Solutions of the Yang-Baxter Equation with Extra Non-Additive Parameters II: Uq(gl(m|n))

The type-I quantum superalgebras are known to admit non-trivial one-parameter families of inequivalent finite dimensional irreps, even for generic q. We apply the recently developed technique to construct new solutions to the quantum Yang-Baxter equation associated with the one-parameter family of irreps of Uq(gl(m|n)), thus obtaining R-matrices which depend not only on a spectral parameter but in addition on further continuous parameters. These extra parameters enter the Yang-Baxter equation in a similar way to the spectral parameter but in a non-additive form. Supported by Habilitationsstipendium der Deutschen Forschungsgemeinschaft. On leave from Department of Physics, Bielefeld University, Germany. Address after Feb. 1, 1995: Yukawa Institute for Theoretical Physics, Kyoto University, Japan. Delius et al: Yang-Baxter Equation with Extra Non-Additive Parameters. II 1 In [1], we developed a systematic method for constructing R-matrices (solutions of the quantum Yang-Baxter equation (QYBE)) associated with the multiplicity-free tensor product of any two affinizable irreps of a quantum algebra. This approach was applied and extended to quantum superalgebras in [2, 3]. For the type-I quantum superalgebra Uq(gl(m|1)) in particular, we were able to obtain R-matrices depending continuously on extra parameters, entering in a similar way as the spectral parameter but in a nonadditive form. In this paper, we continue this study to construct new such R-matrices for the type-I quantum superalgebra Uq(gl(m|n)) for any m ≥ n. The freedom of having extra continuous parameters in R-matrices opens up new and exciting possibilities. For example, in [4], by using the R-matrix associated with the one-parameter family of 4-dimensional irreps of Uq(gl(2|1)), we derived a new exactly solvable lattice model of strongly correlated electrons on the unrestricted 4L-dimensional electronic Hilbert space ⊗n=1C 4 (where L is the lattice length), which is a gl(2|1) supersymmetric generalization of the Hubbard model with the Hubbard on-site interaction coupling coefficient related to the parameter carried by the 4-dimensional irrep. The origin of the extra parameters in our solutions are the parameters which are carried by the irreps themselves of the associated quantum superalgebra. As is well known [5], type-I superalgebras admit nontrivial one-parameter families of finite-dimensional irreps which deform to provide one-parameter families of finite-dimensional irreps of the corresponding type-I quantum superalgebras, for generic q [6]. Note however that for quantum simple bosonic Lie algebras families of finite-dimensional representations are possible only when the deformation parameter q is a root of unity. Therefore our solutions are not related to the chiral Potts model R-matrices which arises from quantum bosonic algebras at q a root of unity only [7, 8]. Let us have a brief review of our general formalism formulated in [2, 3]. Let G denote a simple Lie superalgebra of rank r with generators {ei, fi, hi} and let αi be its simple roots. Then the quantum superalgebra Uq(G) can be defined with the structure of a Z2-graded quasi-triangular Hopf algebra. We will not give the full defining relations of Uq(G) here but mention that Uq(G) has a coproduct structure given by ∆(qi) = qi ⊗ qi , ∆(a) = a⊗ qi + qi ⊗ a , a = ei, fi. (1) The multiplication rule for the tensor product is defined for elements