Lie Algebras in a sentence

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.

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

There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.

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.

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

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.

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.

Nonetheless, much of the terminology that was developed in the theory of associative rings or associative algebras is commonly applied to Lie algebras.

Conversely, it can be proven that any semisimple Lie algebra is the direct sum of its minimal ideals, which are canonically determined simple Lie algebras.

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

Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.

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

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

Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras.

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.

But there are also just five "exceptional Lie algebras" that do not fall into any of these families.

For Lie algebras the definition is slightly different.

Hence the question arises: what are the simple Lie algebras of compact groups?

However, the classification of solvable Lie algebras is a 'wild' problem, and cannot be accomplished in general.