LieAlgebraAktion

LieAlgebraAktion refers to the way a Lie algebra acts on another mathematical object, most commonly a vector space or a smooth manifold. In the vector space setting, a Lie algebra action is a representation of a Lie algebra g on a vector space V: a linear map rho from g to End(V) such that rho([x,y]) = rho(x) rho(y) - rho(y) rho(x) for all x,y in g. Equivalently, each x in g assigns a linear operator x_V on V, and V becomes a g-module. The kernel of the action consists of elements of g that act trivially, and the subspace V^g of invariants contains vectors v with rho(x)v = 0 for all x in g.

On a smooth manifold M, a Lie algebra action is a homomorphism from g to the Lie algebra of vector fields on M, x |-> X_x, with [X_x, X_y] = X_[x,y]. Such actions can often be integrated to an action of a Lie group G with Lie algebra g, at least locally, on M.

Key examples include the adjoint action ad: g -> End(g), defined by ad_x(y) = [x,y], the standard representation of sl_2 on R^2, and actions of the one-dimensional abelian Lie algebra on V via a fixed linear operator. Concretely, these actions correspond to representations of g or, more broadly, modules over the universal enveloping algebra U(g).

Important properties involve invariants and decomposition: V^g gives the fixed points, and, for semisimple g over a field of characteristic zero, representations exhibit complete reducibility. The study of Lie algebra actions connects representation theory with geometry, as they describe symmetries of algebraic, differential, and geometric structures, and underpin the analysis of homogeneous spaces and equivariant maps.