Lie algebra
In more detail: for any Lie group, the multiplication operation near the identity element 1 is commutative to first order.
It is a remarkable fact that these second-order terms (the Lie algebra) completely determine the group structure of G near the identity.
The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group was used.
Thus bilinearity and the alternating property together imply It is customary to denote a Lie algebra by a lower-case fraktur letter such as
As with normal subgroups in groups, ideals in Lie algebras are precisely the kernels of homomorphisms.
[9] Informally speaking, the space of derivations of A is the Lie algebra of the automorphism group of A.
For this reason, spaces of derivations are a natural way to construct Lie algebras: they are the "infinitesimal automorphisms" of A.
Indeed, writing out the condition that (where 1 denotes the identity map on A) gives exactly the definition of D being a derivation.
So the Lie bracket of vector fields describes the non-commutativity of the diffeomorphism group.
, one can recover the Lie group as the subgroup generated by the matrix exponential of elements of
Kenkichi Iwasawa extended this result to finite-dimensional Lie algebras over a field of any characteristic.
The representation theory of Lie algebras plays an important role in various parts of theoretical physics.
There, one considers operators on the space of states that satisfy certain natural commutation relations.
An example is the angular momentum operators, whose commutation relations are those of the Lie algebra
In the study of the hydrogen atom, for example, quantum mechanics textbooks classify (more or less explicitly) the finite-dimensional irreducible representations of the Lie algebra
A more general class of Lie algebras is defined by the vanishing of all commutators of given length.
[25] Every finite-dimensional Lie algebra over a field has a unique maximal solvable ideal, called its radical.
[22] A finite-dimensional Lie algebra over a field of characteristic zero is called reductive if its adjoint representation is semisimple.
is more complicated, but it was also solved by Cartan (see simple Lie group for an equivalent classification).
were classified by Richard Earl Block, Robert Lee Wilson, Alexander Premet, and Helmut Strade.
Every connected Lie group is isomorphic to its universal cover modulo a discrete central subgroup.
[35] Lie theory also does not work so neatly for infinite-dimensional representations of a finite-dimensional group.
can usually not be differentiated to produce a representation of its Lie algebra on the same space, or vice versa.
[36] The theory of Harish-Chandra modules is a more subtle relation between infinite-dimensional representations for groups and Lie algebras.
For example, the homotopy groups of a simply connected topological space form a graded Lie algebra, using the Whitehead product.
In a related construction, Daniel Quillen used differential graded Lie algebras over the rational numbers
Lie rings are used in the study of finite p-groups (for a prime number p) through the Lazard correspondence.
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; see the example below.
[41] The definition of a Lie algebra can be reformulated more abstractly in the language of category theory.
Namely, one can define a Lie algebra in terms of linear maps—that is, morphisms in the category of vector spaces—without considering individual elements.
