フロベニウス多元環のソースを表示

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'''フロベニウス多元環'''（フロベニウスたげんかん、{{lang-en-short|Frobenius algebra}}）、あるいは'''フロベニウス代数'''とは、[[数学]]の[[表現論]]や[[加群論]]において[[次元 (線型代数学)|有限次元]]な[[単位的環|単位的]][[結合多元環]]のうち、良い双対理論を与える特別な[[双線型形式]]を持つものをいう。

フロベニウス多元環は1930年代に [[Richard Brauer|Brauer]] と [[Cecil J. Nesbitt|Nesbitt]] によって[[有限群]]の[[モジュラー表現論|モジュラー表現]]の一般化として研究され始め{{sfn|Weibel|1994|loc=Definition 4.2.5|p={{google books quote|id=flm-dBXfZ_gC|page=96|96}}}}、[[ゲオルク・フロベニウス|Frobenius]] にちなんで名づけられた。[[中山正|中山]]は {{harv|Nakayama|1939}} および特に {{harv|Nakayama|1941}} において豊かな双対理論を初めて発見した。[[ジャン・デュドネ|デュドネ]]はこれを用いて {{harv|Dieudonné|1958}} においてフロベニウス多元環を特徴づけ、フロベニウス多元環のこの性質を ''perfect duality'' と呼んだ。フロベニウス多元環は[[準フロベニウス環]]（右[[正則表現 (数学)|正則表現]]が[[移入加群|移入的な]][[ネーター環]]）へと一般化された。最近では、フロベニウス多元環への関心は、[[位相的場の理論]]との関連からも高まっている。

体上の有限次元多元環に対しては以下のようなクラスの階層がある。
:[[自己移入環|自己入射多元環]] &sup; フロベニウス多元環 &sup; 対称多元環 &sup; [[半単純多元環]] &sup; [[単純環|単純多元環]] &sup; [[可除多元環]]

== 定義 ==
[[可換体|体]] {{mvar|k}} 上の[[次元 (線型代数学)|有限次元]]な[[単位的環|単位的]][[結合多元環]] {{mvar|A}} が'''フロベニウス多元環'''であるとは、[[移入加群|移入]]右 {{mvar|A}} 加群 {{math|''DA'' :{{=}} Hom<sub>''k''</sub>(''A'', ''k'')}} が右[[正則表現 (数学)|正則表現]] {{mvar|A}} に同型であることである{{sfn|Lam|1999|p=66}}。これは[[非退化形式|非退化]][[双線型形式]] {{math|''&sigma;'' : ''A'' &times; ''A'' &rarr; ''k''}} で
:{{math|''&sigma;''(''ab'', ''c'') {{=}} ''&sigma;''(''a'', ''bc'')}}
を満たすものが存在することと同値であり{{sfn|Lam|1999|p=67|loc=Theorem 3.15}}、したがってフロベニウス多元環は「左右対称な」概念である。他の同値な特徴づけとしては線型写像 {{math|''&epsilon;'' : ''A'' &rarr; ''k''}} で {{math|[[核 (代数学)|ker]](''&epsilon;'')}} がゼロでない左[[イデアル]]を含まないものが存在することがある。この線型写像 {{mvar|&epsilon;}} は'''フロベニウス形式''' ({{lang-en-short|Frobenius form}}) と呼ばれる{{sfn|Kock|2003|p={{google books quote|id=6dZZW08Z04MC|page=94|94}}|loc=&sect; 2.2.1}}。

フロベニウス多元環は {{math|''&epsilon;''(''ab'') {{=}} ''&epsilon;''(''ba'')}} を満たすフロベニウス形式 {{mvar|&epsilon;}} をもつとき、'''対称多元環''' ({{lang-en-short|symmetric algebra}}) と呼ばれる。（[[ベクトル空間]]の[[対称代数]]というほとんど関係ない異なる概念もある。）

