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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 Hall algebra ： ウィキペディア英語版 Hall algebra In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian ''p''-groups. It was first discussed by E. but forgotten until it was rediscovered by , both of whom published no more than brief summaries of their work. The Hall polynomials are the structure constants of the Hall algebra. The Hall algebra plays an important role in the theory of Kashiwara–Lusztig's canonical bases in quantum groups. generalized Hall algebras to more general categories, such as the category of representations of a quiver. == Construction == A finite abelian ''p''-group ''M'' is a direct sum of cyclic ''p''-power components $C\_(p)$ be the number of subgroups ''N'' of ''M'' such that ''N'' has type $\backslash nu$ and the quotient ''M/N'' has type $\backslash mu$. Hall proved that the functions ''g'' are polynomial functions of ''p'' with integer coefficients. Thus we may replace ''p'' with an indeterminate ''q'', which results in the Hall polynomials : $g^\backslash lambda\_(q)\backslash in\backslash mathbb().\; \backslash ,$ Hall next constructs an associative ring $H$ over $\backslash mathbb()$, now called the Hall algebra. This ring has a basis consisting of the symbols $u\_\backslash lambda$ and the structure constants of the multiplication in this basis are given by the Hall polynomials: :$u\_\backslash mu\; u\_\backslash nu\; =\; \backslash sum\_\backslash lambda\; g^\backslash lambda\_(q)\; u\_\backslash lambda.\; \backslash ,$ It turns out that ''H'' is a commutative ring, freely generated by the elements $u\_$ corresponding to the elementary ''p''-groups. The linear map from ''H'' to the algebra of symmetric functions defined on the generators by the formula : $u\_\; \backslash mapsto\; q^e\_n\; \backslash ,$ (where ''e''''n'' is the ''n''th elementary symmetric function) uniquely extends to a ring homomorphism and the images of the basis elements $u\_\backslash lambda$ may be interpreted via the Hall–Littlewood symmetric functions. Specializing ''q'' to 1, these symmetric functions become Schur functions, which are thus closely connected with the theory of Hall polynomials. 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「Hall algebra」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.