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| {{dablink|"Frobenius algebra" is also an archaic name for the [[group ring]] of a finite group}}
| | You really will enjoy the above soup recipes because they are really easy to make. This is an important point to be notices that the paleo diet menu was so restrictive and was without any taste as it had a less variety in it. Following a healthy diet signifies changing your diet plan, lowering the amounts but still getting all the proper vitamins and minerals. Recipe books usually have a substantial index where pots and pans are divided simply type of scheduled meal (breakfast, lunch, snacks, dinner, desserts, etc. Nutritious vitamin backed shake recipes top with your favorite nuts or seeds. <br><br>So specifically exactly what really does one particular distinct eat and also by no implies eat right after getting Paleo Recipe Book. We feel sluggish and poor simply because we are not giving our bodies fuel. Le - Bron Was Inspired By Ray Allen's Paleo Diet Transformation. In addition, food is something that we can't turn down yet healthy foods is another matter to discuss. Following a low-carb diet dramatically reduced the spread of cancer, said Martin, who published his findings in the medical journal Cell. <br><br>And what's my Paleo Diet weight loss result so far. The results showed the Paleo dieters lost significantly more weight (14 pounds) compared to the low-fat dieters (5. There are hundreds of pills for losing weight in the market. One thing that you will want to do is keep track of your weight to see if you are on the right rack to a healthier life. For example, changes in your activity level and sleep are highly encouraged in addition to dietary changes. <br><br>Study participants were measured for weight, cholesterol and blood sugars after six months on their respective diets, and again after two years. Attempt to stop cashews as well as nuts as they are theoretically beans and not the main fan family. a Mediterranean style of eating combined with physical activity is the optimal lifestyle plan for preventing a second heart attack. In order to describe this in more detail, we are going to break it down into the athlete’s five stages of daily eating while training. If you are really committed to the Paleo Diet plan, then gone are the times of buying for "staples" like bread and milk. <br><br>On the other hand, coconut and almond milk contain calcium and a host of other vitamins and minerals. In order to satisfy vegetable segment of your diet, there are numerous vegetables to select from. It is infinitely customizable to meet YOUR specific needs and goals. If better taste is required then start from rabbit or goat meat. Following the Paleo will also call for you to give up some things which may perhaps be within your diet program at the moment. <br><br>That’s more than are affected by Multiple Sclerosis, Crohn’s Disease, Cystic Fibrosis, and Parkinson’s combined. Here are some items that you will surely love about this. However, there are some exceptions, along with some specific foods that this diet is centered around; furthermore, it is crucial to stick to these foods. Paleo diet recipes are the Back to Mother Nature diet for the whole family. One of these may be the foods by it tend to be scrumptious.<br><br>If you have any sort of concerns relating to where and ways to utilize [http://gritsandgroceries.info/sitemap/ easy paleo chicken recipes], you could contact us at the webpage. |
| In [[mathematics]], especially in the fields of [[representation theory]] and [[module theory]], a '''Frobenius algebra''' is a [[dimension (vector space)|finite dimensional]] [[unital ring|unital]] [[associative algebra]] with a special kind of [[bilinear form]] which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by [[Richard Brauer|Brauer]] and [[Cecil J. Nesbitt|Nesbitt]] and were named after [[Ferdinand Georg Frobenius|Frobenius]]. [[Tadashi Nakayama (mathematician)|Nakayama]] discovered the beginnings of a rich duality theory in his {{harv|Nakayama|1939}} and especially in his {{harv|Nakayama|1941}}. [[Jean Dieudonné|Dieudonné]] used this to characterize Frobenius algebras in his {{harv|Dieudonné|1958}} where he called this property of Frobenius algebras a ''perfect duality''. Frobenius algebras were generalized to [[quasi-Frobenius ring]]s, those [[Noetherian ring]]s whose right [[regular representation]] is [[injective module|injective]]. In recent times, interest has been renewed in Frobenius algebras due to connections to [[topological quantum field theory]].
