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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Glossary of algebraic groups ： ウィキペディア英語版 Glossary of algebraic groups There are a number of mathematical notions to study and classify algebraic groups. In the sequel, ''G'' denotes an algebraic group over a field ''k''. _n || Every affine algebraic group is isomorphic to a linear algebraic group, and vice-versa |- | affine algebraic group || An algebraic group which is an affine variety || $\_n$, non-example: elliptic curve || The notion of affine algebraic group stresses the independence from any embedding in $\_n$ |- | commutative || The underlying (abstract) group is abelian. || $\_a$ (the additive group), $\_m$ (the multiplicative group),〔These two are the only connected one-dimensional linear groups, 〕 any complete algebraic group (see abelian variety) || |- | diagonalizable group ||A closed subgroup of $(\backslash mathbb\_m)^n$, the group of diagonal matrices (of size ''n''-by-''n'') || || |- | simple algebraic group || A connected group which has no non-trivial connected normal subgroups || $\_n$ || |- | semisimple group || An affine algebraic group with trivial radical || $\_n$, $\_n$ || In characteristic zero, the Lie algebra of a semisimple group is a semisimple Lie algebra |- | reductive group || An affine algebraic group with trivial unipotent radical || Any finite group, $\_n$ || Any semisimple group is reductive |- | unipotent group || An affine algebraic group such that all elements are unipotent || The group of upper-triangular ''n''-by-''n'' matrices with all diagonal entries equal to 1 || Any unipotent group is nilpotent |- | torus || A group that becomes isomorphic to $(\backslash mathbb\_m)^n$ when passing to the algebraic closure of ''k''. || $\_2$ || ''G'' is said to be ''split'' by some bigger field ''k' '', if ''G'' becomes isomorphic to Gm''n'' as an algebraic group over ''k'.'' |- | character group ''X''∗(''G'') || The group of characters, i.e., group homomorphisms $G\backslash rightarrow\_m$ || $X^$ *(\mathbb_m)\cong\mathbb || |- | Lie algebra Lie(''G'') || The tangent space of ''G'' at the unit element. || $(\_n)$) is the space of all ''n''-by-''n'' matrices || Equivalently, the space of all left-invariant derivations. |} ==References== 〔 * * 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「Glossary of algebraic groups」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.