Английская Википедия:Glossary of Lie groups and Lie algebras

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Шаблон: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.

Шаблон:Lie groupsNotations:

• Throughout the glossary, <math>( \cdot, \cdot )</math> denotes the inner product of a Euclidean space E and <math>\langle \cdot, \cdot \rangle</math> denotes the rescaled inner product

<math>\langle \beta, \alpha \rangle = \frac{(\beta, \alpha)}{(\alpha, \alpha)} \, \forall \alpha, \beta \in E. </math>

C

Шаблон:Term Шаблон:Defn(X) := \{x \in \mathfrak{g} | [x, X] = \{0\} \}</math>.}}

L

Шаблон:Term Шаблон:Defn Шаблон:Defn {{defn|no=3|1=A Lie algebra is a vector space <math>\mathfrak{g}</math> over a field <math>F</math> with a binary operation [·, ·] (called the Lie bracket or abbr. bracket) , which satisfies the following conditions: <math>\forall a,b \in F, x,y,z \in \mathfrak{g}</math>,

1. <math>[ax+by,z] = a[x,z] + b[y,z]</math> (bilinearity)
2. <math>[x,x] = 0</math> (alternating)
3. <math>[[x,y], z ] + [[y,z],x] + [[z,x],y] = 0</math> (Jacobi identity)}}

N

Шаблон:Term Шаблон:Defn(K) := \{x \in \mathfrak{g} | [x, K] \subseteq K \}</math>.}}

S

Classical Lie algebras:

Special linear algebra	<math>A_l \ (l \ge 1)</math>	<math>l^2 + 2l</math>	Tr(x) = 0 \}</math> (traceless matrices)
Orthogonal algebra	<math>B_l \ (l \ge 1)</math>	<math>2 l^2 + l</math>	s x = - x^t s , s = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & I_l \\ 0 & I_l & 0 \end{pmatrix}\}</math>
Symplectic algebra	<math>C_l \ (l \ge 2)</math>	<math>2 l^2 - l</math>	s x = - x^t s, s = \begin{pmatrix} 0 & I_l \\ -I_l & 0 \end{pmatrix}\}</math>
Orthogonal algebra	<math>D_l (l \ge 1)</math>	<math>2 l^2 + l</math>	s x = - x^t s, s = \begin{pmatrix} 0 & I_l \\ I_l & 0 \end{pmatrix}\}</math>

Exceptional Lie algebras:

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

U

• Unitarian trick

V

• Verma module

References

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