== 例 ==
# 体 {{mvar|k}} 上の[[行列環]]は {{math|''&epsilon;''(''a'') {{=}} [[跡 (線型代数学)|tr]](''a'')}} をフロベニウス形式にもつフロベニウス多元環である。とくに体は恒等写像をフロベニウス形式にもつフロベニウス多元環である。
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# 任意の有限次元単位的結合多元環 ''A'' は自身の自己準同型環 End(''A'') への自然な準同型を持つ。双線型形式は例1のようにして ''A'' 上定義できる。この双線型形式が非退化であれば、これによって ''A'' はフロベニウス多元環の構造を持つ。
-->
# 体 {{mvar|k}} と[[有限群]] {{mvar|G}} に対して、[[群環]] {{math|''k''[''G'']}} は単位元の係数を取り出す線型写像 {{math|''&epsilon;''(&sum; ''a{{sub|g}}g'') {{=}} ''a''{{sub|1}}}} をフロベニウス形式にもつフロベニウス多元環である{{sfn|Kock|2003|p=100|loc=&sect; 2.2.18}}。
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これは例2の特別な場合である。
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# [[複素数体]] {{mvar|'''C'''}} は実部 {{math|''&epsilon;''(''z'') {{=}} Re(''z'')}} をフロベニウス形式にもつ（{{mvar|'''R'''}} 上の）フロベニウス多元環である{{sfn|Kock|2003|p=99|loc=&sect; 2.2.14}}。
# 体 {{mvar|k}} に対して、4次元の {{mvar|k}} 代数 {{math|''k''[''x'', ''y'']/(''x''<sup>2</sup>, ''y''<sup>2</sup>)}} はフロベニウス多元環である{{sfn|Lam|1999|p=68|loc=Example 3.15B}}。これは以下で述べる可換局所フロベニウス環の特徴づけから従う。この環は {{mvar|x}} と {{mvar|y}} で生成されるイデアルを極大イデアルとする局所環で、{{mvar|xy}} で生成される唯一の極小イデアルを持つからである。
# 体 {{mvar|k}} に対して、3次元の {{mvar|k}} 代数 {{math|''A'' {{=}} ''k''[''x'', ''y'']/(''x'', ''y'')<sup>2</sup>}} はフロベニウス多元環'''ではない'''{{sfn|Lam|1999|p=68|loc=Example 3.15B'}}。<math>x \mapsto y</math> から誘導される {{mvar|xA}} から {{mvar|A}} への {{mvar|A}} 準同型は、{{mvar|A}} から {{mvar|A}} への {{mvar|A}} 準同型に拡張できず、したがって環が自己移入的でなく、フロベニウスでない。

==性質==
* フロベニウス多元環の[[環の直積|直積]]や[[多元環のテンソル積|テンソル積]]はフロベニウス多元環である{{sfn|Lam|1999|p=114|loc=Exercise 3.12}}。
* 体上の有限次元[[可換環|可換]][[局所環|局所]]多元環が、フロベニウスであることと、右[[正則加群]]が移入的であることと、多元環が唯一つの[[極小イデアル]]を持つことは同値である{{sfn|Lam|1999|pp=114–115|loc=Exercise 3.14}}。
* 可換局所フロベニウス多元環は、ちょうど、[[クルル次元|0次元]]局所[[ゴレンシュタイン環]]であって[[剰余体]]を含み剰余体上有限次元であるようなものである。
* フロベニウス多元環は{{仮リンク|準フロベニウス多元環|en|quasi-Frobenius ring}}であり、とくに、左（右）[[アルティン環]]かつ左（右）[[自己移入環]]である。
* 無限体 ''k'' に対し、有限次元の単位的結合多元環は、極小[[右イデアル]]が有限個しかなければ、フロベニウスである{{sfn|Lam|1999|p=67|loc=Corollary 3.15'}}。
* ''F'' が ''k'' の有限次[[拡大体]]であれば、有限次元 ''F''-多元環は{{仮リンク|係数の制限|en|restriction of scalars}}によって自然に有限次元 ''k''-多元環であり、これがフロベニウス ''F''-多元環であることとフロベニウス ''k''-多元環であることは同値である。言い換えると、フロベニウス性は多元環が有限次元多元環である限り体に依存しない。
* 同様に、''F'' が ''k'' の有限次拡大体であれば、すべての ''k''-多元環 ''A'' は自然に ''F''-多元環 ''F'' ⊗<sub>''k''</sub> ''A'' を生じ、''A'' がフロベニウス ''k''-多元環であることと ''F'' ⊗<sub>''k''</sub> ''A'' がフロベニウス ''F''-多元環であることは同値である。
* 右正則表現が移入的な有限次元の単位的結合多元環の中では、フロベニウス多元環 ''A'' はちょうど、その[[単純加群]] ''M'' がその ''A'' 双対 Hom<sub>''A''</sub>(''M'', ''A'') と同じ次元を持つような多元環である。これらの多元環の中では、単純加群の ''A'' 双対は常に単純である。
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==Category-theoretical definition==
In [[category theory]], the notion of '''Frobenius object''' is an abstract definition of a Frobenius algebra in a category. A Frobenius object <math>(A,\mu,\eta,\delta,\varepsilon)</math> in a [[monoidal category]] <math>(C,\otimes,I)</math> consists of an object ''A'' of ''C'' together with four morphisms

