Lie algebra, local

A Lie algebra whose elements are smooth functions on a smooth real manifold (or, more generally, are smooth sections of a smooth vector bundle over ), and the commutation operation is continuous in the -topology and has a local character, that is,

where is the support of the function (section) . A complete classification of local Lie algebras is known for bundles with one-dimensional fibre (in particular, for ordinary functions) (see [3]). Namely, the commutation operation in this case is a bidifferential operator of the first order, that is, it has the form

where are the partial derivatives with respect to local coordinates on . Next, let be the subspace of the tangent space to at a point generated by the vectors

Then the distribution is integrable, so decomposes into the union of integral manifolds. The commutation operation commutes with restriction to , and the structures of local Lie algebras that arise in this way on are transitive in the sense that , for any point , coincides with the tangent space to the integral manifold containing .

Every transitive local Lie algebra is defined locally by the dimension of the underlying manifold up to a change of variables in the base and fibre. For an even-dimensional manifold it is isomorphic to the algebra of Poisson brackets, and for odd-dimensional manifolds it is isomorphic to the algebra of Lagrange brackets (cf. Lagrange bracket, see also [1]).

An example of a local Lie algebra that illustrates the general theory is the structure of the Lie algebra in in which

where are the structure constants of an -dimensional Lie algebra (see [2]). In this case the manifold is naturally identified with the space dual to , and the partition into submanifolds coincides with the partition of into orbits of the coadjoint representation.

Local Lie algebras arise as the Lie algebras of certain infinite-dimensional Lie groups. In particular, they are Lie algebras of differential groups in the sense of J.F. Ritt [4]. From [5] there follows a description of all local Lie algebras connected with bundles on a line with two-dimensional fibre. All such local Lie algebras are extensions of the algebra of Lagrange brackets (which in this case coincides with the Lie algebra of vector fields) by means of a trivial local Lie algebra with one-dimensional fibre. A classification of "simple" local Lie algebras has been announced [6].

References

[1]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[2]	F.A. Berezin, "Some remarks about the associative envelope of a Lie algebra" Funct. Anal. Appl. , 1 : 2 (1967) pp. 91–102 Funktsional. Anal. i Prilozhen. , 1 : 2 (1967) pp. 1–14
[3]	A.A. Kirillov, "Local Lie algebras" Russian Math. Surveys , 31 : 4 (1976) pp. 55–75 Uspekhi Mat. Nauk , 31 : 4 (1976) pp. 57–76
[4]	J.F. Ritt, "Differential groups and formal Lie theory for an infinite number of variables" Ann. of Math. (2) , 52 (1950) pp. 708–726
[5]	J.F. Ritt, "Differential groups of order two" Ann. of Math. (2) , 53 (1951) pp. 491–519
[6]	B. Weisfeiler, "On Lie algebras of differential formal groups of Ritt" Bull. Amer. Math. Soc. , 84 : 1 (1978) pp. 127–130

Comments

For an account of the role of local Lie algebras (and related structures) in the deformation-theoretic approach to quantization cf. [a1].

References

[a1]	A. Lichnerowicz, "Applications of the deformations of algebraic structures to geometry and mathematical physics" M. Hazewinkel (ed.) M. Gerstenhaber (ed.) , Deformation theory of algebras and structures and applications , Kluwer (1988) pp. 855–896