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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 Semi-invariant of a quiver ： ウィキペディア英語版 Semi-invariant of a quiver In mathematics, given a quiver Q with set of vertices Q0 and set of arrows Q1, a representation of Q assigns a vector space ''V''''i'' to each vertex and a linear map ''V''(''α''): ''V''(''s''(''α'')) → ''V''(''t''(''α'')) to each arrow ''α'', where ''s''(''α''), ''t''(''α'') are, respectively, the starting and the ending vertices of α. Given an element d ∈ ℕQ0, the set of representations of Q with dim ''V''''i'' = d(i) for each ''i'' has a vector space structure. It is naturally endowed with an action of the algebraic group ∏i∈ Q0 GL(d(''i'')) by simultaneous base change. Such action induces one on the ring of functions. The ones which are invariants up to a character of the group are called semi-invariants. They form a ring whose structure reflects representation-theoretical properties of the quiver. == Definitions == Let Q = (Q0,Q1,''s'',''t'') be a quiver. Consider a dimension vector d, that is an element in ℕQ0. The set of d-dimensional representations is given by : $\backslash operatorname(Q,\backslash mathbf):=\backslash (i)\backslash \}$ Once fixed bases for each vector space ''V''''i'' this can be identified with the vector space : $\backslash bigoplus\_\; \backslash operatorname\_k(k^,\; k^)$ Such affine variety is endowed with an action of the algebraic group GL(d) := ∏''i''∈ Q0 GL(d(''i'')) by simultaneous base change on each vertex: : $$ \begin GL(\mathbf) \times \operatorname(Q,\mathbf) & \longrightarrow & \operatorname(Q,\mathbf)\\ \Big((g_i), (V_i, V(\alpha))\Big) & \longmapsto & (V_i,g_\cdot V(\alpha)\cdot g_^ ) \end By definition two modules ''M'',''N'' ∈ Rep(Q,d) are isomorphic if and only if their GL(d)-orbits coincide. We have an induces action on the coordinate ring ''k''() by defining: : $$ \begin GL(\mathbf) \times k() & \longrightarrow & k()\\ (g, f) & \longmapsto & g\cdot f(-):=f(g^. -) \end 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「Semi-invariant of a quiver」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.