:<math>\mu:A\otimes A\to A,\qquad \eta:I\to A,\qquad\delta:A\to A\otimes A\qquad\mathrm{and}\qquad\varepsilon:A\to I</math>

such that

* <math>(A,\mu,\eta)\,</math> is a [[monoid object]] in ''C'',
* <math>(A,\delta,\varepsilon)</math> is a [[comonoid object]] in ''C'',

* the diagrams
:[[Image:Frobenius obj coh 1.png]]
and
:[[Image:Frobenius obj coh 2.png]]
commute (for simplicity the diagrams are given here in the case where the monoidal category ''C'' is strict). 

More compactly, a Frobenius algebra in '''C''' is a so-called Frobenius monoidal functor A:'''1''' → '''C''', where '''1''' is the category consisting of one object and one arrow.

A Frobenius algebra is called '''isometric''' or '''special''' if <math>\mu\circ\delta = \mathrm{Id}_A</math>.

==Applications==
Frobenius algebras originally were studied as part of an investigation into the [[representation theory of finite groups]], and have contributed to the study of [[number theory]], [[algebraic geometry]], and [[combinatorics]].  They have been used to study [[Hopf algebra]]s, [[coding theory]], and [[cohomology ring]]s of [[compact space|compact]] [[orientability|oriented]] [[manifold]]s.

=== Topological quantum field theories ===
[[File:Pair of pants cobordism (pantslike).svg|thumb|The product and coproduct on a Frobenius algebra can be interpreted as the functor of a (1+1)-dimensional [[topological quantum field theory]], applied to a [[pair of pants (mathematics)|pair of pants]].]]
{{details|Topological quantum field theory}}

Recently, it has been seen that they play an important role in the algebraic treatment and axiomatic foundation of [[topological quantum field theory]].  A commutative Frobenius algebra determines uniquely (up to isomorphism) a (1+1)-dimensional TQFT.  More precisely, the [[category (category theory)|category]] of commutative Frobenius ''K''-algebras is [[equivalence of categories|equivalent]] to the category of [[symmetric monoidal functor|symmetric strong monoidal functors]] from 2-'''Cob''' (the category of 2-dimensional [[cobordism]]s between 1-dimensional manifolds) to '''Vect'''<sub>''K''</sub> (the category of [[vector space]]s over ''K'').

The correspondence between TQFTs and Frobenius algebras is given as follows:
* 1-dimensional manifolds are disjoint unions of circles: a TQFT associates a vector space with a circle, and the tensor product of vector spaces with a disjoint union of circles,
* a TQFT associates (functorially) to each cobordism between manifolds a map between vector spaces,
* the map associated with a [[pair of pants (mathematics)|pair of pants]] (a cobordism between 1 circle and 2 circles) gives a product map ''V'' ⊗ ''V'' → ''V'' or a coproduct map ''V'' → ''V'' ⊗ ''V'', depending on how the boundary components are grouped – which is commutative or cocommutative, and
* the map associated with a disk gives a counit (trace) or unit (scalars), depending on grouping of boundary.

== Generalization: Frobenius extension ==
Let ''B'' be a subring sharing the identity element of a unital associative ring ''A''. This is also known as ring extension ''A'' | ''B''. Such a ring extension is called '''Frobenius''' if

* There is a linear mapping ''E'': ''A'' → ''B'' satisfying the bimodule condition ''E(bac)'' = ''bE(a)c'' for all ''b,c'' ∈ ''B'' and ''a'' ∈ ''A''.
*There are elements in ''A'' denoted <math>\{x_i \}^n_{i=1}</math> and <math> \{y_i \}^n_{i=1} </math> such that for all ''a'' ∈ ''A'' we have:

:<math> \sum_{i=1}^n E(ax_i) y_i = a = \sum_{i=1}^n x_i E(y_i a)</math>

The map ''E'' is sometimes referred to as a Frobenius homomorphism and the elements <math>x_i, y_i</math> as dual bases. (As an exercise it is possible to give an equivalent definition of Frobenius extension as a Frobenius algebra-coalgebra object in the category of ''B''-''B''-bimodules, where the equations just given become the counit equations for the counit ''E''.)  

