Glossary of Lie groups and Lie algebras

Template:Short description 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.

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Throughout the glossary, $(\cdot, \cdot)$ denotes the inner product of a Euclidean space E and $⟨ \cdot, \cdot ⟩$ denotes the rescaled inner product

⟨ β, α ⟩ = \frac{(β, α)}{(α, α)} \forall α, β \in E .

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Classical Lie algebras:

Special linear algebra	$A_{l} (l \geq 1)$	$l^{2} + 2 l$	$𝔰 𝔩 (l + 1, F) = {x \in 𝔤 𝔩 (l + 1, F) \| T r (x) = 0}$ (traceless matrices)
Orthogonal algebra	$B_{l} (l \geq 1)$	$2 l^{2} + l$	$𝔬 (2 l + 1, F) = {x \in 𝔤 𝔩 (2 l + 1, F) \| s x = - x^{t} s, s = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & I_{l} \\ 0 & I_{l} & 0 \end{matrix})}$
Symplectic algebra	$C_{l} (l \geq 2)$	$2 l^{2} - l$	$𝔰 𝔭 (2 l, F) = {x \in 𝔤 𝔩 (2 l, F) \| s x = - x^{t} s, s = (\begin{matrix} 0 & I_{l} \\ - I_{l} & 0 \end{matrix})}$
Orthogonal algebra	$D_{l} (l \geq 1)$	$2 l^{2} + l$	$𝔬 (2 l, F) = {x \in 𝔤 𝔩 (2 l, F) \| s x = - x^{t} s, s = (\begin{matrix} 0 & I_{l} \\ I_{l} & 0 \end{matrix})}$

Exceptional Lie algebras:

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

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Unitarian trick

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Verma module

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References

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Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. Template:ISBN
Humphreys, James E. Introduction to Lie Algebras and Representation Theory, Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1978. Template:ISBN
Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. Template:Isbn
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Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, Template:Isbn.
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J.-P. Serre, "Lie algebras and Lie groups", Benjamin (1965) (Translated from French)

Glossary of Lie groups and Lie algebras

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Glossary of Lie groups and Lie algebras

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