Hermitean Oscillator-like Realizations of Classical Algebras and Superalgebras in Hilbert Space with Positive Definite Metric (original) (raw)

1986, Fortschritte der Physik

The hermitean oscillator-like realizations of classical algebras in terms of bosonic and fermionic creation and annihilation operators are given. The hermitean realizations of classical superalgebras using boson-fermion oscillators are explicitely described. The assumption of positive definite metric in a Hilbert space of the oscillators states is exploited. Due to this fact, the realizations of superalgebras in the Hilbert space can be constructed only for: the real orthosymplectic superalgebra osp ( N ; 2 M ; R); the unitary compact superalgebra su ( N ; M ) ; the unitary noncompact one S U ( N ; K , M ) ; and the quaternionic unitary superalgebra uu,(N; M ; H ) . classical Lie superalgebras (i.e. simple Lie superalgebras whose Lie subalgebra is reductive) can be divided into four classes : a) standard classical Lie superalgebras A(%, m), B(n, m ) , C(n) and B(n, m ) ; b) exceptional Lie superalgebras F(4), G(3) ; c ) strange Lie superalgebras P(n), &(n) ; d) one-parameter family of deformations of D(2, 1) denoted by . The standard classical Lie superalgebras are supersymmetric analogues of Cartan classical Lie algebras. The classification of real forms of classical Lie superalgebras are given in [3]. Recently, the realizations of supersymmetry algebras using the oscillator operators was proposed. It is connected with the problem of bosonization of the fermionic systems [4, 51 as well as the description of unitary irreducible representations of noncompact supersymmetries [6--81. By the oscillator method, using bosonic and fermionic oscillators there were constructed unitary irreducible representations of : i) anti-de Sitter superalgebra osp (2; 4 ; R) in [9]; ii) extended anti-de Sitter superalgebra osp ( N ; 4; R) in [lo]; *) On leave of absence from Institute of Teacher's Training-ODN, 50-527 Wroclaw, ul. Dawida la, Poland.