Lie algebra
Lie algebra
![{\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}},\ (x,y)\mapsto [x,y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/320c0c8f1de0f51896d88d7a3cacdc2edd4cc30f)
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.
![{\displaystyle [x,y]=x\times y.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07242b4e095b05677a80bd7d7fde5f2fb3d76233)
![{\displaystyle [x,x]=x\times x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/288c71f1e200894aee3e95818f3cf34f95b02eb6)
History
Lie algebras were introduced to study the concept of infinitesimal transformations by Marius Sophus Lie in the 1870s,[2] and independently discovered by Wilhelm Killing[3] in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group is used.
Definitions
Definition of a Lie algebra
![{\displaystyle [\,\cdot \,,\cdot \,]:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6296f764a7944984cdc7eaa85de74a9566993bdd)
Bilinearity,
![{\displaystyle [ax+by,z]=a[x,z]+b[y,z],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f568240b82f64c1f483ced15389429ad34a18bde)
![{\displaystyle [z,ax+by]=a[z,x]+b[z,y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e8fe7285a34bcfbd5e9412bc16bf857d443864a)
Alternativity,
![[x,x]=0\](https://wikimedia.org/api/rest_v1/media/math/render/svg/07b140cc32f46dd51382217a0fe40d7941d248b0)
The Jacobi identity,
![[x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0\](https://wikimedia.org/api/rest_v1/media/math/render/svg/67603b4c461b8f6b937a4ace3e4ff0ce058309b0)
![[x+y,x+y]](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c0808f3e1cacefc5287673604694a416f91361)
![[x,y]+[y,x]=0\](https://wikimedia.org/api/rest_v1/media/math/render/svg/55d2a80df91d0e16cfa14924cfc729627f3234af)
Anticommutativity,
![{\displaystyle [x,y]=-[y,x],\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c392773a54647318dc10eaadf3f60e84760a8980)
Generators and dimension
See the classification of low-dimensional real Lie algebras for other small examples.
Subalgebras, ideals and homomorphisms
![[[x,y],z]](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d1355c94372444268d5200cf3079e4b2e8c5510)
![[x,[y,z]]](https://wikimedia.org/api/rest_v1/media/math/render/svg/99c3a3b210ab676378107460425cdcd01b90d839)
![{\displaystyle [{\mathfrak {g}},{\mathfrak {i}}]\subseteq {\mathfrak {i}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8242a1cf36864dc503b45192ace093a8a3bfae85)
A Lie algebra homomorphism is a linear map compatible with the respective Lie brackets:
![{\displaystyle \phi :{\mathfrak {g}}\to {\mathfrak {g'}},\quad \phi ([x,y])=[\phi (x),\phi (y)]\ {\text{for all}}\ x,y\in {\mathfrak {g}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22e91ebab9c2aaf52ae7c5bd89f5d986038fd27b)
![{\displaystyle [x,y]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bb3ac5611c95c794cec85401f417c5e0146185d)
![{\displaystyle {\mathfrak {z}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}\ \mid \ [x,s]=0\ {\text{ for all }}s\in S\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/beef7ab1fd4087d8f3c02729e51a1642e8467411)
![{\displaystyle {\mathfrak {n}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}\ \mid \ [x,s]\in S\ {\text{ for all}}\ s\in S\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1054639612ed3d9a3b8c1c4fbe652f15e3769355)
Direct sum and semidirect product
![{\displaystyle [(x,x'),(y,y')]=([x,y],[x',y']),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00719cf5b8674e4be63e88fdd9403e9122d3a3ac)
![{\displaystyle [(x,0),(0,x')]=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe813c102ddf8d403025225b2661eb68f25e0ad5)
Levi's theorem says that a finite-dimensional Lie algebra is a semidirect product of its radical and the complementary subalgebra (Levi subalgebra).
