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Oscillator representations for orthosymplectic algebras. (English) Zbl 0688.17002
An $$\epsilon$$-graded Lie algebra ($$\epsilon$$ GLA) is defined as a $$\Gamma$$-graded algebra with a commutation factor $$\epsilon$$ such that the “$$\epsilon$$ skew symmetry” and “$$\epsilon$$-Jacobi identity” are satisfied. When $$\epsilon (\alpha,\beta)=1$$, the definition reduces to that of an ordinary graded Lie algebra; when $$\Gamma ={\mathbb{Z}}_ 2$$ and $$\epsilon (\alpha,\beta)=(-1)^{\alpha \beta}$$ for $$\alpha,\beta \in {\mathbb{Z}}_ 2$$, it reduces to the definition of a Lie superalgebra [V. G. Kac, Adv. Math. 26, 8-96 (1977; Zbl 0366.17012)]. One of the series of simple Lie superalgebras is the orthosymplectic series. In classical Lie algebra theory, the spin representation of the orthogonal Lie algebra and the metaplectic representation of the symplectic Lie algebra are of great importance. Here, a similar construction unifies these representations and leads to the oscillator representation (or the singleton representation) of the orthosymplectic Lie superalgebra $${\mathfrak osp}(p,2q).$$
An $$\epsilon$$-Heisenberg algebra is introduced by extending the usual definition. Then a Clifford-Weyl algebra is defined for the case of $$\epsilon$$ GLA’s; if $$\epsilon$$ is symmetric it includes a Clifford algebra and if $$\epsilon$$ is skew symmetric it includes a Weyl algebra. Representations of the Clifford-Weyl algebra are obtained from those of the $$\epsilon$$-Heisenberg algebra. Then the orthosymplectic Lie superalgebra is shown to be realizable in the Clifford-Weyl algebra, thus proving that representations of $${\mathfrak osp}(p,2q)$$ can be obtained from those of the $$\epsilon$$-Heisenberg algebra. Unitarity and super unitarity of the representations are investigated.
Reviewer: J.Van der Jeugt

##### MSC:
 17A70 Superalgebras 17B70 Graded Lie (super)algebras
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