Polynomial super-gl(n) algebras

We introduce a class of finite dimensional nonlinear superalgebras L = L0 + L1 providing gradings of L0 = 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 Qa, Q , and even generators Eb, 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.