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| ==Definition==
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| A finite dimensional, unital, associative algebra ''A'' defined over a [[field (mathematics)|field]] ''k'' is said to be a '''Frobenius algebra''' if ''A'' is equipped with a [[nondegenerate bilinear form|nondegenerate]] [[bilinear form]] σ:''A'' × ''A'' → ''k'' that satisfies the following equation: ''σ''(''a''·''b'',''c'')=''σ''(''a'',''b''·''c''). This bilinear form is called the '''Frobenius form''' of the algebra.
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| Equivalently, one may equip ''A'' with a [[linear functional]] ''λ'':''A''→''k'' such that the [[kernel (algebra)|kernel]] of ''λ'' contains no nonzero left [[ideal (ring theory)|ideal]] of ''A''.
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| A Frobenius algebra is called '''symmetric''' if ''σ'' is [[symmetric bilinear form|symmetric]], or equivalently ''λ'' satisfies ''λ''(''a''·''b'') = ''λ''(''b''·''a'').
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| There is also a different, mostly unrelated notion of the [[symmetric algebra]] of a [[vector space]].
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| ==Examples==
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| # Any [[matrix ring|matrix algebra]] defined over a field ''k'' is a Frobenius algebra with Frobenius form ''σ''(''a'',''b'')=tr(''a''·''b'') where tr denotes the [[trace (linear algebra)|trace]].
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| # Any finite-dimensional unital associative algebra ''A'' has a natural homomorphism to its own endomorphism ring End(''A''). A bilinear form can be defined on ''A'' in the sense of the previous example. If this bilinear form is nondegenerate, then it equips ''A'' with the structure of a Frobenius algebra.
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| # Every [[group ring]] of a [[finite group]] over a field is a Frobenius algebra, with Frobenius form ''σ''(''a'',''b'') the coefficient of the identity element in ''a''·''b''. This is a special case of example 2.
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| # For a field ''k'', the four-dimensional ''k''-algebra ''k''[''x'',''y'']/ (''x''<sup>2</sup>, ''y''<sup>2</sup>) is a Frobenius algebra. This follows from the characterization of commutative local Frobenius rings below, since this ring is a local ring with its maximal ideal generated by ''x'' and ''y'', and unique minimal ideal generated by ''xy''.
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| # For a field ''k'', the three-dimensional ''k''-algebra ''A''=''k''[''x'',''y'']/ (''x'', ''y'')<sup>2</sup> is '''not''' a Frobenius algebra. The ''A'' homomorphism from ''xA'' into ''A'' induced by ''x'' ↦ ''y'' cannot be extended to an ''A'' homomorphism from ''A'' into ''A'', showing that the ring is not self-injective, thus not Frobenius.
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| ==Properties==
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| * The [[product of rings|direct product]] and [[tensor product]] of Frobenius algebras are Frobenius algebras.
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| * A finite-dimensional [[commutative ring|commutative]] [[local ring|local]] algebra over a field is Frobenius if and only if the right [[regular module]] is injective, if and only if the algebra has a unique [[minimal ideal]].
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| * Commutative, local Frobenius algebras are precisely the [[Krull dimension|zero-dimensional]] local [[Gorenstein ring]]s containing their [[residue field]] and finite dimensional over it.
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| * Frobenius algebras are [[quasi-Frobenius ring]]s, and in particular, they are left and right [[Artinian ring|Artinian]] and left and right [[self-injective ring|self-injective]].
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| * For a field ''k'', a finite-dimensional, unital, associative algebra is Frobenius if and only if the injective right ''A''-module Hom<sub>''k''</sub>(''A'',''k'') is isomorphic to the right [[regular representation]] of ''A''.
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| * For an infinite field ''k'', a finite dimensional, unitial, associative ''k''-algebra is a Frobenius algebra if it has only finitely many minimal [[right ideal]]s.
