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We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and axiomatize these algebras. Furthermore, we define a geometric cobordism categories whose functors to the category of vector spaces are parameterized by these algebras. The theory is also extended to the graded and super-graded cases. As an application, we consider Frobenius algebras having some additional properties making them more tractable. These properties are present in Frobenius algebras arising as quotients of Jacobian ideal, such as those having their origin in quasi-homogeneous singularities and their symmetries., Comment: 42 pages LaTex. Paper version of talks at WAGP2000 in Oct 2000 and at``Mathematical apspects of orbifold string theoy'' in May 2001 Revised version: Typos corrected, slight revisions in the last section

Pseudo-effect algebras are partial algebras (E; +, 0, 1) with a partially defined addition + which is not necessary commutative and with two complements, left and right ones. We define central elements of a pseudo-effect algebra and the centre, which in the case of MV-algebras coincides with the set of Boolean elements and in the case of effect algebras with the Riesz decomposition property central elements are only characteristic elements. If E satisfies general comparability, then E is a pseudo MV-algebra. Finally, we apply central elements to obtain a variation of the Cantor-Bernstein theorem for pseudo-effect algebras.

In this paper we generalize the max algebra system of nonnegative matrices to the class of nonnegative tensors and derive its fundamental properties. If $\mathbb{A} \in \Re_ + ^{\left[ {m,n} \right]}$ is a nonnegative essentially positive tensor such that satisfies the condition class NC, we prove that there exist $\mu \left( \mathbb{A} \right)$ and a corresponding positive vector $x$ such that $\mathop {\max }\limits_{1 \le{i_2}\cdots {i_m} \le n} \left\{ {{a_{i{i_2}\cdots {i_m}}}{x_{{i_2}}}\cdots {x_{{i_m}}}} \right\}=\mu \left( \mathbb{A} \right) x_i^{m - 1},\,\,\,\,i = 1,2,\cdots ,n.$ This theorem, is well known as the max algebra version of Perron--Frobenius theorem for this new system.

We introduce the notion of Q-systems as Frobenius algebras in a C* tensor category, enjoying a standardness property. Q-systems in the category of endomorphisms of an infinite factor \(N\) completely characterize extensions \(N\subset M\). Modules and bimodules of Q-systems are equivalent to homomorphisms \(N\rightarrow M\) resp. \(M_1\rightarrow M_2\).

It could well be argued that the previous chapters were like an introductory course in a foreign language. Much attention was devoted to learning the new vocabulary and relating it to the vocabulary in a previously known language. Then there were a lot of exercises to learn how the new vocabulary is used. So that’s the grammar. But, to continue the analogy, there was no poetry. Or to put it into colloquial mathematical terminology, there were no real theorems. That is about to change. The Frobenius theorem is a real theorem.

The Fundamental Theorem of Algebra (stated below) provides an ideal case study for illustrating the roles of alternative proofs in mathematical practice. Like the Pythagorean Theorem, the Fundamental Theorem of Algebra has been proved in many different ways since its enunciation by Euler in 1739. Unlike the Pythagorean Theorem, however, early attempts to prove the Fundamental Theorem of Algebra are not shrouded in the mists of antiquity, so we know how the adequacy of those attempts was evaluated by mathematicians of the time. We can see how criticisms of earlier efforts to prove the theorem led to alternative proof strategies, and we can analyze why the proof given by Gauss in his 1799 inaugural dissertation was the first to be accorded general acceptance, though it too would later be deemed not fully rigorous.

We prove that if E 1 and E 2 are σ-complete effect algebras such that E 1 is a factor of E 2 and E 2 is a factor of E 1, then E 1 and E 2 are isomorphic.

Let A be a central simple algebra of degree n over a field of characteristic different from 2 and let B ? A be a maximal commutative subalgebra. We show that if there is an involution on A that preserves B and such that the socle of each local component of B is a homogeneous C 2 -module for this action, then B is a Frobenius algebra.

Let H be a finite Frobenius group with a perfect Frobenius complement G. Two new proofs that G is isomorphic to SL2(5) are given.

In 1994, the following infinite family of congruences was conjectured for the partition functioncΦ2(n)which counts the number of2-colored Frobenius partitions ofn: for alln≥0andα≥1,cΦ2(5αn+λα)≡0(mod5α), whereλαis the least positive reciprocal of12modulo5α. In this paper, the first four cases of this family are proved.

In the paper, it is shown how, by using the minimal means and without introducing complex numbers, one can prove the fundamental theorem of algebra for real polynomials. The fundamental theorem for polynomials with complex coefficients is an immediate corollary of the theorem mentioned.

