Representation of a Lie algebra
Encyclopedia
In the mathematical
field of representation theory
, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra
as a set of matrices
(or endomorphism
s of a vector space
) in such a way that the Lie bracket is given by the commutator
.
The notion is closely related to that of a representation of a Lie group
. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra.
is a Lie algebra homomorphism
from
to the Lie algebra of endomorphism
s on a vector space
(with the commutator
as the Lie bracket), sending an element x of
to an element ρx of 
.
Explicitly, this means that
for all
in 
. The vector space 
, together with the representation 
, is called a 
-module. (Many authors abuse terminology and refer to 
itself as the representation).
One can equivalently define a
-module as a vector space 
together with a bilinear map 
such that
for all
in 
and 
in 
. This is related to the previous definition by setting 
is a homomorphism
of Lie group
s, and
and 
are the Lie algebra
s of
and 
respectively, then the induced map 
on tangent space
s is a Lie algebra homomorphism. In particular, a representation of Lie groups
determines a Lie algebra homomorphism
from
to the Lie algebra of the general linear group
, i.e. the endomorphism algebra of 
.
A partial converse to this statement says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated simply connected Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras.
. This follows from the universal property
of that construction.
If the Lie algebra is semisimple, then all reducible
representations are decomposable
. Otherwise, that's not true in general.
If we have two representations, with V1 and V2 as their underlying vector spaces and ·[·]1 and ·[·]2 as the representations, then the product of both representations would have
as the underlying vector space and
If L is a real Lie algebra and
is a complex representation of it, we can construct another representation of L called its dual representation as follows.
Let V∗ be the dual vector space of V. In other words, V∗ is the set of all linear maps from V to C with addition defined over it in the usual linear way, but scalar multiplication defined over it such that
for any z in C, ω in V∗ and X in V. This is usually rewritten as a contraction with a sesquilinear form ⟨·,·⟩. i.e. ⟨ω,X⟩ is defined to be ω[X].
We define
as follows:
for any A in L, ω in V∗ and X in V. This defines
uniquely.
Briefly, finite-dimensional representations of a semisimple Lie algebra are completely reducible, so it suffices to classify irreducible (simple) representations. Semisimple Lie algebras are classified in terms of the weights
of the adjoint representation, the so-called root system
; in a similar manner all finite-dimensional irreducible representations can be understood in terms of weights; see weight (representation theory)
for details.
algebra
A which is a representation of L as a Z2 graded vector space
and in addition, the elements of L acts as derivation
s/antiderivations.
More specifically, if H is a pure element of L and x and y are pure elements of A,
Also, if A is unital, then
Now, for the case of a representation of a Lie algebra, we simply drop all the gradings and the (−1) to the some power factors.
A Lie (super)algebra is an algebra and it has an adjoint representation
of itself. This is a representation on an algebra: the (anti)derivation property is the superJacobi identity
.
If a vector space is both an associative algebra
and a Lie algebra
and the adjoint representation of the Lie algebra on itself is a representation on an algebra (i.e., acts by derivations on the associative algebra structure), then it is a Poisson algebra
. The analogous observation for Lie superalgebras gives the notion of a Poisson superalgebra
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
field of representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
as a set of matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
(or endomorphism
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
s of a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
) in such a way that the Lie bracket is given by the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
.
The notion is closely related to that of a representation of a Lie group
Representation of a Lie group
In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie...
. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the universal cover of a Lie group are the integrated form of the representations of its Lie algebra.
Formal definition
A representation of a Lie algebraLie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
is a Lie algebra homomorphismfrom
to the Lie algebra of endomorphismEndomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
s on a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
(with the commutatorCommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
as the Lie bracket), sending an element x of
to an element ρx of .Explicitly, this means that
for all
in . The vector space , together with the representation , is called a -module. (Many authors abuse terminology and refer to itself as the representation).One can equivalently define a
-module as a vector space together with a bilinear map such thatfor all
in and in . This is related to the previous definition by setting Infinitesimal Lie group representations
If is a homomorphismHomomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
of Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, and
and are the Lie algebraLie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s of
and respectively, then the induced map on tangent spaceTangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
s is a Lie algebra homomorphism. In particular, a representation of Lie groups
determines a Lie algebra homomorphism
from
to the Lie algebra of the general linear groupGeneral linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
, i.e. the endomorphism algebra of .A partial converse to this statement says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated simply connected Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras.
