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We review the modern development of the theory of linear irreducible infinite dimensional representations of noncompact groups. Such representations arise on spaces of states, spaces of observables, spaces of classical fields, etc. We connect the unitarity problem in mathematics to the elimination of ghosts in formalisms using indefinite Hermitian metrics.
For conformally flat Riemannian manifolds of dimension n ≥ 3 ,we describe an explicit resolution of the sheaf Θc of conformal Killing vector fields which is formally self-adjoint, and we deduce a duality theorem for the cohomology of X with values inΘc.
Young diagrams and tableaux are defined for covariant tensor irreducible representations of U(m) and are then generalised to cover the case of mixed tensors. The extension of these techniques to U(m,n) is described, including a discussion of characters, supercharacters, branching rules and Kronecker products.
The SO(4,1) symmetry of the Dirac equation is constructed. It is realized on the full space of solutions of the Dirac equation. Generalizations and possible physical implications are noted.
The Casimir operators of the following groups are explicitely constructed: The method is based on a particular fibre bundle structure of the generic orbits generated by the co-adjoint representation of a semi-direct product.
Group orthogonality relations are presented in various coordinate-free, and possibly new, guises; it is not assumed that the ground field is algebraically closed. One of the more unlikely guises is used to give a basis-free proof of the “generalized Frobenius-Schur criterion” for the Wigner type of a corepresentation.
We study in this communication, the basic differences in the properties of Lie supergroups and graded Lie groups with regard to uniqueness, representations and classification.
An infinite dimensional symmetry algebra of the Heisenberg spin chain is described and some of its properties are discussed. It is shown that the principle of gauge equivalence of Lax pairs leads to the existence of such symmetry algebras even in models which do not have a global non-Abelian symmetry. This is explained for the examples of the non-linear Schrödinger equation and the complex...
A geometrical treatment of supersymmetric σ-models on ordinary and graded manifolds is given: It is shown that the SuSy σ-models on graded manifolds can be reduced to give generalizations of the SuSy sine-Gordon equation, endowed with an associate linear set. In particular the SuSy generalization of the Complex sine-Gordon model is briefly discussed.
A group theoretical approach to the separation of variables is applied to the Hamilton-Jacobi and Laplace-Beltrami equation in the hermitian hyperbolic space HH(2). Symmetry reduction by maximal abelian subgroups of the isometry group SU(2,1) leads to completely integrable systems defined in a Minkowski space and involving nontrivial interactions.
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