Polynomial super-gl(n) algebras

Abstract

We introduce a class of finite dimensional nonlinear superalgebras L = L 0 + L 1 providing gradings of L 0 = gl(n) ≃ sl(n) + gl(1). Odd generators close by anticommutation on polynomials (of degree > 1) in the gl(n) generators. Specifically, we investigate 'type I' super-gl(n) algebras, having odd generators transforming in a single irreducible representation of gl(n) together with its contragredient. Admissible structure constants are discussed in terms of available gl(n) couplings, and various special cases and candidate superalgebras are identified and exemplified via concrete oscillator constructions. For the case of the n-dimensional defining representation, with odd generators Q a , Q b , and even generators E a b , a, b = 1, . . . , n, a three parameter family of quadratic super-gl(n) algebras (deformations of sl(n/1)) is defined. In general, additional covariant Serre-type conditions are imposed, in order that the Jacobi identities be fulfilled. For these quadratic super-gl(n) algebras, the construction of Kac modules, and conditions for atypicality, are briefly considered. Applications in quantum field theory, including Hamiltonian lattice QCD and space-time supersymmetry, are discussed. † Alexander von Humboldt Fellowwhere there may be lower degree terms in the last line. Thus {Q α } form a tensor operator under the action of L 0 , and the anticommutator {Q α , Q β } transforms in the tensor product of the relevant representations of the even subalgebra.12 Further details of the generalisation of these Serre-type relations to the cases gl 2 (n/{1 3 } + {1 3 }) and gl 2 (n/{3} + {3}), as well as the present case, are given in §A.4.

Matrix elements for type 1 unitary irreducible representations of the Lie superalgebra gl(m|n)

“…We are certainly of the opinion that these characteristic identities are a valuable yet underestimated (perhaps even unknown) mathematical tool, and over the course of this series of papers we aim to convince readers of their usefulness and importance. Characteristic identities associated to Lie superalgebras have been studied in the work of Green and Jarvis [10,11] and Gould [12].…”

Hidden supersymmetry and quadratic deformations of the space-time conformal superalgebra

“…We begin with a review of the class of quadratic deformations of Lie superalgebras introduced in [13] and [12].…”

“…A.1 gl 2 (n/1) α,c structure constants and isomorphisms For completeness we here reiterate from [13,12] the parametric form of the quadratic superalgebra gl 2 (n/1), and illustrate various scaling and other constraints determining isomorphic forms. Recall that the most general nontrivial odd anticommutator bracket compatible with gl(n) covariance is given by the expansion…”

On the structure of the observable algebra of QCD on the lattice

“…For that purpose, let us introduce the following tensor operators (totally antisymmetric in both upper and lower indices) built from j's: Using these relations and keeping in mind the nilpotency properties, one can calculate arbitrary (even order) polynomials in w and w * in terms of the above tensor operators. In particular, taking the sum of these two relations, we get the following anticommutator for the baryonic observables: In fact, these relations (A.8) together with (5.5), (5.6), (5.7) and the mutual anticommutativity of w and w * can again be taken as the defining relations for a type of polynomial superalgebra generalising gl(12N/1), this time with odd generators of antisymmetric type (see [8] for details, and also the discussion above).…”

“…Such 'nonlinear' extensions of Lie algebras and superalgebras have been recognised in other contexts in recent literature. An initial investigation of them in the case of generalisations of gl(4N/1) (or more generally of type I Lie superalgebras) has been given in [8] (see also the related remarks in the appendix).…”