Infinitely divisible projective representations of the Lie algebras

Extract

Infinitely divisible group representations were first defined by Streater(1) as an important concept closely related to continuous tensor product. Araki(2) analysed the factorizable representations of Lie groups and obtained a generalization of the Levy–Khinchine formula. A similar concept for Lie algebras was defined and studied by Streater in (3). Although the definition is not strictly an infinitesimal analogue of infinitely divisible representations of Lie groups, the results of (3) in the cohomological formulation are very similar to Araki's main theorem. Parthasarathy and Schmidt(4) generalized the concept of infinite divisibifity to the projective representations of locally compact groups and obtained a one-to-one correspondence between infinitely divisible projective representations and 1-co-cycles in the group cohomology with coefficients in a Hubert space. A similar generalization for Lie algebras is studied in the present paper. Infinitely divisible projective representations of Lie algebras are studied by a purely algebraic method, independently of (4) (since not all our projective representations are necessarily integrable). As expected, a one-to-one relation is obtained between the infinitely divisible projective representations and 1-co-cycles in the cohomology on the corresponding enveloping algebra with coefficients in a Hilbert space. The present problem is simpler than the group case since there is no continuity condition on the multiplier in a Lie algebra. A similar algebraic method was used in a discussion of infinitely divisible representations of canonical anticommutation relations (9).