Glossary of semisimple groups

This is a glossary for the terminology applied in the mathematical theories of semisimple Lie groups. It also covers terms related to their Lie algebras, their representation theory, and various geometric, algebraic and combinatorial structures that occur in connection with the development of what is a central theory of contemporary mathematics.

A

The adjoint representation of any Lie group is its action on its Lie algebra, derived from the conjugation action of the group on itself.

An affine Lie algebra is a particular type of Kac–Moody algebra.

B

C

D

The irreducible representations of a simply-connected compact Lie group are indexed by their highest weight. These dominant weights form the lattice points in an orthant in the weight lattice of the Lie group.

E

F

For the irreducible representations of a simply-connected compact Lie group there exists a set of fundamental weights, indexed by the vertices of the Dynkin diagram of G, such that dominant weights are simply non-negative integer linear combinations of the fundamental weights.

The corresponding irreducible representations are the fundamental representations of the Lie group. In particular, from the expansion of a dominant weight in terms of the fundamental weights, one can take a corresponding tensor product of the fundamental representations and extract one copy of the irreducible representation corresponding to that dominant weight.

In the case of the special unitary group SU(n), the n − 1 fundamental representations are the wedge products

consisting of alternating tensors, for k=1,2,...,n-1.

G

H

I

K

L

M

N

Elements in a semisimple Lie algebra that are represented in every linear representation by a nilpotent endomorphism.

P

R

S

A Schur polynomial is a symmetric function, of a type occurring in the Weyl character formula applied to unitary groups.

A simple Lie group is simply laced when its Dynkin diagram is without multiple edges

U

V

W

A Weyl chamber is one of the connected components of the complement in V, a real vector space on which a root system is defined, when the hyperplanes orthogonal to the root vectors are removed.

The Weyl character formula gives in closed form the characters of the irreducible complex representations of the simple Lie groups.

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