Lie Groups in a sentence

The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the related matter of the representation theory of Lie groups.

This correspondence between Lie groups and Lie algebras allows one to study Lie groups in terms of Lie algebras.

It is however not an equivalence of categories : different Lie groups may have isomorphic Lie algebras (for example SO(3) and SU(2) ), and there are (infinite dimensional) Lie algebras that are not associated to any Lie group.

Quotients of Lie groups If G is a Lie group and N is a normal Lie subgroup of G, the quotient G / N is also a Lie group.

Constructions There are several standard ways to form new Lie groups from old ones: *The product of two Lie groups is a Lie group.

Relation to Lie groups seeAlso Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups.

Lie's fundamental theorems describe a relation between Lie groups and Lie algebras.

Lie theory with its Lie groups and Lie algebras became one of the major areas of study.

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.

Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily.

The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group.

A construction on Lie groups On an n-dimensional Lie group, Haar measure can be constructed easily as the measure induced by a left-invariant n-form.

For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras.

Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.

However, even though su(2) and so(3) are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups.

Lie groups A Lie group is a group in the category of smooth manifolds.

Moreover, every homomorphism between Lie algebras lifts to a unique homomorphism between the corresponding simply connected Lie groups.

The language of category theory provides a concise definition for Lie groups: a Lie group is a group object in the category of smooth manifolds.

Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism.

Diffeomorphism groups of compact manifolds of larger dimension are regular Fréchet Lie groups ; very little about their structure is known.