For example, a Frobenius algebra ''A'' over a commutative ring ''K'', with associative nondegenerate bilinear form (-,-) and projective K-bases <math> x_i, y_i</math> is a Frobenius extension ''A'' | ''K'' with ''E(a)'' = (''a'',1). Other examples of Frobenius extensions are pairs of group algebras associated to a subgroup of finite index, Hopf subalgebras of a semisimple Hopf algebra, Galois extensions and certain von Neumann algebra subfactors of finite index. Another source of examples of Frobenius extensions (and twisted versions) are certain subalgebra pairs of Frobenius algebras, where the subalgebra is stabilized by the symmetrizing automorphism of the overalgebra. 

The details of the [[group ring]] example are the following application of elementary notions in [[group theory]]. Let ''G'' be a group and ''H'' a subgroup of finite index ''n'' in ''G''; let ''g''<sub>1</sub>, ..., ''g<sub>n</sub>''. be left coset representatives, so that ''G'' is a disjoint union of the cosets ''g''<sub>1</sub>''H'', ..., ''g<sub>n</sub>H''. Over any commutative base ring k define the group algebras ''A'' = ''k[G]'' and ''B'' = ''k[H]'', so ''B'' is a subalgebra of ''A''. Define a Frobenius homomorphism ''E'': ''A'' → ''B'' by letting ''E(h)'' = ''h'' for all ''h'' in ''H'', and ''E(g)'' = 0 for ''g'' not in ''H'' : extend this linearly from the basis group elements to all of ''A'', so one obtains the ''B''-''B''-bimodule projection 

:<math>E \left (\sum_{g \in G} n_g g \right ) = \sum_{h \in H} n_h h \ \ \ \text{ for } n_g \in k </math>

(The orthonormality condition <math>E(g_i^{-1}g_j) = \delta_{ij} 1</math> follows.)  The dual base  is given by <math>x_i = g_i, y_i = g_i^{-1} </math>, since 

:<math> \sum_{i=1}^n g_i E(g_i^{-1} \sum_{g \in G} n_g g) = \sum_i \sum_{h \in H} n_{g_ih} g_ih = \sum_{g \in G} n_g g </math>

The other dual base equation may be derived from the observation that G is also a disjoint union of the right cosets <math> Hg_1^{-1},\ldots,Hg_n^{-1}</math>.  

Also Hopf-Galois extensions are Frobenius extensions by a theorem of Kreimer and Takeuchi from 1989.  A simple example of this is a finite group ''G'' acting by automorphisms on an algebra ''A'' with subalgebra of invariants: 

:<math>B = \{ x \in A | \forall g \in G, g(x) = x \}.</math>

By DeMeyer's criterion ''A'' is ''G''-Galois over ''B'' if there are elements <math>\{ a_i \}_{i=1}^n,  \{ b_i \}_{i=1}^n </math> in ''A'' satisfying: 

:<math> \forall g \in G: \ \  \sum_{i=1}^n a_i g(b_i) = \delta_{g,1_G}1_A </math>

whence also 

:<math> \forall g \in G: \ \ \sum_{i=1}^n g(a_i) b_i = \delta_{g,1_G}1_A.</math>

Then ''A'' is a Frobenius extension of ''B'' with ''E'': ''A'' → ''B'' defined by 

:<math> E(a) = \sum_{g \in G} g(a)</math>

which satisfies 

:<math> \forall x \in A: \ \ \sum_{i=1}^n E(xa_i)b_i = x = \sum_{i=1}^n a_i E(b_i x). </math>

(Furthermore an example of a [[separable algebra]] extension since <math>e = \sum_{i=1}^n a_i \otimes_B b_i</math> is a separability element satisfying ''ea = ae'' for all ''a'' in ''A'' as well as <math>\sum_{i=1}^n a_i b_i = 1</math>. Also an example of a [[depth two subring]] (''B'' in ''A'') since 

:<math> a \otimes_B 1 = \sum_{g \in G} t_g g(a)</math>

where 

:<math> t_g = \sum_{i=1}^n a_i \otimes_B g(b_i)</math>

for each ''g'' in ''G'' and ''a'' in ''A''.)    

Frobenius extensions have a well-developed theory of induced representations investigated in papers by  Kasch and Pareigis, Nakayama and Tzuzuku in the 1950s and 1960s. For example, for each ''B''-module ''M'', the induced module ''A'' ⊗<sub>''B''</sub> ''M'' (if ''M'' is a left module) and co-induced module Hom<sub>''B''</sub>(''A, M'') are naturally isomorphic as ''A''-modules (as an exercise one defines the isomorphism given ''E'' and dual bases). The endomorphism ring theorem of Kasch from 1960 states that if ''A'' | ''B'' is a Frobenius extension, then so is ''A'' → End(''A<sub>B</sub>'') where the mapping is given by ''a'' ↦ ''λ<sub>a</sub>(x)'' and ''λ<sub>a</sub>(x) = ax'' for each ''a,x'' ∈ ''A''. Endomorphism ring theorems and converses were investigated later by Mueller, Morita, Onodera and others.
-->

