M. Havlíček

Boson–fermion representations of Lie superalgebras: The example of osp(1,2)

A method for constructing infinite‐dimensional representations of Lie superalgebras employing boson representations of their Lie subalgebras is outlined. As an example the osp(1,2) superalgebra is considered; explicit formulae for its generators in terms of one pair of boson operators, at most one pair of fermion ones, and at most one parameter are obtained, the Casimir operator being represented by a multiple of unity. The restriction of these representations to the real form of osp(1,2) is skew‐symmetric in the even part and can be regarded as a natural generalization of skew‐symmetric representations of real Lie algebras. Some other aspects of the presented construction are discussed.

Quantum-mechanical pseudo-hamiltonians

The question of incorporating the phenomenological description of a quantum-mechanical system via non-self-adjoint HamiltonianHp into the standard formalism of the quantum theory is discussed using the theory of minimal unitary dilations of contractive semigroups. It is shown that this problem can satisfactorily be solved if the approximative character of the phenomenological description is taken into account. The notions of approximative description and of pseudo-HamiltonianHp are strictly defined, the definitions being motivated by the requirements of physical adequacy and applicability of the unitary dilations theory. A criterion for an abstract closed densely defined operatorHp to be a pseudo-Hamiltonian is formulated. Various sufficient conditions are then obtained for the physically interesting case of Schrödinger operators onL2(RN) with complex potentials.

Note on the description of an unstable system

The paper collects some general properties of the description of an unstable particle considered byHorwitz andMarchand. It is proved generally, that the initial decay rate of finite energy states equals to zero.

An embedding of the Poincaré Lie algebra into an extension of the Lie field of SO0(1,4)

The * isomorphism between algebraic extensions of the Lie fields of SO0(1,4) and the Poincare group are considered herein. It is shown that the principal series of unitary ray representations of SO0(1,4) are associated, via the * isomorphism, with real mass, positive and negative energy representations of the Poincare Lie algebra with arbitrary spin. Results on the most degenerate exceptional series of SO0(1,4) representations are also given.

Highest-weight representations ofsl(2, ℂ) andsl(3, ℂ) via canonical realizations

The infinite-dimensional representations of thesl(n+1, ℂ) Lie algebras (maximal representations) constructed in our previous paper are studied on the two simplest examplesn = 1,2. The sufficient condition for irreducibility of the maximal representations is proved to be also necessary in these cases. It is further shown, that our method allows us to construct other set of infinite-dimensional highest-weight representations ofsl(3, ℂ), so calledmixed representations which are irreducible in some cases when the maximal as well as the standard highest-weight representations (Verma modules) are reducible.

Matrix canonical realizations of the Lie algebra o(m, n). II. Casimir operators

The matrix canonical realizations of the Lie algebra of pseudo-orthogonal group O(m, n) described in the first part of this paper are further investigated. The explicit formulae for values of the Casimir operators (which are multiples of identity in these realizations) are obtained.

Highest-weight representations of the sl(n+1,C) algebras: Maximal representations

Representations of the sl(n+1, C) Lie algebras are constructed with the help of canonical (boson) realisations of these algebras. For every weight Lambda on the standard Cartan subalgebra of sl(n+1, C) the authors obtain a representation rho Lambda (n+1) (called the maximal representation) which contains an irreducible subrepresentation with Lambda as the highest weight. It is shown that for a major part of the weights Lambda the representations rho Lambda (n+1) themselves are irreducible. The standard construction of the highest-weight representations of semi-simple Lie algebras is based on the so-called elementary representations; comparing with them, the authors maximal representations are given in the explicit form.

A complete set of irreducible highest-weight representations forsl (3, ℂ)

Infinitedimensional highest-weight representations (HWRs) ofsl (3, ℂ) are constructed using canonical (boson) realizations of this algebra. The method applies to all weights except for the cases in which the irreducible HWRs are finitedimensional; the obtained representations are irreducible and such that matrix elements of their generators are easily calculated.