a, b, c, d ∈ Uq(G) by (a⊗ b)(c⊗ d) = (−1)(ac⊗ bd) (2) where [a] ∈ Z2 denotes the degree of the element a. Let πα be a one-parameter family of irreps of Uq(G) afforded by the irreducible module V (Λα) in such a way that the highest weight of the irrep depends on the parameter α. Assume for any α that the irrep πα is affinizable, i.e. it can be extended to an irrep of the corresponding quantum affine superalgebra Uq(Ĝ). Consider an operator (R-matrix) R(x|α, β) ∈ End(V (Λα)⊗ V (Λβ)), Delius et al: Yang-Baxter Equation with Extra Non-Additive Parameters. II 2 where x ∈ C is the usual spectral parameter and πα, πβ are two irreps from the one-parameter family. It has been shown by Jimbo [9] that a solution to the linear equations R(x|α, β)∆(a) = ∆̄(a)R(x|α, β) , ∀a ∈ Uq(G), R(x|α, β) ( xπα(e0)⊗ πβ(q 0) + πα(q 0)⊗ πβ(e0) ) = ( xπα(e0)⊗ πβ(q 0) + πα(q 0)⊗ πβ(e0) ) R(x|α, β) (3) satisfies the QYBE in the tensor product module V (Λα)⊗ V (Λβ)⊗ V (Λγ) of three irreps from the one-parameter family: R12(x|α, β)R13(xy|α, γ)R23(y|β, γ) = R23(y|β, γ)R13(xy|α, γ)R12(x|α, β). (4) In the above, ∆̄ = T ·∆, with T the twist map defined by T (a⊗b) = (−1)[a][b]b⊗a , ∀a, b ∈ Uq(G) and ∆αβ(a) = (πα ⊗ πβ)∆(a); also, if R(x|α, β) = ∑ i πα(ai) ⊗ πβ(bi), then R12(x|α, β) = ∑ i πα(ai) ⊗ πβ(bi) ⊗ I etc. Jimbo also showed that the solution to (3) is unique, up to scalar functions. The multiplicative spectral parameter x can be transformed into an additive spectral parameter u by x = exp(u). In all our equations we implicitly use the “graded” multiplication rule of eq. (2). Thus the R-matrix of a quantum superalgebra satisfies a “graded” QYBE which, when written as an ordinary matrix equation, contains extra signs: (R(x|α, β)) j ij (R(xy|α, γ)) ik ik (R(y|β, γ)) jk jk (−1) [i][j]+[k][i]+[k][j] = (R(y|β, γ)) k jk (R(xy|α, γ)) ik ik (R(x|α, β)) ij ij (−1) ′′′. (5) However after a redefinition ( R̃(·|α, β) )i′j′ ij = (R(·|α, β)) j ij (−1) [i][j] (6) the signs disappear from the equation. Thus any solution of the “graded” QYBE arising from the R-matrix of a quantum superalgebra provides also a solution of the standard QYBE after the redefinition in eq. (6). Introduce the graded permutation operator Pαβ on the tensor product module V (Λα)⊗V (Λβ) such that P(vα ⊗ vβ) = (−1) vβ ⊗ vα , ∀vα ∈ V (Λα) , vβ ∈ V (Λβ) (7) and set Ř(x|α, β) = PR(x|α, β). (8) Then (3) can be rewritten as Ř(x|α, β)∆(a) = ∆(a)Ř(x|α, β) , ∀a ∈ Uq(G), Ř(x|α, β) ( xπα(e0)⊗ πβ(q 0) + πα(q 0)⊗ πβ(e0) ) = ( πβ(e0)⊗ πα(q 0) + xπβ(q 0)⊗ πα(e0) ) Ř(x|α, β) (9) Delius et al: Yang-Baxter Equation with Extra Non-Additive Parameters. II 3 and in terms of Ř(x|α, β) the QYBE becomes (I ⊗ Ř(x|α, β))(Ř(xy|α, γ) ⊗ I)(I ⊗ Ř(y|β, γ)) = (Ř(y|β, γ) ⊗ I)(I ⊗ Ř(xy|α, γ))(Ř(x|α, β) ⊗ I) (10) both sides of which act from V (Λα) ⊗ V (Λβ) ⊗ V (Λγ) to V (Λγ) ⊗ V (Λβ) ⊗ V (Λα). Note that this equation, if written in matrix form, does not have extra signs. This is because the definition of the graded permutation operator in eq. (7) includes the signs of eq. (6). In the following we will normalize the R-matrix Ř(x|α, β) in such a way that Ř(x|α, β)Ř(x−1|β, α) = I, which is usually called the unitarity condition in the literature. For three special values of x: x = 0, x = ∞ and x = 1, Ř(·|α, β) satisfies the spectral-free, but extra non-additive-parameter-dependent QYBE, (I ⊗ Ř(·|α, β))(Ř(·|α, γ) ⊗ I)(I ⊗ Ř(·|β, γ)) = (Ř(·|β, γ) ⊗ I)(I ⊗ Ř(·|α, γ)(Ř(·|α, β) ⊗ I). (11) In the case of a multiplicity-free tensor product decomposition V (Λα)⊗ V (Λβ) = ⊕