Infinitely Divisible Representations of Lie Algebras

by R. F. Streater. Z. Wahr. u. Ver. Geb., 19, 67-80, 1971.

ABSTRACT

We define the concept of infinitely divisible representation of a Lie algebra, associated with a given cyclic vector. We show that any such representation can be embedded in a Fock space. This is the infinitesimal analogue of the result proved by the author in a previous paper [1], which says that any infinitely divisible representation of a group can be embedded in a Fock space. Independently, Araki has also worked out the case for groups. We relate the classification of infinitely divisible representations to the existence of a one-cocycle for the Lie algebra, related to the method of Araki by taking the group elements to be close to the identity. From such a representation we construct a quantum Markov process. In the case when the algebra is the real line, the theory is connected to that of infinitely divisible measures. In lectures at MaPhySto, Aarhus, I explain that cocycles of the Lie algebra R are one-to-one to the class of infinitely divisible measures defined by Kolmogorov, whereas for the Lie group R, the cocycles are one-to-one with the larger class of Levy measures. The coboundaries correspond to the measures discovered by de Finetti.

[1] R. F. Streater, Current commutation relations, continuous tensor products and infinitely divisible group representations, in LOCAL QUANTUM THEORY, Ed. R. Jost, Academic Press, 1969. This was the proceedings of the Varenna Summer School, 1968.

It is shown that any infinitely divisible representation of a Lie algebra (or of the enveloping algebra) is associated with a conditionally positive semi-definite form on the algebra. This is shown to be given by a positive semi-definite form on the truncated functions (omitting the identity from the algebra). This result has been rederived by several authors in similar contexts: G. C. Hegerfeldt treated the case of the Borchers polynomial algebra of test functions, and Schurmann the general case of a co-algebra. The case of projective representations of a Lie algebra was solved early by Daniela Mathon in "Infinitely divisible projective representations of the Lie algebras", Proc. Camb. Phil. Soc., 72, 357-368, 1972.

The subject has been taken further by Luigi Accardi, Uwe Franz and Michael Skeide. Uwe Franz constructs Lévy processes from continuous tensor products here.