless e ≥ max(i, k) and f ≥ 0. From these conditions it follows that Theorem 11.The RLLL relation (107) has a non-trivial solution if and only if d := log q − µ1 µ3 ∈ Z. Up to overall normalization it is given by R a,b,c i,j,k = R(d) a,b,c i,j,k specified by ( 114) and (115).The proof is similar to Theorem 9. R is not locally finite.
When q is a primitive root of unity of odd degree N ≥ 3, it follows that R a,b,c i,j,k = 0 if max(i, j, k) ≥ N and max(a, b, c) < N .It implies that the subspace i,j,k≥0,max(i,j,k)≥N C|i ⊗|j ⊗|k ⊂ F + ⊗F + ⊗F + is invariant under R.This fact was originally shown in [11,Th.2.2.1b] for each tensor component by resorting to the recursion relations, where an important consequence on the quotient was also pointed out in Proposition 2.3.2 therein.The above proof based on (120) is an illuminating simplification and has a natural generalization to the quantized coordinate rings of other types [14, eqs.(5.75), (8.39)].

Solutions of RLLL relation for
In this section we deal with the RLLL relations which contain L Z (19) and L X (20).As mentioned after (15), the parameters r, s, t, w are assumed to be generic, hence the boundary conditions like (62), ( 71), (80), (92), ( 102) and (113) need not be considered.We shall only treat the types XXZ, ZXX and XZX, and leave ZZX, XZZ, ZXZ and XXX cases for future study as they are considerably more complicated.Throughout the section, R ∈ End(F ⊗ F ⊗ F ) with the sum (25) extending over a, b, c ∈ Z, and the RLLL relation holds 4.1.XXZ type.We consider the RLLL relation where L X 124 and L X 135 are given by ( 20) with (r, s, t, w) = (r 1 , s 1 , t 1 , w 1 ) and (r 2 , s 2 , t 2 , w 2 ), respectively, and 236 is given by ( 19) with (r, s, t, w) = (r 3 , s 3 , t 3 , w 3 ).
Theorem 13.Recursion relations derived from (121) consists of only those R a,b,c i,j,k ' s having the same parity of a − c + j + k.Each subsystem specified by h admits a unique solution up to normalization, which is given by ( 127)-(128).
Proof.The former assertion on the parity can be verified directly.Solving a partial set of recursion relations already leads to (127)-(128), proving the uniqueness.Then it is straightforward to check that it actually satisfies all the remaining recursion relations.R is not locally finite.Let us compare the 3D R (127) for XXZ with (90) for OOZ.To fit L X in (121) to L O , we specialize the parameters as Using the notation e, f in (91), where g = e − 1 2 (h + d), and assuming d ∈ Z is so chosen that e ∈ Z, this is rewritten as Note that the condition e ∈ Z is equivalent to h + d ∈ 2Z.Therefore, if R 0,0,0 0,0,h (h = 0, 1) are taken as in the limit µ 1 → µ 2 q d , the 3D R (90) for the OOZ case is formally reproduced.
Theorem 14. Recursion relations derived from (133) consists of only those R a,b,c i,j,k ' s having the same parity of −a + c + i + j.Each subsystem specified by h admits a unique solution up to normalization, which is given by ( 139)-(140).
The proof is similar to Theorem 13.R is not locally finite.As the XXZ type, by specializing the parameters as (129) with i = 2, 3 and taking the limit µ 3 → µ 2 q d with appropriate tuning of R 0,0,0 h,0,0 (h = 0, 1), one can reproduce the 3D R (103) for ZOO from (139).
Theorem 15.Recursion relations derived from (141) consists of only those R a,b,c i,j,k ' s having the same parity of −b + i + j + k.Each subsystem specified by h admits a unique solution up to normalization, which is given by ( 147)-(148).

5.
Relation to the representation theory of the quantized coordinate ring.
5.1.Quantized coordinate ring A q (sl 3 ).The algebra A q (sl 3 ) is a Hopf algebra dual to the quantized universal enveloping algebra U q (sl 3 ).See for example [8,20,10,6,13] and the references therein.It is generated by t ij (1 ≤ i, j ≤ 3) with the relations [t ik , t jl ] = 0 (i < j, k > l), (q − q −1 )t jk t il (i < j, k < l), (152) where S 3 denotes the symmetric group of degree 3 and l(σ) is the length of the permutation σ.The coproduct ∆ : A q (sl 3 ) → A q (sl 3 ) ⊗N is given by the matrix product form ∆t The following maps define the algebra homomorphisms to the q-Weyl algebra (6): Here, u i , g i , h i are arbitrary parameters.We let ρ Z,i = π Z • ρ i and ρ X,i = π X • ρ i denote the representations A q (sl 3 ) → End(F ) obtained by the compositions with π Z and π X in ( 8) and (9).5.2.3D R of type OOO as an intertwiner of ρ O,i .From the remark after ( 16), one can restrict ρ X,i with (u i , g i , h i ) = (1, µ i , µ −1 i ) from End(F ) to End(F + ).The resulting representation will be denoted by ρ O,i : A q (sl 3 ) → End(F + ).