==脚注==
{{reflist|2}}

== 参考文献 ==
*{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | last2=Nesbitt | first2=C. | author2-link=Cecil J. Nesbitt | title=On the regular representations of algebras. | pmid=16588158 | pmc=1076908 | doi=10.1073/pnas.23.4.236 | year=1937 | journal=Proc. Nat. Acad. Sci. USA | volume=23 | issue=4 | pages=236–240}}
* {{Citation | last1=DeMeyer, F. | first1=Ingraham, E. | title=Separable Algebras over Commutative Rings | publisher=Springer | series=Lect. Notes Math 181 | year=1971}}
* {{Citation | last1=Dieudonné | first1=Jean | author1-link=Jean Dieudonné | title=Remarks on quasi-Frobenius rings | mr=0097427  | year=1958 | journal=Illinois Journal of Mathematics | issn=0019-2082 | volume=2 | pages=346–354}}
* {{Citation | last1=Frobenius | first1=Ferdinand Georg | author1-link=Ferdinand Georg Frobenius | title=Theorie der hyperkomplexen Größen I | language=German | year=1903 | journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften | pages=504–537 | jfm=34.0238.02}}
* {{Citation | last1=Kock | first1=Joachim | title=Frobenius Algebras and 2D Topological Quantum Field Theories | publisher=Cambridge University Press | location=Cambridge | series=London Mathematical Society Student Texts | isbn=0-521-83267-5 | year=2003 | url={{google books|6dZZW08Z04MC|plainurl=yes}} }}
* {{Citation | last1=Lam | first1=T. Y. | title=Lectures on Modules and Rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | year=1999 | url={{google books|6iLUBwAAQBAJ|Lectures on modules and rings|page=406|plainurl=yes}}}}
* {{Citation | last=Lurie | first=Jacob | title = On the Classification of Topological Field Theories
|url = http://www-math.mit.edu/~lurie/papers/cobordism.pdf }}
* {{Citation | last1=Nakayama | first1=Tadasi | author1-link=Tadashi Nakayama (mathematician) | title=On Frobeniusean algebras. I | doi=10.2307/1968946 | mr=0000016  | year=1939 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | volume=40 | pages=611–633 | issue=3 | publisher=Annals of Mathematics | jstor=1968946}}
* {{Citation | last1=Nakayama | first1=Tadasi | author1-link=Tadashi Nakayama (mathematician) | title=On Frobeniusean algebras. II | doi=10.2307/1968984 | mr=0004237  | year=1941 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | volume=42 | pages=1–21 | issue=1 | publisher=Annals of Mathematics | jstor=1968984}}
* {{Citation | last1=Nesbitt | first1=C. | author1-link=Cecil J. Nesbitt | title=On the regular representations of algebras | doi=10.2307/1968639 | mr=1503429  | year=1938 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=39 | issue=3 | pages=634–658 | jstor=1968639}}
* {{Citation | last1=Onodera | first1=T. | title=Some studies on projective Frobenius extensions | year=1964 | journal=[[Hokkaido Univ. Ser. 1]] | volume=18 | pages=89–107}}
* {{cite book
|last1      = Weibel
|first1     = Charles A. 
|year       = 1994
|title      = An Introduction to Homological Algebra
|url        = {{google books|flm-dBXfZ_gC|An introduction to homological algebra|plainurl=yes}}
|publisher  = Cambridge University Press
|isbn       = 0-521-43500-5
|ref        = harv
}}

==関連項目==
{{colbegin}}
* [[双代数]]
* {{仮リンク|フロベニウス圏|en|Frobenius category}}
* [[フロベニウスノルム]]
* [[フロベニウス内積]]
* [[ホップ代数]]
* [[準フロベニウスリー環]]
* {{仮リンク|ダガーコンパクト圏|en|Dagger compact category}}
{{colend}}

==外部リンク==
* Ross Street, [http://www.maths.mq.edu.au/~street/FAMC.pdf Frobenius algebras and monoidal categories]

{{DEFAULTSORT:ふろへにうすたけんかん}}
[[Category:表現論]]
[[Category:多元環論]]
[[Category:加群論]]
[[Category:モノイド圏]]
[[Category:フェルディナント・ゲオルク・フロベニウス]]
[[Category:数学に関する記事]]
[[Category:数学のエポニム]]