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Lie bracket : ウィキペディア英語版
Lie algebra

In mathematics, a Lie algebra (, not ) is a vector space together with a non-associative multiplication called "Lie bracket" (y ). It was introduced to study the concept of infinitesimal transformations. Hermann Weyl introduced the term "Lie algebra" (after Sophus Lie) in the 1930s. In older texts, the name "infinitesimal group" is used.
Lie algebras are closely related to Lie groups which are groups that are also smooth manifolds, with the property that the group operations of multiplication and inversion are smooth maps. Any Lie group gives rise to a Lie algebra. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering (Lie's third theorem). This correspondence between Lie groups and Lie algebras allows one to study Lie groups in terms of Lie algebras.
== Definitions ==

A Lie algebra is a vector space \,\mathfrak over some field ''F'' together with a binary operation (): \mathfrak\times\mathfrak\to\mathfrak called the Lie bracket that satisfies the following axioms:
* Bilinearity,
:: (x + b y, z ) = a (z ) + b (z ), \quad (a x + b y ) = a(x ) + b (y )
:for all scalars ''a'', ''b'' in ''F'' and all elements ''x'', ''y'', ''z'' in \mathfrak.
* Alternativity,
:: ()=0\
:for all ''x'' in \mathfrak.
* The Jacobi identity,
:: ] + and using alternativity shows that () + ()=0\ for all elements ''x'', ''y'' in \mathfrak, showing that bilinearity and alternativity together imply
*Anticommutativity,
::,
:for all elements ''x'', ''y'' in \mathfrak. Anticommutativity only implies the alternating property if the field's characteristic is not 2.〔Humphreys p. 1〕
It is customary to express a Lie algebra in lower-case fraktur, like \mathfrak. If a Lie algebra is associated with a Lie group, then the spelling of the Lie algebra is the same as that Lie group. For example, the Lie algebra of SU(''n'') is written as \mathfrak(n).

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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