Lie algebra, local

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

$$ \supp [ f _ {1} , f _ {2} ] \subset \ \supp f _ {1} \cap \supp f _ {2} , $$

where $ \supp f $ is the support of the function (section) $ f $. A complete classification of local Lie algebras is known for bundles $ E $ 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

$$ [ f _ {1} , f _ {2} ] = \sum _ { i,j } c ^ {ij} ( x) \partial _ {i} f _ {1} \ \partial _ {j} f _ {2} + \sum _ { k } a ^ {k} ( x) ( f _ {1} \partial _ {k} f _ {2} - f _ {2} \partial _ {k} f _ {1} ) , $$

where $ \partial _ {i} = \partial / \partial x ^ {i} $ are the partial derivatives with respect to local coordinates on $ M $. Next, let $ P ( x) $ be the subspace of the tangent space $ T _ {x} M $ to $ M $ at a point $ x \in M $ generated by the vectors

$$ a ( x) = \sum _ { k } a ^ {k} ( x) \ \partial _ {k} \ \textrm{ and } \ c ^ {i} ( x) = \ \sum _ { j } c ^ {ij} ( x) \partial _ {j} , $$

$$ i = 1 \dots n . $$

Then the distribution $ \{ {P ( x) } : {x \in M } \} $ is integrable, so $ M $ decomposes into the union $ \cup _ {\alpha \in A } M _ \alpha $ of integral manifolds. The commutation operation commutes with restriction to $ M _ \alpha $, and the structures of local Lie algebras that arise in this way on $ M _ \alpha $ are transitive in the sense that $ P ( x) $, for any point $ x $, coincides with the tangent space to the integral manifold $ M _ \alpha $ containing $ x $.

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 $ C ^ \infty ( \mathbf R ^ {n} ) $ in which

$$ [ f _ {1} , f _ {2} ] = \sum _ { i,j,k } c _ {k} ^ {ij} x ^ {k} \ \partial _ {i} f _ {1} \partial _ {j} f _ {2} , $$

where $ c _ {k} ^ {ij} $ are the structure constants of an $ n $- dimensional Lie algebra $ \mathfrak g $( see [2]). In this case the manifold $ M = \mathbf R ^ {n} $ is naturally identified with the space $ \mathfrak g ^ {*} $ dual to $ \mathfrak g $, and the partition into submanifolds $ M _ \alpha $ coincides with the partition of $ \mathfrak g ^ {*} $ 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