Super Yangian Double and its Central Extension

We introduce super Yangian double DYh̄[gl(m|n)] and its central extension ̂ DYh̄[gl(m|n)]. We give their defining relations in terms of current generators and obtain Drinfeld comultiplication. Queen Elizabeth II Fellow; Email: [email protected] This paper concerns with Drinfeld current realization [1] of super Yangian double DYh̄[gl(m|n)] and its central extension ̂ DYh̄[gl(m|n)]. The Yangian double [2] DYh̄(G) of a simple bosonic Lie algebra G is a quantum double of the Yangian Yh̄(G) [1]. It is a deformation of the entire loop algebra and has important applications in massive integrable models [3, 4]. The Yangian double with center (or central extension of the Yangian double) ̂ DYh̄(G) for G = gl(n), sl(n) were introduced in [5, 6] in terms of Drinfeld current generators. The philosophy behind this paper is to introduce super Yangian double DYh̄[gl(m|n)] and its central extension ̂ DYh̄[gl(m|n)]. This is achieved by generalizing the Reshetikhin and Semenov-Tian-Shansky (RS) construction [7] to the supersymmetric case. Using this super RS construction and Gauss decomposition [8], we obtain the defining relations for ̂ DYh̄[gl(m|n)] in terms of super current generators. The computation in this paper is parallel to our recent work [9] on Drinfeld current realization of quantum affine superalgebra Uq[gl(m|n) ] (see also [10] for the special case of m = n = 1), which in some sense is a superization of work [11]. The graded Yang-Baxter equation (YBE) with spectral-parameter dependence takes the form R12(u− v)R13(u)R23(v) = R23(v)R13(u)R12(u− v), (1) where R(u) ∈ End(V ⊗ V ) with V being graded vector space and obeys the weight conservation condition: R(u) β αβ 6= 0 only when [α ] + [β ] + [α] + [β] = 0 mod2. The multiplication rule for the tensor product is defined for homogeneous elements a, b, c, d of a quantum superalgebra by (a⊗ b)(c⊗ d) = (−1) (ac⊗ bd) (2) where [a] ∈ Z2 denotes the grading of the element a. Introduce the graded permutation operator P on the tensor product module V ⊗ V such that P (vα ⊗ vβ) = (−1) (vβ ⊗ vα) , ∀vα, vβ ∈ V . In most cases R-matrix enjoys, among others, the following properties (i) P12R12(u)P12 = R21(u), (3) (ii) R12(u)R21(−u) = 1. (4) The graded YBE, when written in matrix form, carries extra signs [13, 12], R(u− v) β αβ R(u) αγ αγ R(v) βγ βγ (−1) [α][β]+[γ][α]+[γ][β] = R(v) γ βγ R(u) αγ αγ R(u− v) αβ αβ (−1) ′′′. (5)