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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
as a set of
matrices (or
endomorphisms of a
vector space) in such a way that the Lie bracket is given by the
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 commutator of two elements, a ...
. In the language of physics, one looks for a vector space
together with a collection of operators on
satisfying some fixed set of commutation relations, such as the relations satisfied by the
angular momentum operators.
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.
In the study of representations of a Lie algebra, a particular
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, called the
universal enveloping algebra, associated with the Lie algebra plays an important role. The universality of this ring says that the
category of representations of a Lie algebra is the same as the category of
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s over its enveloping algebra.
Formal definition
Let
be a Lie algebra and let
be a vector space. We let
denote the space of endomorphisms of
, that is, the space of all linear maps of
to itself. We make
into a Lie algebra with bracket given by the commutator:
for all ''ρ,σ'' in
. Then a representation of
on
is a
Lie algebra homomorphism
:
.
Explicitly, this means that
should be a linear map and it should satisfy
:
for all ''X, Y'' in
. The vector space ''V'', together with the representation ''ρ'', is called a
-module. (Many authors abuse terminology and refer to ''V'' itself as the representation).
The representation
is said to be faithful if it is injective.
One can equivalently define a
-module as a vector space ''V'' together with a
bilinear map such that
:
for all ''X,Y'' in
and ''v'' in ''V''. This is related to the previous definition by setting ''X'' ⋅ ''v'' = ''ρ''(''X'')(''v'').
Examples
Adjoint representations
The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra
on itself:
:
Indeed, by virtue of the
Jacobi identity,
is a Lie algebra homomorphism.
Infinitesimal Lie group representations
A Lie algebra representation also arises in nature. If
: ''G'' → ''H'' is a
homomorphism of (real or complex)
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s, and
and
are the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s of ''G'' and ''H'' respectively, then the
differential on
tangent spaces at the identities is a Lie algebra homomorphism. In particular, for a finite-dimensional vector space ''V'', a
representation of Lie groups
:
determines a Lie algebra homomorphism
:
from
to the Lie algebra of the
general linear group GL(''V''), i.e. the endomorphism algebra of ''V''.
For example, let
. Then the differential of
at the identity is an element of
. Denoting it by
one obtains a representation
of ''G'' on the vector space
. This is the
adjoint representation of ''G''. Applying the preceding, one gets the Lie algebra representation
. It can be shown that
, the adjoint representation 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
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras.
In quantum physics
In quantum theory, one considers "observables" that are self-adjoint operators on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. The commutation relations among these operators are then an important tool. The
angular momentum operators, for example, satisfy the commutation relations
:
.
Thus, the span of these three operators forms a Lie algebra, which is isomorphic to the Lie algebra so(3) of the
rotation group SO(3). Then if
is any subspace of the quantum Hilbert space that is invariant under the angular momentum operators,
will constitute a representation of the Lie algebra so(3). An understanding of the representation theory of so(3) is of great help in, for example, analyzing Hamiltonians with rotational symmetry, such as the
hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen consti ...
. Many other interesting Lie algebras (and their representations) arise in other parts of quantum physics. Indeed, the history of representation theory is characterized by rich interactions between mathematics and physics.
Basic concepts
Invariant subspaces and irreducibility
Given a representation
of a Lie algebra
, we say that a subspace
of
is invariant if
for all
and
. A nonzero representation is said to be irreducible if the only invariant subspaces are
itself and the zero space
. The term ''simple module'' is also used for an irreducible representation.
Homomorphisms
Let
be a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
. Let ''V'', ''W'' be
-modules. Then a linear map
is a homomorphism of
-modules if it is
-equivariant; i.e.,
for any
. If ''f'' is bijective,
are said to be equivalent. Such maps are also referred to as intertwining maps or morphisms.
Similarly, many other constructions from module theory in abstract algebra carry over to this setting: submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc.
Schur's lemma
A simple but useful tool in studying irreducible representations is Schur's lemma. It has two parts:
*If ''V'', ''W'' are irreducible
-modules and
is a homomorphism, then
is either zero or an isomorphism.
*If ''V'' is an irreducible
-module over an algebraically closed field and
is a homomorphism, then
is a scalar multiple of the identity.
Complete reducibility
Let ''V'' be a representation of a Lie algebra
. Then ''V'' is said to be completely reducible (or semisimple) if it is isomorphic to a direct sum of irreducible representations (cf.
semisimple module
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself ...
). If ''V'' is finite-dimensional, then ''V'' is completely reducible if and only if every invariant subspace of ''V'' has an invariant complement. (That is, if ''W'' is an invariant subspace, then there is another invariant subspace ''P'' such that ''V'' is the direct sum of ''W'' and ''P''.)
If
is a finite-dimensional
semisimple Lie algebra over a field of characteristic zero and ''V'' is finite-dimensional, then ''V'' is semisimple; this is
Weyl's complete reducibility theorem. Thus, for semisimple Lie algebras, a classification of irreducible (i.e. simple) representations leads immediately to classification of all representations. For other Lie algebra, which do not have this special property, classifying the irreducible representations may not help much in classifying general representations.
A Lie algebra is said to be
reductive if the adjoint representation is semisimple. Certainly, every (finite-dimensional) semisimple Lie algebra
is reductive, since ''every'' representation of
is completely reducible, as we have just noted. In the other direction, the definition of a reductive Lie algebra means that it decomposes as a direct sum of ideals (i.e., invariant subspaces for the adjoint representation) that have no nontrivial sub-ideals. Some of these ideals will be one-dimensional and the rest are simple Lie algebras. Thus, a reductive Lie algebra is a direct sum of a commutative algebra and a semisimple algebra.
Invariants
An element ''v'' of ''V'' is said to be
-invariant if
for all
. The set of all invariant elements is denoted by
.
Basic constructions
Tensor products of representations
If we have two representations of a Lie algebra
, with ''V''
1 and ''V''
2 as their underlying vector spaces, then the tensor product of the representations would have ''V''
1 ⊗ ''V''
2 as the underlying vector space, with the action of
uniquely determined by the assumption that
:
for all
and
.
In the language of homomorphisms, this means that we define
by the formula
:
.
In the physics literature, the tensor product with the identity operator is often suppressed in the notation, with the formula written as
:
,
where it is understood that
acts on the first factor in the tensor product and
acts on the second factor in the tensor product. In the context of representations of the Lie algebra su(2), the tensor product of representations goes under the name "addition of angular momentum." In this context,
might, for example, be the orbital angular momentum while
is the spin angular momentum.
Dual representations
Let
be a Lie algebra and
be a representation of
. Let
be the dual space, that is, the space of linear functionals on
. Then we can define a representation
by the formula
:
where for any operator
, the transpose operator
is defined as the "composition with
" operator:
:
The minus sign in the definition of
is needed to ensure that
is actually a representation of
, in light of the identity
If we work in a basis, then the transpose in the above definition can be interpreted as the ordinary matrix transpose.
Representation on linear maps
Let
be
-modules,
a Lie algebra. Then
becomes a
-module by setting
. In particular,
; that is to say, the
-module homomorphisms from
to
are simply the elements of
that are invariant under the just-defined action of
on
. If we take
to be the base field, we recover the action of
on
given in the previous subsection.
Representation theory of semisimple Lie algebras
See
Representation theory of semisimple Lie algebras.
Enveloping algebras
To each Lie algebra
over a field ''k'', one can associate a certain
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
called the universal enveloping algebra of
and denoted
. The universal property of the universal enveloping algebra guarantees that every representation of
gives rise to a representation of
. Conversely, the
PBW theorem tells us that
sits inside
, so that every representation of
can be restricted to
. Thus, there is a one-to-one correspondence between representations of
and those of
.
The universal enveloping algebra plays an important role in the representation theory of semisimple Lie algebras, described above. Specifically, the finite-dimensional irreducible representations are constructed as quotients of
Verma modules, and Verma modules are constructed as quotients of the universal enveloping algebra.
The construction of
is as follows. Let ''T'' be the
tensor algebra of the vector space
. Thus, by definition,
and the multiplication on it is given by
. Let
be the
quotient ring of ''T'' by the ideal generated by elements of the form
:
.
There is a natural linear map from
into
obtained by restricting the quotient map of
to degree one piece. The
PBW theorem implies that the canonical map is actually injective. Thus, every Lie algebra
can be embedded into an associative algebra
in such a way that the bracket on
is given by
in
.
If
is
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
, then
is the symmetric algebra of the vector space
.
Since
is a module over itself via adjoint representation, the enveloping algebra
becomes a
-module by extending the adjoint representation. But one can also use the left and right
regular representation to make the enveloping algebra a
-module; namely, with the notation
, the mapping
defines a representation of
on
. The right regular representation is defined similarly.
Induced representation
Let
be a finite-dimensional Lie algebra over a field of characteristic zero and
a subalgebra.
acts on
from the right and thus, for any
-module ''W'', one can form the left
-module
. It is a
-module denoted by
and called the
-module induced by ''W''. It satisfies (and is in fact characterized by) the universal property: for any
-module ''E''
:
.
Furthermore,
is an exact functor from the category of
-modules to the category of
-modules. These uses the fact that
is a free right module over
. In particular, if
is simple (resp. absolutely simple), then ''W'' is simple (resp. absolutely simple). Here, a
-module ''V'' is absolutely simple if
is simple for any field extension
.
The induction is transitive:
for any Lie subalgebra
and any Lie subalgebra
. The induction commutes with restriction: let
be subalgebra and
an ideal of
that is contained in
. Set
and
. Then
.
Infinite-dimensional representations and "category O"
Let
be a finite-dimensional semisimple Lie algebra over a field of characteristic zero. (in the solvable or nilpotent case, one studies
primitive ideals of the enveloping algebra; cf. Dixmier for the definitive account.)
The category of (possibly infinite-dimensional) modules over
turns out to be too large especially for homological algebra methods to be useful: it was realized that a smaller subcategory
category O is a better place for the representation theory in the semisimple case in zero characteristic. For instance, the category O turned out to be of a right size to formulate the celebrated BGG reciprocity.
Why the BGG category O?
/ref>
(g,K)-module
One of the most important applications of Lie algebra representations is to the representation theory of real reductive Lie groups. The application is based on the idea that if is a Hilbert-space representation of, say, a connected real semisimple linear Lie group ''G'', then it has two natural actions: the complexification and the connected maximal compact subgroup ''K''. The -module structure of allows algebraic especially homological methods to be applied and -module structure allows harmonic analysis to be carried out in a way similar to that on connected compact semisimple Lie groups.
Representation on an algebra
If we have a Lie superalgebra ''L'', then a representation of ''L'' on an algebra is a (not necessarily associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
) Z2 graded algebra ''A'' which is a representation of ''L'' as a Z2 graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be th ...
and in addition, the elements of ''L'' acts as derivations/antiderivation In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring or a field ''K'', a ''K''-derivation is a ''K''-linear map that satisfies Le ...
s on ''A''.
More specifically, if ''H'' is a pure element
A chemical element is a species of atoms that have a given number of protons in their nuclei, including the pure substance consisting only of that species. Unlike chemical compounds, chemical elements cannot be broken down into simpler subst ...
of ''L'' and ''x'' and ''y'' are pure element
A chemical element is a species of atoms that have a given number of protons in their nuclei, including the pure substance consisting only of that species. Unlike chemical compounds, chemical elements cannot be broken down into simpler subst ...
s of ''A'',
:''H'' 'xy''= (''H'' 'x''''y'' + (−1)''xH''''x''(''H'' 'y''
Also, if ''A'' is unital, then
:''H'' = 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 of itself. This is a representation on an algebra: the (anti)derivation property is the super
Super may refer to:
Computing
* SUPER (computer program), or Simplified Universal Player Encoder & Renderer, a video converter / player
* Super (computer science), a keyword in object-oriented programming languages
* Super key (keyboard butt ...
Jacobi identity.
If a vector space is both an associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
and a Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
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 In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central i ...
. The analogous observation for Lie superalgebras gives the notion of a Poisson superalgebra In mathematics, a Poisson superalgebra is a Z2- graded generalization of a Poisson algebra. Specifically, a Poisson superalgebra is an (associative) superalgebra ''A'' with a Lie superbracket
: cdot,\cdot: A\otimes A\to A
such that (''A'', ,· is ...
.
See also
* Representation of a Lie group
* Weight (representation theory)
* Weyl's theorem on complete reducibility
* Root system
* Weyl character formula
* Representation theory of a connected compact Lie group
* Whitehead's lemma (Lie algebras)
* Kazhdan–Lusztig conjectures
*Quillen's lemma In algebra, Quillen's lemma states that an endomorphism of a simple module over the enveloping algebra of a finite-dimensional Lie algebra over a field ''k'' is algebraic over ''k''. In contrast to a version of Schur's lemma due to Dixmier, it ...
- analog of Schur's lemma
Notes
References
*Bernstein I.N., Gelfand I.M., Gelfand S.I., "Structure of Representations that are generated by vectors of highest weight," Functional. Anal. Appl. 5 (1971)
*.
*A. Beilinson and J. Bernstein, "Localisation de g-modules," Comptes Rendus de l'Académie des Sciences, Série I, vol. 292, iss. 1, pp. 15–18, 1981.
*
*
*
* D. Gaitsgory
Geometric Representation theory, Math 267y, Fall 2005
*
*
*
* Ryoshi Hotta, Kiyoshi Takeuchi, Toshiyuki Tanisaki, ''D-modules, perverse sheaves, and representation theory''; translated by Kiyoshi Takeuch
*
*N. Jacobson, ''Lie algebras'', Courier Dover Publications, 1979.
*
*
* (elementary treatment for SL(2,C))
*
Further reading
*
{{DEFAULTSORT:Lie Algebra Representation