Murphy elements from the double-row transfer matrix

arXiv:0812.0898v3 [math-ph] 10 Mar 2009 Murphy elements from the double-row transfer matrix Anastasia Doikou1 University of Patras, Department of Engineering Sciences, GR-26500 Patras, Greece Abstract We consider the double-row (open) transfer matrix constructed from generic tensor-type representations of various types of Hecke algebras. For different choices of boundary conditions for the relevant integrable lattice model we express the double-row transfer matrix solely in terms of generators of the corresponding Hecke algebra (tensor-type realizations). We then expand the open transfer matrix and extract the associated Murphy elements from the first/last terms of the expansion. Suitable combinations of the Murphy elements as has been shown commute with the corresponding Hecke algebra. Keywords: algebraic structures of integrable models, integrable spin chains (vertex models), solvable lattice models 1adoikou@upatras.gr 1 Introduction There has been much activity lately associated to algebraic structures underlying inte- grable lattice models. On the one hand there is an immediate connection between these models and realizations of the braid group [1]–[7], given that spin chain models may be constructed as tensorial representations of quotients of the braid group called Hecke al- gebras. On the other hand integrable lattice models provide perhaps the most natural framework for the study of quantum groups [8, 9]. The symmetry algebras underlying these models may be seen as deformations of the usual Lie algebras [1, 10], and their defining relations emanate directly from the fundamental relations ruling such models, that is the Yang-Baxter [11] and reflection equations [12]. Several studies have been devoted on the uncovering of the symmetries of open spin chain models as well as on connecting the associated Hecke algebras with the underlying quantum group symmetries, and in most cases it turns out that the exact symmetries – quantum algebras– commute with the Hecke algebra (see e.g. [13, 14, 4]). In the spin chain context the transfer matrices may be usually expressed in terms of the quantum algebra elements in a universal manner, i.e. independent of the choice of representation of the quantum algebra. However, such generic expressions in terms of Hecke algebra elements are missing, with the exception of generic formulas of integrable Hamiltonians (see e.g. [3, 4] for computational details). In the present investigation we provide generic expressions, of double-row transfer matrices [15] in terms of generators of Hecke algebras (tensor type representations). It is worth noting that such generic expressions starting from Sklyanin’s transfer matrix [15] offer an immediate link between spin chain like systems and other integrable lattice models such as Potts models and in general face type models [16, 17, 18]. Having such expressions at our disposal we are then able to extract from the double-row transfer matrix the so-called Murphy elements, which commute with the Hecke algebras (see [19, 20] and references therein). The outline of this paper is as follows. In the next section we give basic definitions regarding the A, B and C-type Hecke algebras. We also define the Murphy elements associated to each one of the aforementioned Hecke algebras. In section 3 starting from the double-row transfer matrix [15] we end up with generic formulas expressed in terms of generators of Hecke algebras (tensor representations). We finally prove that the Murphy elements are directly obtained from suitable double-row transfer matrices of varying di- mension. In the last section we briefly discuss the findings of this study, and also propose possible directions for future investigations. 1 2 Hecke algebras: definitions We shall review in this section basic definitions regarding various types of Hecke algebras, and the associated Murphy elements (see also [20]–[28]). Definition 2.1. The A-type Hecke algebra HN(q) is defined by the generators gl, l = 1, . . . , N −1 satisfying the following relations: gl gl+1 gl = gl+1 gl gl+1, (2.1) h gl, gm i = 0, |l −m| > 1 (2.2) (gl −q) (gl + q−1) = 0. (2.3) Definition 2.2. The B-type Hecke algebra BN(q, Q0) is defined by generators gl, l ∈ {1, . . . , N −1}, satisfying the Hecke relations (2.1)-(2.3) and g0 obeying: g1 g0 g1 g0 = g0 g1 g0 g1, (2.4) h g0, gl i = 0, l > 1 (2.5) (g0 −Q0) (g0 + Q−1 0 ) = 0. (2.6) The algebra above is apparently an extension of the Hecke algebra defined in (2.3). Also the B-type Hecke algebra is a quotient of the affine Hecke algebra, which is defined by generators gi, g0 that satisfy (2.1)-(2.5). Definition 2.3. The C-type Hecke algebra CN(q, Q0, QN), is defined by the generators gl, l ∈{1, . . . , N −1}, g0 satisfying (2.1)-(2.6) and an extra generator gN, obeying gN gN−1 gN gN−1 = gN−1 gN gN−1 gN (2.7) h gN, gi i , 0 ≤i ≤N −2 (2.8) (gN −QN) (gN + Q−1 N ) = 0. (2.9) There is also a quotient of the C-type Hecke algebra called the two boundary Temperley- Lieb algebra [29]–[32], [2, 3]

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