Representation theory of semisimple Lie algebras

[1] The theory gives the structural description and classification of a finite-dimensional representation of a semisimple Lie algebra (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

Hence, the representation theory of semisimple Lie algebras marks the starting point for the general theory of representations of connected compact Lie groups.

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 Fulton & Harris (1991), Hall (2015), and Humphreys (1972).

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.

Classification of the finite-dimensional irreducible representations of a semisimple Lie algebra

The first step amounts to analysis of hypothesized representations resulting in a tentative classification.

Working over the complex numbers in addition admits nicer bases.

[2] Moreover, a complex semisimple Lie algebra has the complete reducibility property.

Step One has the side benefit that the structure of the irreducible representations is better understood.

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.

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.

There are several standard ways of constructing irreducible representations: The Lie algebra sl(2,C) of the special linear group SL(2,C) is the space of 2x2 trace-zero matrices with complex entries.

This claim follows from the general result on complete reducibility of semisimple Lie algebras,[11] or from the fact that sl(2,C) is the complexification of the Lie algebra of the simply connected compact group SU(2).

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.

; 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

There is a similar theory[16] classifying the irreducible representations of sl(3,C), which is the complexified Lie algebra of the group SU(3).

We may work with a basis consisting of the following two diagonal elements together with six other matrices

That is to say, in this setting, a "dominant integral element" is precisely a pair of non-negative integers.

actually arises the highest weight of some irreducible representation (Step Two in the overview above).

copies of the dual of the standard representation, and extracts an irreducible invariant subspace.

In the case of the Lie algebra sl(3,C), the construction can be done in an elementary way, as described above.

is any weight, not necessarily dominant or integral, one can construct an infinite-dimensional representation

Finally, there is also a formula for the eigenvalue of the Casimir element, which acts as a scalar in each irreducible representation.

This approach is particularly helpful in understanding Weyl's theorem on complete reducibility.