Properties
Representations of a Lie algebra are in one-to-one correspondence with algebra representations of the associated universal enveloping algebraUniversal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
. This follows from the universal property
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
of that construction.
If the Lie algebra is semisimple, then all reducible
Reduction (mathematics)
In mathematics, reduction refers to the rewriting of an expression into a simpler form. For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible is called "reducing a fraction"...
representations are decomposable
Indecomposable module
In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules.Indecomposable is a weaker notion than simple module:simple means "no proper submodule" N...
. Otherwise, that's not true in general.
If we have two representations, with V1 and V2 as their underlying vector spaces and ·[·]1 and ·[·]2 as the representations, then the product of both representations would have
as the underlying vector space andIf L is a real Lie algebra and
is a complex representation of it, we can construct another representation of L called its dual representation as follows.Let V∗ be the dual vector space of V. In other words, V∗ is the set of all linear maps from V to C with addition defined over it in the usual linear way, but scalar multiplication defined over it such that
for any z in C, ω in V∗ and X in V. This is usually rewritten as a contraction with a sesquilinear form ⟨·,·⟩. i.e. ⟨ω,X⟩ is defined to be ω[X].We define
as follows:- ⟨
(A)[ω],X⟩ + ⟨ω,
A[X]⟩ = 0,
for any A in L, ω in V∗ and X in V. This defines
uniquely.Finite-dimensional representations of semisimple Lie algebras
Similarly to how semisimple Lie algebras can be classified, the finite-dimensional representations of semisimple Lie algebras can be classified. This is a classical theory, widely regarded as beautiful, and a standard reference is .Briefly, finite-dimensional representations of a semisimple Lie algebra are completely reducible, so it suffices to classify irreducible (simple) representations. Semisimple Lie algebras are classified in terms of the weights
Weight (representation theory)
In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F – a linear functional – or equivalently, a one dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group...
of the adjoint representation, the so-called root system
Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...
; in a similar manner all finite-dimensional irreducible representations can be understood in terms of weights; see weight (representation theory)
Weight (representation theory)
In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F – a linear functional – or equivalently, a one dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group...
for details.
Representation on an algebra
If we have a Lie superalgebra L, then a representation of L on an algebra is a (not necessarily associative) Z2 gradedGraded algebra
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
algebra
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
A which is a representation of L as a Z2 graded vector space
Graded vector space
In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.-N-graded vector spaces:...
and in addition, the elements of L acts as derivation
Derivation
Derivation may refer to:* Derivation , a function on an algebra which generalizes certain features of the derivative operator* Derivation * Derivation in differential algebra, a unary function satisfying the Leibniz product law...
s/antiderivations.
More specifically, if H is a pure element of L and x and y are pure elements of A,
- H[xy] = (H[x])y + (−1)xHx(H[y])
Also, if A is unital, then
- H[1] = 0
Now, for the case of a representation of a Lie algebra, we simply drop all the gradings and the (−1) to the some power factors.
A Lie (super)algebra is an algebra and it has an adjoint representation
Adjoint endomorphism
In mathematics, the adjoint endomorphism or adjoint action is an endomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras and Lie groups....
of itself. This is a representation on an algebra: the (anti)derivation property is the superJacobi identity
Jacobi identity
In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...
.
If a vector space is both an associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
and a Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
and the adjoint representation of the Lie algebra on itself is a representation on an algebra (i.e., acts by derivations on the associative algebra structure), then it is a Poisson algebra
Poisson algebra
In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups...
. The analogous observation for Lie superalgebras gives the notion of a Poisson superalgebra
Poisson superalgebra
In mathematics, a Poisson superalgebra is a Z2-graded generalization of a Poisson algebra. Specifically, a Poisson superalgebra is an superalgebra A with a Lie superbracket[\cdot,\cdot] : A\otimes A\to A...
.