Glossary of Lie groups and Lie algebras

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, ${\displaystyle (\cdot ,\cdot )}$ denotes the inner product of a Euclidean space E and ${\displaystyle \langle \cdot ,\cdot \rangle }$ denotes the rescaled inner product

${\displaystyle \langle \beta ,\alpha \rangle ={\frac {(\beta ,\alpha )}{(\alpha ,\alpha )}}\,\forall \alpha ,\beta \in E.}$