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| * If ''F'' is a finite dimensional [[extension field]] of ''k'', then a finite dimensional ''F''-algebra is naturally a finite dimensional ''k''-algebra via [[restriction of scalars]], and is a Frobenius ''F''-algebra if and only if it is a Frobenius ''k''-algebra. In other words, the Frobenius property does not depend on the field, as long as the algebra remains a finite dimensional algebra.
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| * Similarly, if ''F'' is a finite dimensional extension field of ''k'', then every ''k''-algebra ''A'' gives rise naturally to a ''F'' algebra, ''F'' ⊗<sub>''k''</sub> ''A'', and ''A'' is a Frobenius ''k''-algebra if and only if ''F'' ⊗<sub>''k''</sub> ''A'' is a Frobenius ''F''-algebra.
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| * Amongst those finite-dimensional, unital, associative algebras whose right regular representation is injective, the Frobenius algebras ''A'' are precisely those whose [[simple module]]s ''M'' have the same dimension as their ''A''-duals, Hom<sub>''A''</sub>(''M'',''A''). Amongst these algebras, the ''A''-duals of simple modules are always simple.
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| ==Category-theoretical definition==
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| 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
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| :<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>
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| such that
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| * <math>(A,\mu,\eta)\,</math> is a [[monoid object]] in ''C'',
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| * <math>(A,\delta,\varepsilon)</math> is a [[comonoid object]] in ''C'',
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| * the diagrams
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| :[[Image:Frobenius obj coh 1.png]]
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| and | |
| :[[Image:Frobenius obj coh 2.png]]
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| commute (for simplicity the diagrams are given here in the case where the monoidal category ''C'' is strict).
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| 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.
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| A Frobenius algebra is called '''isometric''' or '''special''' if <math>\mu\circ\delta = \mathrm{Id}_A</math>.
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| ==Applications==
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| 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.
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| === Topological quantum field theories ===
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| [[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]].]]
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| {{details|Topological quantum field theory}}
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| 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'').
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| The correspondence between TQFTs and Frobenius algebras is given as follows:
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| * 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,
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| * a TQFT associates (functorially) to each cobordism between manifolds a map between vector spaces,
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| * 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
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| * the map associated with a disk gives a counit (trace) or unit (scalars), depending on grouping of boundary.
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| == Generalization: Frobenius extension ==
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| 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
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| * 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''.
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| *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:
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| :<math> \sum_{i=1}^n E(ax_i) y_i = a = \sum_{i=1}^n x_i E(y_i a)</math>
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| 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''.)
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| 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 Nakayama automorphism of the overalgebra.
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| 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
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| :<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>
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| (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
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| :<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>
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| 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>.
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| 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:
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| :<math>B = \{ x \in A | \forall g \in G, g(x) = x \}.</math>
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| 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:
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| :<math> \forall g \in G: \ \ \sum_{i=1}^n a_i g(b_i) = \delta_{g,1_G}1_A </math>
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| whence also
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| :<math> \forall g \in G: \ \ \sum_{i=1}^n g(a_i) b_i = \delta_{g,1_G}1_A.</math>
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| Then ''A'' is a Frobenius extension of ''B'' with ''E'': ''A'' → ''B'' defined by
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| :<math> E(a) = \sum_{g \in G} g(a)</math>
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| which satisfies
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| :<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>
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| (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
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| :<math> a \otimes_B 1 = \sum_{g \in G} t_g g(a)</math>
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| where
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| :<math> t_g = \sum_{i=1}^n a_i \otimes_B g(b_i)</math>
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| for each ''g'' in ''G'' and ''a'' in ''A''.)
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| 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.
