Representations of centrally extended Lie superalgebra $\mathfrak {psl}(2|2)$psl(2|2)

Abstract

The symmetries provided by representations of the centrally extended Lie superalgebra psl(2|2) are known to play an important role in the spin chain models originated in the planar anti-de Sitter/conformal field theory correspondence and onedimensional Hubbard model. We give a complete description of finite-dimensional irreducible representations of this superalgebra thus extending the work of Beisert which deals with a generic family of representations. Our description includes a new class of modules with degenerate eigenvalues of the central elements. Moreover, we construct explicit bases in all irreducible representations by applying the techniques of Mickelsson-Zhelobenko algebras.As discovered by Beisert [1, 2, 3], certain spin chain models originated in the planar anti-de Sitter/conformal field theory (AdS/CFT) correspondence admit hidden symmetries provided by the action of the Yangian Y(g) associated with the centrally extended Lie super-This is a semi-direct product of the simple Lie superalgebra psl(2|2) of type A(1, 1) and the abelian Lie algebra C 3 spanned by elements C, K and P which are central in g. Due to the results of [6], psl(2|2) is distinguished among the basic classical Lie superalgebras by the existence of a three-dimensional central extension. It was pointed out in [4] that this phenomenon originates in some special Serre relations. A new R-matrix associated with the extended Lie superalgebra g is found by Yamane [15]. Furthermore, g can be obtained from the Lie superalgebras of type D(2, 1; α) by a particular limit with respect to the parameter α.The Yangian symmetries of the one-dimensional Hubbard model associated with Y(g) were considered in [2]; they extend those provided by the direct sum of two copies of the Yangian for sl(2) previously found in [14]. An extensive review of the Yangian symmetries in the spin chain models can be found in [13].These applications motivate the study of representations of both the Lie superalgebra g and its Yangian. In this paper we aim to prove a classification theorem for finitedimensional irreducible representation of g. Generic representations of g were already described by Beisert [3]. As we demonstrate below, beside these generic modules, the complete classification includes some degenerate representations which were not considered in [3]. In more detail, if L is a finite-dimensional irreducible representation of the Lie superalgebra g, then each of the central elements C, K and P acts in L as multiplication by a scalar. We will let the lower case letters denote the corresponding scalars,

${\mathrm{AdS}}_{3}/{\mathrm{CFT}}_{2}$ andq-Poincaré superalgebras

“…These modes are similar to a type of representations which were then later on studied in AdS 5 in[22]. There, such representations are non-physical, and they have been dubbed middle multiplets.…”

Ghost center and representations of the diagonal reduction algebra of $\mathfrak{osp}(1|2)$

“…Excitingly, the study of reduction super algebras also achieves results in the representation theory of Lie super algebras [MM14]. For example, in this paper we illustrate how one can use reduction algebras to decompose certain (infinite-dimensional) tensor product representations of osp(1|2).…”

Classical Lie bialgebras for AdS/CFT integrability by contraction and reduction

“…The eigenvalue equation ∆Υ = λΥ can be understood as an equation of motion for the free scalar field Υ where λ = µ 2 − 1 describes its AdS mass µ. 13 The eigenfunctions for eigenvalue λ = µ 2 − 1 are given by the following hypergeometric functions 14 Υ µ,ω,κ (τ, ψ, φ) ∼ e ıωτ+ ıκφ sin κ (ψ) cos 1+µ (ψ) (47)…”

“…Equipped with these algebraic tools, scattering matrices for some higher representations [6,[12][13][14] have been constructed [15][16][17]. Importantly, also the overall phase for the scattering matrix could be pinned down by consistency considerations of the quantum algebra together with considerations of the underlying physical system [18][19][20][21][22][23].…”