This is a glossary for the terminology applied in the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

theories of

Lie groups In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...

and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see

Glossary of representation theory This is a glossary of representation theory in mathematics. The term "module" is often used synonymously for a representation; for the module-theoretic terminology, see also glossary of module theory. See also Glossary of Lie groups and Lie algeb ...

. Because of the lack of other options, the glossary also includes some generalizations such as

quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...

. Notations: *Throughout the glossary,

( \cdot, \cdot )

denotes the

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

of a Euclidean space ''E'' and

\langle \cdot, \cdot \rangle

denotes the rescaled inner product ::

\langle \beta, \alpha \rangle = \frac \, \forall \alpha, \beta \in E.

A

B

C

D

E

F

G

H

I

J

K

L

N

M

P

Q

R

S

Classical Lie algebras: Exceptional Lie algebras:

T

U

Unitarian trick In mathematics, the unitarian trick is a device in the representation theory of Lie groups, introduced by for the special linear group and by Hermann Weyl for general semisimple groups. It applies to show that the representation theory of some g ...

V

Verma module Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Spe ...

W

References

* * Erdmann, Karin & Wildon, Mark. ''Introduction to Lie Algebras'', 1st edition, Springer, 2006. * Humphreys, James E. ''Introduction to Lie Algebras and Representation Theory'', Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1978. * Jacobson, Nathan, ''Lie algebras'', Republication of the 1962 original. Dover Publications, Inc., New York, 1979. * *

Claudio Procesi Claudio Procesi (born 31 March 1941 in Rome) is an Italian mathematician, known for works in algebra and representation theory. Career Procesi studied at the Sapienza University of Rome, where he received his degree (Laurea) in 1963. In 1966 he ...

(2007) ''Lie Groups: an approach through invariants and representation'', Springer, . *. *J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965) (Translated from French) {{DEFAULTSORT:Glossary Of Lie Algebras Lie Algebra Wikipedia glossaries using description lists