Derivations
![\delta ([x,y])=[\delta (x),y]+[x,\delta (y)]](https://wikimedia.org/api/rest_v1/media/math/render/svg/f87aef094b6b41ea9f7592abc397e6c714b87f7d)
![{\displaystyle \mathrm {ad} _{x}(y):=[x,y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/883a92018bbd927d508c0901ce8c4b5f2ac35dd8)
![{\displaystyle [\delta _{1},\delta _{2}]\ =\ \delta _{1}\circ \delta _{2}-\delta _{2}\circ \delta _{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38ad2f1638cd9f0521716f2c89c34907ae379745)
Split Lie algebra
Examples
Vector spaces
Associative algebra with commutator bracket
On an associative algebra over a field with multiplication , a Lie bracket may be defined by the commutator . With this bracket, is a Lie algebra.[9] The associative algebra A is called an enveloping algebra of the Lie algebra . Every Lie algebra can be embedded into one that arises from an associative algebra in this fashion; see universal enveloping algebra.
The associative algebra of endomorphisms of an F-vector space with the above Lie bracket is denoted .
For a finite dimensional vector space , the previous example becomes the Lie algebra of n × n matrices, denoted or ,[10] with the bracket , where denotes matrix multiplication. This is the Lie algebra of the general linear group, consisting of invertible matrices.
Special matrices
The matrices of trace zero form the special linear Lie algebra , the Lie algebra of the special linear group .[11]
The skew-hermitian matrices form the unitary Lie algebra , the Lie algebra of the unitary group U(n).
Matrix Lie algebras
![{\displaystyle [X,Y]=XY-YX}](https://wikimedia.org/api/rest_v1/media/math/render/svg/838f73010b4f791eeaf245317fb4b6e07c45d741)
The following are examples of Lie algebras of matrix Lie groups:[12]
The special linear group , consisting of all n × n matrices with determinant 1. Its Lie algebra consists of all n × n matrices with real entries and trace 0. Similarly, one can define the corresponding real Lie group and its Lie algebra .
The unitary group consists of n × n unitary matrices (satisfying ). Its Lie algebra consists of skew-self-adjoint matrices ().
The special orthogonal group , consisting of real determinant-one orthogonal matrices (). Its Lie algebra consists of real skew-symmetric matrices (). The full orthogonal group , without the determinant-one condition, consists of and a separate connected component, so it has the same Lie algebra as . Similarly, one can define a complex version of this group and algebra, simply by allowing complex matrix entries.
Two dimensions
On any field there is, up to isomorphism, a single two-dimensional nonabelian Lie algebra with generators x, y, and bracket defined as . It generates the affine group in one dimension.
- This can be realized by the matrices:
Three dimensions
The Heisenberg algebra is a three-dimensional Lie algebra generated by elements x, y and z with Lie brackets
- .
![[x,y]=z,\quad [x,z]=0,\quad [y,z]=0](https://wikimedia.org/api/rest_v1/media/math/render/svg/4360eb6f88b60ae723e133e389c76b208658a816)
- Any element of theHeisenberg groupis thus representable as a product of group generators, i.e.,matrix exponentialsof these Lie algebra generators,
The Lie algebra of the group SO(3) is spanned by the three matrices[13]
![{\displaystyle [F_{1},F_{2}]=F_{3},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d357a07d3fb64786da9f02bb9ad5f0a5027c3bf)
![{\displaystyle [F_{2},F_{3}]=F_{1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/356514684addccaaccc0b2872fa1521294d21727)
![{\displaystyle [F_{3},F_{1}]=F_{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37b8356e46a3dcb952c873b955d8c2b92a6b8d20)
Infinite dimensions
An important class of infinite-dimensional real Lie algebras arises in differential topology. The space of smooth vector fields on a differentiable manifold M forms a Lie algebra, where the Lie bracket is defined to be the commutator of vector fields. One way of expressing the Lie bracket is through the formalism of Lie derivatives, which identifies a vector field X with a first order partial differential operator L**X acting on smooth functions by letting L**X(f) be the directional derivative of the function f in the direction of X. The Lie bracket [X,Y] of two vector fields is the vector field defined through its action on functions by the formula:
![L_{[X,Y]}f=L_{X}(L_{Y}f)-L_{Y}(L_{X}f).\,](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2744f4fa1f6829787912d0f27fd27fe5047422d)
Kac–Moody algebras are a large class of infinite-dimensional Lie algebras whose structure is very similar to the finite-dimensional cases above.
The Moyal algebra is an infinite-dimensional Lie algebra that contains all classical Lie algebras as subalgebras.
