Lie Algebra in a sentence

The definition of finite groups of Lie type due to Chevalley involves restricting from a Lie algebra over the complex numbers to a Lie algebra over the integers, and the reducing modulo p to get a Lie algebra over a finite field.

Representation theory * The universal enveloping algebra of a Lie algebra is an associative algebra that can be used to study the given Lie algebra.

The exponential map (Lie theory) from the Lie algebra to the Lie group, : provides a one-to-one correspondence between small enough neighborhoods of the origin of the Lie algebra and neighborhoods of the identity element of the Lie group.

A Lie algebra is solvable if and only if Classification The Levi decomposition expresses an arbitrary Lie algebra as a semidirect sum of its solvable radical and a semisimple Lie algebra, almost in a canonical way.

For every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra.

Informally we can think of elements of the Lie algebra as elements of the group that are " infinitesimally close" to the identity, and the Lie bracket of the Lie algebra is related to the commutator of two such infinitesimal elements.

On the Lie algebra side of affairs, things are simpler since the qualifying criteria for the prefix Lie in Lie algebra are purely algebraic.

So every abstract Lie algebra is the Lie algebra of some (linear) Lie group.

The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones.

This Lie algebra is finite-dimensional and it has the same dimension as the manifold G. The Lie algebra of G determines G up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element.

Ado's theorem says every finite-dimensional Lie algebra is isomorphic to a matrix Lie algebra.

Generators and dimension Elements of a Lie algebra are said to be generators of the Lie algebra if the smallest subalgebra of containing them is itself.

The dimension of a Lie algebra is its dimension as a vector space over F. The cardinality of a minimal generating set of a Lie algebra is always less than or equal to its dimension.

Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering ( Lie's third theorem ).

Lie algebras: Assigning to every real (complex) Lie group its real (complex) Lie algebra defines a functor.

It can be used, however, to define the tensor product of two representations of a Lie algebra (rather than of an associative algebra).

Any Lie algebra is an example of a Lie ring.

Any Lie group gives rise to a Lie algebra.

Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket.

As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup.