In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the
theory of Lie groups and Lie algebras. The theory was worked out mainly by
E. Cartan and
H. Weyl and because of that, the theory is also known as the Cartan–Weyl theory. The theory gives the structural description and classification of a finite-dimensional
representation of a
semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals ...
(over
); in particular, it gives a way to parametrize (or classify) irreducible finite-dimensional representations of a semisimple Lie algebra, the result known as the
theorem of the highest weight.
There is a natural one-to-one correspondence between the finite-dimensional representations of a simply connected compact Lie group ''K'' and the finite-dimensional representations of the complex semisimple Lie algebra
that is the complexification of the Lie algebra of ''K'' (this fact is essentially a special case of the
Lie group–Lie algebra correspondence
In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are Isomorphism, isomorphic to each other have Lie algebra ...
). Also, finite-dimensional representations of a connected compact Lie group can be studied through finite-dimensional representations of the universal cover of such a group. Hence, the representation theory of semisimple Lie algebras marks the starting point for the general theory of
representations of connected compact Lie groups.
The theory is a basis for the later works of
Harish-Chandra
Harish-Chandra (né Harishchandra) FRS (11 October 1923 – 16 October 1983) was an Indian-American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups.
Early ...
that concern (infinite-dimensional) representation theory of
real reductive groups.
Classifying finite-dimensional representations of semisimple Lie algebras
There is a beautiful theory classifying the finite-dimensional representations of a semisimple Lie algebra over
. The finite-dimensional ''irreducible'' representations are described by a
theorem of the highest weight. The theory is described in various textbooks, including , , and .
Following an overview, the theory is described in increasing generality, starting with two simple cases that can be done "by hand" and then proceeding to the general result. The emphasis here is on the representation theory; for the geometric structures involving root systems needed to define the term "dominant integral element," follow the above link on weights in representation theory.
Overview
Classification of the finite-dimensional irreducible representations of a semisimple Lie algebra
over
or
generally consists of two steps. The first step amounts to analysis of hypothesized representations resulting in a tentative classification. The second step is actual realization of these representations.
A real Lie algebra is usually complexified enabling analysis in an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
. Working over the complex numbers in addition admits nicer bases. The following theorem applies: A real-linear finite-dimensional representation of a real Lie algebra extends to a complex-linear representation of its complexification. The real-linear representation is irreducible if and only if the corresponding complex-linear representation is irreducible. Moreover, a complex semisimple Lie algebra has the ''
complete reducibility property''. This means that every finite-dimensional representation decomposes as a direct sum of
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
s.
:Conclusion: ''Classification amounts to studying irreducible complex linear representations of the (complexified) Lie algebra.''
Classification: Step One
The first step is to ''hypothesize'' the existence of irreducible representations. That is to say, one hypothesizes that one has an irreducible representation
of a complex semisimple Lie algebra
without worrying about how the representation is constructed. The properties of these hypothetical representations are investigated, and conditions ''necessary'' for the existence of an irreducible representation are then established.
The properties involve the
weights of the representation. Here is the simplest description. Let
be a Cartan subalgebra of
, that is a maximal commutative subalgebra with the property that
is diagonalizable for each
, and let
be a basis for
. A ''weight''
for a representation
of
is a collection of simultaneous
eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s
:
for the commuting operators
. In basis-independent language,
is a linear functional
on
such that there exists a nonzero vector
such that
for every
.
A
partial ordering
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; ...
on the set of weights is defined, and the notion of ''highest weight'' in terms of this partial ordering is established for any set of weights. Using the structure on the Lie algebra, the notions
dominant element and
integral element
In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over a subring ''A'' of ''B'' if ''b'' is a root of a polynomial, root of some monic polynomial over ''A''.
If ''A'', ''B'' are field (mathematics), fields ...
are defined. Every finite-dimensional representation must have a maximal weight
, i.e., one for which no strictly higher weight occurs. If
is irreducible and
is a weight vector with weight
, then the entire space
must be generated by the action of
on
. Thus,
is a "highest weight cyclic" representation. One then shows that the weight
is actually the ''highest'' weight (not just maximal) and that every highest weight cyclic representation is irreducible. One then shows that two irreducible representations with the same highest weight are isomorphic. Finally, one shows that the highest weight
must be dominant and integral.
:Conclusion: ''Irreducible representations are classified by their highest weights, and the highest weight is always a dominant integral element.''
Step One has the side benefit that the structure of the irreducible representations is better understood. Representations decompose as direct sums of ''weight spaces'', with the weight space corresponding to the highest weight one-dimensional. Repeated application of the representatives of certain elements of the Lie algebra called ''lowering operators'' yields a set of generators for the representation as a vector space. The application of one such operator on a vector with definite weight results either in zero or a vector with ''strictly lower'' weight. ''Raising operators'' work similarly, but results in a vector with ''strictly higher'' weight or zero. The representatives of the Cartan subalgebra acts diagonally in a basis of weight vectors.
Classification: Step Two
Step Two is concerned with constructing the representations that Step One allows for. That is to say, we now fix a dominant integral element
and try to ''construct'' an irreducible representation with highest weight
.
There are several standard ways of constructing irreducible representations:
*Construction using
Verma modules. This approach is purely Lie algebraic. (Generally applicable to complex semisimple Lie algebras.)
*The ''
compact group approach'' using the
Peter–Weyl theorem. If, for example,
, one would work with the simply connected compact group
. (Generally applicable to complex semisimple Lie algebras.)
*Construction using the
Borel–Weil theorem, in which holomorphic representations of the group corresponding to
are constructed. (Generally applicable to complex semisimple Lie algebras.)
*Performing standard operations on ''known'' representations, in particular applying
Clebsch–Gordan decomposition to
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
s of representations. (Not generally applicable.)
[This approach is used heavily for ]classical Lie algebra The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types A_n , B_n , C_n and D_n , where for \mathfrak(n) the general linear Lie algebra and I_n the n \times n identity matrix:
...
s in . In the case
, this construction is described below.
*In the simplest cases, construction from scratch.
:Conclusion: Every ''dominant integral element of a complex semisimple Lie algebra gives rise to an irreducible, finite-dimensional representation. These are the only irreducible representations.''
The case of sl(2,C)
The Lie algebra sl(2,C) of the
special linear group
In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix ...
SL(2,C) is the space of 2x2 trace-zero matrices with complex entries. The following elements form a basis:
:
These satisfy the commutation relations
:
.
Every finite-dimensional representation of sl(2,C) decomposes as a direct sum of irreducible representations. This claim follows from the general result on complete reducibility of semisimple Lie algebras, or from the fact that sl(2,C) is the complexification of the Lie algebra of the simply connected compact group SU(2). The irreducible representations
, in turn, can be classified by the largest eigenvalue of
, which must be a non-negative integer ''m''. That is to say, in this case, a "dominant integral element" is simply a non-negative integer.
The irreducible representation with largest eigenvalue ''m'' has dimension
and is spanned by eigenvectors for
with eigenvalues
. The operators
and
move up and down the chain of eigenvectors, respectively. This analysis is described in detail in the
representation theory of SU(2) (from the point of the view of the complexified Lie algebra).
One can give a concrete realization of the representations (Step Two in the overview above) in either of two ways. First, in this simple example, it is not hard to write down an explicit basis for the representation and an explicit formula for how the generators
of the Lie algebra act on this basis. Alternatively, one can realize the representation with highest weight
by letting
denote the space of homogeneous polynomials of degree
in two complex variables, and then defining the action of
,
, and
by
:
Note that the formulas for the action of
,
, and
do not depend on
; the subscript in the formulas merely indicates that we are restricting the action of the indicated operators to the space of homogeneous polynomials of degree
in
and
.
The case of sl(3,C)
There is a similar
theory
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
classifying the irreducible representations of sl(3,C), which is the complexified Lie algebra of the group SU(3). The Lie algebra sl(3,C) is eight dimensional. We may work with a basis consisting of the following two diagonal elements
:
,
together with six other matrices
and
each of which has a 1 in an off-diagonal entry and zeros elsewhere. (The
's have a 1 above the diagonal and the
's have a 1 below the diagonal.)
The strategy is then to simultaneously diagonalize
and
in each irreducible representation
. Recall that in the sl(2,C) case, the action of
and
raise and lower the eigenvalues of
. Similarly, in the sl(3,C) case, the action of
and
"raise" and "lower" the eigenvalues of
and
. The irreducible representations are then classified by the largest eigenvalues
and
of
and
, respectively, where
and
are non-negative integers. That is to say, in this setting, a "dominant integral element" is precisely a pair of non-negative integers.
Unlike the representations of sl(2,C), the representation of sl(3,C) cannot be described explicitly in general. Thus, it requires an argument to show that ''every'' pair
actually arises the highest weight of some irreducible representation (Step Two in the overview above). This can be done as follows. First, we construct the "fundamental representations", with highest weights (1,0) and (0,1). These are the three-dimensional standard representation (in which
) and the dual of the standard representation. Then one takes a tensor product of
copies of the standard representation and
copies of the dual of the standard representation, and extracts an irreducible invariant subspace.
Although the representations cannot be described explicitly, there is a lot of useful information describing their structure. For example, the dimension of the irreducible representation with highest weight
is given by
:
There is also a simple pattern to the multiplicities of the various weight spaces. Finally, the irreducible representations with highest weight
can be realized concretely on the space of homogeneous polynomials of degree
in three complex variables.
The case of a general semisimple Lie algebras
Let
be a
semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals ...
and let
be a
Cartan subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
of
, that is, a maximal commutative subalgebra with the property that ad
''H'' is diagonalizable for all ''H'' in
. As an example, we may consider the case where
is sl(''n'',C), the algebra of ''n'' by ''n'' traceless matrices, and
is the subalgebra of traceless diagonal matrices. We then let ''R'' denote the associated
root system
In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and ...
. We then choose a base (or system of
positive simple roots)
for ''R''.
We now briefly summarize the structures needed to state the
theorem of the highest weight; more details can be found in the article on
weights in representation theory.
We choose an inner product on
that is invariant under the action of the
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections t ...
of ''R'', which we use to identify
with its dual space. If
is a representation of
, we define a
weight
In science and engineering, the weight of an object is a quantity associated with the gravitational force exerted on the object by other objects in its environment, although there is some variation and debate as to the exact definition.
Some sta ...
of ''V'' to be an element
in
with the property that for some nonzero ''v'' in ''V'', we have
for all ''H'' in
. We then define one weight
to be ''higher'' than another weight
if
is expressible as a linear combination of elements of
with non-negative real coefficients. A weight
is called a
''highest weight'' if
is higher than every other weight of
. Finally, if
is a weight, we say that
is
dominant if it has non-negative inner product with each element of
and we say that
is
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
if
is an integer for each
in ''R''.
Finite-dimensional representations of a semisimple Lie algebra are
completely reducible, so it suffices to classify irreducible (simple) representations. The irreducible representations, in turn, may be classified by the "theorem of the highest weight" as follows:
*Every irreducible, finite-dimensional representation of
has a highest weight, and this highest weight is dominant and integral.
*Two irreducible, finite-dimensional representations with the same highest weight are isomorphic.
*Every dominant integral element arises as the highest weight of some irreducible, finite-dimensional representation of
.
The last point of the theorem (Step Two in the overview above) is the most difficult one. In the case of the Lie algebra sl(3,C), the construction can be done in an elementary way, as described above. In general, the construction of the representations may be given by using
Verma modules.
Construction using Verma modules
If
is ''any'' weight, not necessarily dominant or integral, one can construct an infinite-dimensional representation
of
with highest weight
known as a
Verma module. The Verma module then has a maximal proper invariant subspace
, so that the
quotient representation is irreducible—and still has highest weight
. In the case that
is dominant and integral, we wish to show that
is finite dimensional.
The strategy for proving finite-dimensionality of
is to show that the set of weights of
is invariant under the action of the
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections t ...
of
relative to the given Cartan subalgebra
. (Note that the weights of the Verma module
itself are definitely not invariant under
.) Once this invariance result is established, it follows that
has only finitely many weights. After all, if
is a weight of
, then
must be integral—indeed,
must differ from
by an integer combination of roots—and by the invariance result,
must be lower than
for every
in
. But there are only finitely many integral elements
with this property. Thus,
has only finitely many weights, each of which has finite multiplicity (even in the Verma module, so certainly also in
). From this, it follows that
must be finite dimensional.
Additional properties of the representations
Much is known about the representations of a complex semisimple Lie algebra
, besides the classification in terms of highest weights. We mention a few of these briefly. We have already alluded to
Weyl's theorem, which states that every finite-dimensional representation of
decomposes as a direct sum of irreducible representations. There is also the
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...
, which leads to the
Weyl dimension formula (a formula for the dimension of the representation in terms of its highest weight), the
Kostant multiplicity formula (a formula for the multiplicities of the various weights occurring in a representation). Finally, there is also a formula for the eigenvalue of the
Casimir element
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum opera ...
, which acts as a scalar in each irreducible representation.
Lie group representations and Weyl's unitarian trick
Although it is possible to develop the representation theory of complex semisimple Lie algebras in a self-contained way, it can be illuminating to bring in a perspective using Lie ''groups''. This approach is particularly helpful in understanding
Weyl's theorem on complete reducibility. It is known that every complex semisimple Lie algebra
has a ''
compact real form''
. This means first that
is the complexification of
:
:
and second that there exists a simply connected compact group
whose Lie algebra is
. As an example, we may consider
, in which case
may be taken to be the special unitary group SU(n).
Given a finite-dimensional representation
of
, we can restrict it to
. Then since
is simply connected, we can integrate the representation to the group
. The method of averaging over the group shows that there is an inner product on
that is invariant under the action of
; that is, the action of
on
is ''unitary''. At this point, we may use unitarity to see that
decomposes as a direct sum of irreducible representations.
[ Section 4.4] This line of reasoning is called the ''
unitarian trick'' and was Weyl's original argument for what is now called Weyl's theorem. There is also a
purely algebraic argument for the complete reducibility of representations of semisimple Lie algebras.
If
is a complex semisimple Lie algebra, there is a unique complex semisimple Lie group
with Lie algebra
, in addition to the simply connected compact group
. (If
then
.) Then we have the following result about finite-dimensional representations.
Statement: The objects in the following list are in one-to-one correspondence:
* Smooth representations of
* Holomorphic representations of
* Real linear representations of
* Complex linear representations of
:Conclusion: ''The representation theory of compact Lie groups can shed light on the representation theory of complex semisimple Lie algebras.''
Remarks
Notes
References
*
*
*
*
*
* .
* {{citation, last=Rossmann, first= Wulf , title=Lie Groups: An Introduction Through Linear Groups, series= Oxford Graduate Texts in Mathematics, publisher= Oxford University Press, isbn= 978-0-19-859683-7, year=2002.
Lie algebras