The Virasoro algebra is of paramount importance in string theory.
Representations
Definitions
A representation is said to be faithful if its kernel is zero. Ado's theorem[14] states that every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space.
Adjoint representation
![\operatorname {ad} (x)(y)=[x,y]](https://wikimedia.org/api/rest_v1/media/math/render/svg/883d113b6039d5f39214bec543d5198c7a16aa6b)
Goals of representation theory
Representation theory in physics
Structure theory and classification
Lie algebras can be classified to some extent. In particular, this has an application to the classification of Lie groups.
Abelian, nilpotent, and solvable
Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras.
- if the Lie bracket vanishes, i.e. [x,y] = 0, for all x and y in
- if the
![{\mathfrak {g}}>[{\mathfrak {g}},{\mathfrak {g}}]>[[{\mathfrak {g}},{\mathfrak {g}}],{\mathfrak {g}}]>[[[{\mathfrak {g}},{\mathfrak {g}}],{\mathfrak {g}}],{\mathfrak {g}}]>\cdots](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fbc7736bc5f43a440623e77833ea7de0cfa99fe)
![\operatorname {ad} (u):{\mathfrak {g}}\to {\mathfrak {g}},\quad \operatorname {ad} (u)v=[u,v]](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1b5c04d185bd495cc049795d628ba972f21ba9)
is nilpotent.
- if the
![{\mathfrak {g}}>[{\mathfrak {g}},{\mathfrak {g}}]>[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]>[[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]],[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]]>\cdots](https://wikimedia.org/api/rest_v1/media/math/render/svg/55be9f812e367366f54c391ec9559604163654ab)
becomes zero eventually.
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.
Simple and semisimple
- if it is isomorphic to a direct sum of simple algebras. There are several equivalent characterizations of semisimple algebras, such as having no nonzero solvable ideals.
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.
Cartan's criterion
![K({\mathfrak {g}},[{\mathfrak {g}},{\mathfrak {g}}])=0.](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ecced3d594ff22f930666328433047354c990ea)
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. Furthermore, semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems. However, the classification of solvable Lie algebras is a 'wild' problem, and cannot be accomplished in general.
Relation to Lie groups
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.
Lie algebra with additional structures
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 ring
![[\cdot ,\cdot ]](https://wikimedia.org/api/rest_v1/media/math/render/svg/28dd4c22d60192519c1c12cf645b040f368db9e9)
Bilinearity:
![[x+y,z]=[x,z]+[y,z],\quad [z,x+y]=[z,x]+[z,y]](https://wikimedia.org/api/rest_v1/media/math/render/svg/165b1dcd70eaf2436d298c473319328d0e72abda)
- for all x, y, z ∈ L.
The Jacobi identity:
![[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0\quad](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5140846419cb8d565520cb9aa5778495f05ef04)
- for all x, y, z in L.
For all x in L:
![[x,x]=0\quad](https://wikimedia.org/api/rest_v1/media/math/render/svg/73f16254787dd7a9c55e04cdcd793322b28dde0e)
![[x,y]=xy-yx](https://wikimedia.org/api/rest_v1/media/math/render/svg/42b4220c8122ebd2a21c517ca80639581679cfa6)
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 the reducing modulo p to get a Lie algebra over a finite field.
Examples
Any Lie algebra over a general ring instead of a field is an example of a Lie ring. Lie rings are not Lie groups under addition, despite the name.
Any associative ring can be made into a Lie ring by defining a bracket operator
![{\displaystyle [x,y]=xy-yx.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adaa95a59e9814f5965d79550c54297626db87a9)
For an example of a Lie ring arising from the study of groups, let be a group with the commutator operation, and let be a central series in — that is the commutator subgroup is contained in for any . Then
- is a Lie ring with addition supplied by the group operation (which will be × in each homogeneous part), and the bracket operation given by
![[xG_{i},yG_{j}]=(x,y)G_{i+j}\](https://wikimedia.org/api/rest_v1/media/math/render/svg/065718391adabdf7f84dfea6bcce72c1b8e62447)
- extended linearly. The centrality of the series ensures the commutatorgives the bracket operation the appropriate Lie theoretic properties.
