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Glossary of semisimple groups

释义

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

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

J

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.

O

P

Parabolic subgroup
Peter–Weyl theorem
Positive root

Q

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

T

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

2 : Glossaries of mathematics|Lie groups

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