3D R of type ZZZ as an intertwiner of ρ
which includes the parameters u i , g i , h i (i = 1, 2).On the other hand, recall that the RLLL relation (28) of ZZZ type contains 18 equations depending on r α , s α , t α , w α (α = 1, 2, 3).Now we state a result analogous to the OOO case in the previous subsection.
Proposition 16.The intertwining relation (158) and the RLLL relation (28) are equivalent provided that the parameters in the former obey the constraint u 1 = u 2 (=: u) and g 1 h 1 = g 2 h 2 (=: p), and those in the latter satisfy where the constants A lm , B lm are given by The correspondence between the indices l, m and a, b, c, i, j, k, a ′ , b ′ , c ′ , i ′ , j ′ , k ′ is specified as follows:   21) which can be regarded as quantized six-vertex models with Boltzmann weights taken from the q-Weyl algebra W q (6) or the q-oscillator algebra O q (10).In each case the solution R has been obtained explicitly whose elements are factorized or expressed in terms of terminating q-hypergeometric type series as in Table 236 R 456 and the basic feature of the solution R = R ABC .We observe the factorization when the number ♯(Z) of Z in ABC is odd.♯(sector) is the dimension of the solution space for the recursion relations of R a,b,c i,j,k .
6.2.On tetrahedron equation of the form RRRR = RRRR.Let us discuss the tetrahedron equation of the form A standard strategy for the proof is to compare the two maneuvers: The underlines indicate the components to be rewritten by the RLLL = LLLR relation or trivial commutativity of the operators acting on distinct set of components.The above relations show that the composition (R 124 R 135 R 236 R 456 ) −1 R 456 R 236 R 135 R 124 commutes with L αβ6 L αγ5 L βγ4 L αδ3 L βδ2 L γδ1 .Therefore if the action of the latter is irreducible, Schur's lemma compels R 124 R 135 R 236 R 456 = (scalar)R 456 R 236 R 135 R 124 and the scalar can be fixed by considering the special case.
In this type of argument, RLLL = LLLR serves as an auxiliary linear problem for RRRR = RRRR, which is analogous to the quantum group symmetry ensuring the Yang-Baxter equation.It indeed works when all the L's are L O , where RLLL = LLLR is identified with the intertwining relation of the quantized coordinate ring A q (sl 3 ).See Section 5.2.The corresponding 3D R of type OOO (118) certainly satisfies the tetrahedron equation R 124 R 135 R 236 R 456 = R 456 R 236 R 135 R 124 [9,3].
The results in this paper suggest a natural generalization where the six L operators in (165) and (166) are taken either as L Z or L O (resp.L Z or L X ) in the context of Section 3 (resp.Section 4).Let us exhibit Figure 2. L = L 1,1,µ −1 ,µ 2 as an O q -valued six-vertex model.The last two relations in (10) is a quantization of the free Fermion condition [1, Fig. 10.1, eq.(10.16.4)| ω7=ω8=0 ].
two integers d and d ′ , we write the pair (d mod 2, d ′ mod 2) ∈ Z 2 × Z 2 simply as (d, d ′ ) mod 2 .Proposition 2. (i) Any recursion relation consists of only those R a,b,c i,j,k 's having the same parity pair (d 1 , d 2 ) mod 2 , where d 1 = a + c − j and d 2 = b − i − k. (ii) Each subsystem of recursion relations corresponding to a given (d 1 , d 2 ) mod 2 allows a solution of dimension at most one.

1 γR.
↔sγ ,tγ →tγ wγ ,wγ →w −1γ = L a ′ b ′ c ′ i ′ j ′ k ′ rγ↔sγ ,tγ →tγ wγ ,wγ →w −This equality follows from (158) by applying the relations (160) and (161) including B lm .6. Discussion 6.1.Summary.In this paper we have studied the tetrahedron equation of the form R 456 L C 236 L B 135 L A 124 = L A 124 L B 135 L C 236 R 456 for the three kinds of 3D L operators L Z , L X , L O in (19)-( 2.3.RLLL relation.Quantized six-vertex model satisfies the quantized Yang-Baxter equation.It is a version of the tetrahedron equation having the form of the Yang-Baxter equation up to conjugation: R 456 L 236 L 135 L 124 = L 124 L 135 L 236 R 456 .
t 11 t 12 t 13 t 21 t 22 t 23 t 31 t 32 t 33 t 11 t 12 t 13 t 21 t 22 t 23 t 31 t 32 t 33 ) Then one can directly check that the relations (159) validate the equalities ijk R.

Table 1 .
1.They are new except for the OOO case.Type ABC of R 456 L C 236 L B 135 L A 124 = L A 124 L B 135 L C