Glossary of Lie groups and Lie algebras explained

This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.

Notations:

• Throughout the glossary,

( ⋅ , ⋅ ) denotes the inner product of a Euclidean space E and \langle ⋅ , ⋅ \rangle denotes the rescaled inner product\langle \beta, \alpha \rangle = ((\beta, \alpha))/((\alpha, \alpha)) \forall \alpha, \beta \in E.

S

Classical Lie algebras:

Special linear algebra	A _{l} (l \ge 1 )	l ^{2} + 2 l	ak{sl}(l+1, F) = \{ x \in ak{gl}(l+1,F) | Tr(x) = 0 \} (traceless matrices)
Orthogonal algebra	B _{l} (l \ge 1 )	2 l ^{2} + l	ak{o}(2l+1, F) = \{ x \in ak{gl}(2l+1,F) | s x = - x ^{t} s , s = \begin{pmatrix} 1 & 0 & 0 \ 0 & 0 & I _{l} \ 0 & I _{l} & 0 \end{pmatrix}\}
Symplectic algebra	C _{l} (l \ge 2 )	2 l ^{2} - l	ak{sp}(2l, F) = \{ x \in ak{gl}(2l,F) | s x = - x ^{t} s, s = \begin{pmatrix} 0 & I _{l} \ - I _{l} & 0 \end{pmatrix}\}
Orthogonal algebra	D _{l} (l \ge 1 )	2 l ^{2} + l	ak{o}(2l, F) = \{ x \in ak{gl}(2l,F) | s x = - x ^{t} s, s = \begin{pmatrix} 0 & I _{l} \ I _{l} & 0 \end{pmatrix}\}

Exceptional Lie algebras:

Root System	dimension
G2	14
F4	52
E6	78
E7	133
E8	248

U

• Unitarian trick

V

• Verma module

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.
• Book: Kac, Victor. Victor Kac

. Victor Kac. Infinite dimensional Lie algebras. 3rd . . 1990. 0-521-46693-8.

• Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, .
• .
• J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965) (Translated from French)