Note on centrally extended su(2/2) and Serre relations

Dobrev, V. K.

doi:10.1002/prop.200900052

Cited by 7 publications

(6 citation statements)

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“…The second subscript n in ξ ± i,n , κ i,n denotes the level, with n = 0 providing a subalgebra isomorphic to psl(2|2) c . Our choice corresponds to the following Chevalley-Serre presentation of psl(2|2) c [33]:…”

Section: The Algebra: Level Onementioning

confidence: 99%

Secret symmetries in AdS/CFT

et al. 2012

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Section: The Algebra: Level Onementioning

confidence: 99%

Secret symmetries in AdS/CFT

et al. 2012

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“…where the two copies of sl(2) are spanned by the elements E 12 , E 21 , h 1 and E 34 , E 43 , h 3 , respectively. Given complex numbers u, v, w, z such that uz − v w = 1, the corresponding automorphism φ : g → g mentioned in the Introduction is determined by the mapping 4) and the condition that each element of the subalgebra sl(2) ⊕ sl(2) is stable under φ. By [3], the images of the central elements C, K, P are then found from the matrix relation…”

Section: Central Extension Of Lie Superalgebra Psl(2|2)mentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 98%

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

Matsumoto

Molev

2014

Journal of Mathematical Physics

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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,

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“…The level n = 0 is a subalgebra which coincides with the original psl(2|2) c Lie superalgebra. The choice corresponds to one of the Chevalley-Serre presentations of psl(2|2) c [32], based on Cartan generators κ i,0 , and positive (negative) simple odd roots…”

Section: The Hopf Superalgebra: Level Onementioning

confidence: 99%