Polynomial super-gl(n) algebras

Abstract

We introduce a class of finite dimensional nonlinear superalgebras $L=L_{\bar{0}}+L_{\bar{1}}$ providing gradings of $L_{\bar{0}}=gl(n)\simeq sl(n)+gl(1)$. 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 $Q_a,\bar{Q}{}^b$, 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.

31 pages, LaTeX, including minor corrections to equation (3) and reference [60]

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Cited by in corpus (5)

• Charge Superselection Sectors for QCD on the Lattice
• On the Stratified Classical Configuration Space of Lattice QCD
• On the Structure of the Observable Algebra of QCD on the Lattice
• A class of quadratic deformations of Lie superalgebras
• Hidden supersymmetry and quadratic deformations of the space-time conformal superalgebra