Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras.
In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.
Lie algebras were introduced to study the concept of infinitesimal transformations by Marius Sophus Lie in the 1870s, and independently discovered by Wilhelm Killing in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group is used.
A Lie algebra homomorphism is a linear map compatible with the respective Lie brackets:
Although the definitions above are sufficient for a conventional understanding of Lie algebras, once this is understood, additional insight can be gained by using notation common to category theory, that is, by defining a Lie algebra in terms of linear maps—that is, morphisms of the category of vector spaces—without considering individual elements. (In this section, the field over which the algebra is defined is supposed to be of characteristic different from two.)
Lie algebras can be classified to some extent. In particular, this has an application to the classification of Lie groups.
Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras.
Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its radical. Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, solvable) Lie algebras.
The concept of semisimplicity for Lie algebras is closely related with the complete reducibility (semisimplicity) of their representations. When the ground field F has characteristic zero, any finite-dimensional representation of a semisimple Lie algebra is semisimple (i.e., direct sum of irreducible representations). In general, a Lie algebra is called reductive if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive.
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. (Such a decomposition exists for a finite-dimensional Lie algebra over a field of characteristic zero.) Furthermore, semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems.
Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups.
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. Every representation of a Lie algebra lifts uniquely to a representation of the corresponding connected, simply connected Lie group, and conversely every representation of any Lie group induces a representation of the group's Lie algebra; the representations are in one-to-one correspondence. Therefore, knowing the representations of a Lie algebra settles the question of representations of the group.
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. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the center, once the classification of Lie algebras is known (solved by Cartan et al. in the semisimple case).
If the Lie algebra is infinite-dimensional, the issue is more subtle. In many instances, the exponential map is not even locally a homeomorphism (for example, in Diff(S1), one may find diffeomorphisms arbitrarily close to the identity that are not in the image of exp). Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.
A Lie algebra can be equipped with some additional structures that are assumed to be compatible with the bracket. For example, a graded Lie algebra is a Lie algebra with a graded vector space structure. If it also comes with differential (so that the underlying graded vector space is a chain complex), then it is called a differential graded Lie algebra.
A simplicial Lie algebra is a simplicial object in the category of Lie algebras; in other words, it is obtained by replacing the underlying set with a simplicial set (so it might be better thought of as a family of Lie algebras).
Lie rings are used in the study of finite p-groups through the Lazard correspondence. The lower central factors of a p-group are finite abelian p-groups, so modules over Z/pZ. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the commutator of two coset representatives. The Lie ring structure is enriched with another module homomorphism, the pth power map, making the associated Lie ring a so-called restricted Lie ring.
Lie rings are also useful in the definition of a p-adic analytic groups and their endomorphisms by studying Lie algebras over rings of integers such as the p-adic integers. 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 then reducing modulo p to get a Lie algebra over a finite field.