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| ==See also==
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| {{colbegin}}
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| * [[Bialgebra]]
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| * [[Frobenius category]]
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| * [[Frobenius norm]]
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| * [[Frobenius inner product]]
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| * [[Hopf algebra]]
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| * [[Quasi-Frobenius Lie algebra]]
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| * [[Dagger compact category]]
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| {{colend}}
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| ==References==
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| *{{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}}
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| * {{Citation | last1=DeMeyer, F. | first1=Ingraham, E. | title=Separable Algebras over Commutative Rings | publisher=Springer | series=Lect. Notes Math 181 | year=1971}}
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| * {{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}}
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| * {{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}}
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| * {{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}}
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| * {{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}}
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| * {{Citation | last=Lurie | first=Jacob | title = On the Classification of Topological Field Theories
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| |url = http://www-math.mit.edu/~lurie/papers/cobordism.pdf }}
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| * {{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}}
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| * {{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}}
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| * {{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}}
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| * {{Citation | last1=Onodera | first1=T. | title=Some studies on projective Frobenius extensions | year=1964 | journal=[[Hokkaido Univ. Ser. 1]] | volume=18 | pages=89–107}}
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| ==External links==
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| * Ross Street, [http://www.maths.mq.edu.au/~street/FAMC.pdf Frobenius algebras and monoidal categories]
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| {{DEFAULTSORT:Frobenius Algebra}}
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| [[Category:Algebras]]
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| [[Category:Module theory]]
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| [[Category:Monoidal categories]]
| |
You really will enjoy the above soup recipes because they are really easy to make. This is an important point to be notices that the paleo diet menu was so restrictive and was without any taste as it had a less variety in it. Following a healthy diet signifies changing your diet plan, lowering the amounts but still getting all the proper vitamins and minerals. Recipe books usually have a substantial index where pots and pans are divided simply type of scheduled meal (breakfast, lunch, snacks, dinner, desserts, etc. Nutritious vitamin backed shake recipes top with your favorite nuts or seeds.
So specifically exactly what really does one particular distinct eat and also by no implies eat right after getting Paleo Recipe Book. We feel sluggish and poor simply because we are not giving our bodies fuel. Le - Bron Was Inspired By Ray Allen's Paleo Diet Transformation. In addition, food is something that we can't turn down yet healthy foods is another matter to discuss. Following a low-carb diet dramatically reduced the spread of cancer, said Martin, who published his findings in the medical journal Cell.
And what's my Paleo Diet weight loss result so far. The results showed the Paleo dieters lost significantly more weight (14 pounds) compared to the low-fat dieters (5. There are hundreds of pills for losing weight in the market. One thing that you will want to do is keep track of your weight to see if you are on the right rack to a healthier life. For example, changes in your activity level and sleep are highly encouraged in addition to dietary changes.
Study participants were measured for weight, cholesterol and blood sugars after six months on their respective diets, and again after two years. Attempt to stop cashews as well as nuts as they are theoretically beans and not the main fan family. a Mediterranean style of eating combined with physical activity is the optimal lifestyle plan for preventing a second heart attack. In order to describe this in more detail, we are going to break it down into the athlete’s five stages of daily eating while training. If you are really committed to the Paleo Diet plan, then gone are the times of buying for "staples" like bread and milk.
On the other hand, coconut and almond milk contain calcium and a host of other vitamins and minerals. In order to satisfy vegetable segment of your diet, there are numerous vegetables to select from. It is infinitely customizable to meet YOUR specific needs and goals. If better taste is required then start from rabbit or goat meat. Following the Paleo will also call for you to give up some things which may perhaps be within your diet program at the moment.
That’s more than are affected by Multiple Sclerosis, Crohn’s Disease, Cystic Fibrosis, and Parkinson’s combined. Here are some items that you will surely love about this. However, there are some exceptions, along with some specific foods that this diet is centered around; furthermore, it is crucial to stick to these foods. Paleo diet recipes are the Back to Mother Nature diet for the whole family. One of these may be the foods by it tend to be scrumptious.
If you have any sort of concerns relating to where and ways to utilize easy paleo chicken recipes, you could contact us at the webpage.