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In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite; this definition is more restrictive and excludes tori, though allowing negative semidefinite includes tori and agrees with the previous definition. A compact Lie algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification. == Definition == Formally, one may define a compact Lie algebra either as the Lie algebra of a compact Lie group, or as a real Lie algebra whose Killing form is negative definite. These definitions do not quite agree:〔 * The Killing form on the Lie algebra of a compact Lie group is negative ''semi''definite, not negative definite in general. * If the Killing form of a Lie algebra is negative definite, then the Lie algebra is the Lie algebra of a compact Lie group. The difference is precisely in whether to include tori (and their corresponding Lie algebra, which is abelian and hence has trivial Killing form) or not: real Lie algebras with negative definite Killing forms correspond to compact semisimple Lie groups, while real Lie algebras with negative semidefinite Killing forms correspond to products of compact semisimple Lie groups and tori. One can distinguish between these by calling a Lie algebra with negative semidefinite Killing form a compact ''reductive'' Lie algebra, and a Lie algebra with negative definite Killing form a compact ''semisimple'' Lie algebra, which corresponds to reductive Lie algebras being direct sums of semisimple and abelian. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Compact Lie algebra」の詳細全文を読む スポンサード リンク
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