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

• Adjoint representation

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

• Affine Lie algebra

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

• Algebraic group

B

• (B, N) pair
• Borel subgroup
• Borel-Bott-Weil theorem
• Bruhat decomposition

C

• Cartan decomposition
• Cartan matrix
• Cartan subalgebra
• Cartan subgroup
• Casimir invariant
• Clebsch–Gordan coefficients
• Compact Lie group
• Complex reflection group
• coroot
• Coxeter group
• Coxeter number
• Cuspidal representation

D

• Discrete series
• Dominant weight

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.

• Dynkin diagram

E

• E6 (mathematics)
• E7 (mathematics)
• E7½ (Lie algebra)
• E8 (mathematics)
• En (Lie algebra)
• Exceptional Lie algebra

F

• F4 (mathematics)
• Flag manifold
• Fundamental representation

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

$Alt^k\ {\mathbb C}^n$

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

• Fundamental Weyl chamber

G

• G2 (mathematics)
• Generalized Cartan matrix
• Generalized Kac–Moody algebra
• Generalized Verma module

H

• Harish-Chandra homomorphism
• Highest weight
• Highest weight module

I

• Iwasawa decomposition

K

• Kac–Moody algebra
• Killing form
• Kirillov character formula

L

• Langlands decomposition
• Langlands dual
• Levi decomposition
• Lie algebra
• Lowest weight

M

• Maximal compact subgroup
• Maximal torus

N

• Nilpotent cone

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

P

• Parabolic subgroup
• Peter–Weyl theorem
• Positive root

R

• Real form
• Reductive group
• Reflection group
• Root datum
• Root system

S

• Schur polynomial

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

• Semisimple Lie algebra
• Semisimple Lie group
• Simple Lie algebra
• Simple Lie group
• Simple root
• Simply laced group

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

• Steinberg representation

U

• Unitarian trick

V

• Verma module

W

• Weight (representation theory)
• Weight module
• Weight space
• Weyl chamber

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.

• Weyl character formula

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

• Weyl group