We study the superoptimal Frobenius operators in several matrix vector spaces and in particular in the circulant algebra, by emphasizing both the algebraic and geometric properties. More specifically we prove a series of "negative" results that explain why this approximation procedure is not competitive with the optimal Frobenius approximation, although it could be used for regularization purposes.

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

We show that the derived center of the category of simplicial algebras over every algebraic theory is homotopically discrete, with the abelian monoid of components isomorphic to the center of the category of discrete algebras. For example, in the case of commutative algebras in characteristic $p$, this center is freely generated by Frobenius. Our proof involves the calculation of homotopy coherent centers of categories of simplicial presheaves as well as of Bousfield localizations. Numerous other classes of examples are discussed., 40 pages

International audience; Bialgebras and Frobenius algebras are different ways in which monoids and comonoids interact as part of the same theory. Such theories feature in many fields: e.g. quantum computing, compositional semantics of concurrency, network algebra and component-based programming. In this paper we study an important sub-theory of Coecke and Duncan's ZX-calculus, related to strongly-complementary observables, where two Frobenius algebras interact. We characterize its free model as a category of ℤ2-vector subspaces. Moreover, we use the framework of PROPs to exhibit the modular structure of its algebra via a universal construction involving span and cospan categories of ℤ2-matrices and distributive laws between PROPs. Our approach demonstrates that the Frobenius structures result from the interaction of bialgebras.

We show how the notion of full Frobenius group of finite Morley rank generalizes that of bad group, and how it seems to be more appropriate when we consider the possible existence (still unknown) of nonalgebraic simple groups of finite Morley rank of a certain type, notably with no involution. We also show how these groups appear as a major obstacle in the analysis of FT-groups, if one tries to extend the Feit-Thompson theorem to groups of finite Morley rank.

The paper continues the study of morphisms of double Frobenius algebras (which include finite-dimensional Hopf algebras, character algebras, and adjacency algebras of association schemes). Here we consider morphisms satisfying a certain regularity condition.

The paper contains new theorems concerning the Frobenius problem, belonging to additive number theory, and using continued fractions.

The relations between several necessary conditions for a square complex matrix A to be m-cyclic are examined. These conditions are known to be equivalent if A is an irreducible nonnegative matrix whose index of imprimitivity is m. In particular, we find that if the digraph of A contains at least one cycle with nonzero signed length, then the following conditions are each equivalent to the m-cyclicity of A: (i) A is diagonally similar to e 2π i /m A ; (ii) all cycles in the digraph of A have signed length an integral multiple of m. In the course of our investigations, we lay down the groundwork of the theory of cyclically m-partite or linearly partite digraphs, and characterize these digraphs in terms of the signed lengths of their cycles. The characterization of diagonal similarity between matrices in terms of matrix cycle products due to Saunders and Schneider plays a key role in our development. Our investigations also lead to a new illuminating conceptual proof for the second part of the Frobenius theorem on an irreducible nonnegative matrix. The connection between the m-cyclicity of a square complex matrix and that of its associated collection of elementary Jordan blocks is studied in the second half of this paper. In particular, for an m-cyclic collection U it is found that the problem of determining all m-tuples (k 1 ,…,k m ) of positive integers for which there exists an m-cyclic matrix A in the superdiagonal (k 1 ,…,k m ) -block form such that U (A)= U is equivalent to determine all row sum vectors of (0,1)-matrices that have a prescribed column sum vector and each of whose column vectors has cyclically consecutive equal components. As a by-product we also obtain an equivalent condition on m given square complex matrices B 1 ,…,B m so that there exist complex rectangular matrices A 1 ,…,A m that satisfy B j =A j ⋯A m A 1 ⋯A j−1 for j=1,…,m , thus completing the work of Flanders for the case m=2 .

Double Frobenius algebras (or dF-algebras) were recently introduced by the author. The concept generalizes finite-dimensional Hopf algebras, adjacency algebras of (non-commutative) association schemes, and C -algebras (or character algebras). This paper studies basic properties of various Nakayama automorphisms in a dF-algebra. As an application the theorem of D. E. Radford, stating that the antipode S of a finite-dimensional Hopf algebra is of finite order, is generalized to dF-algebras, and so is his formula for S 4 n in terms of certain group-like elements.

We study a certain 1-parameter family of non-commutative graded regular algebras of global dimension four which were introduced by Vancliff, Van Rompay and Willaert in [12]. Most members of the family have a singleton point scheme of multiplicity twenty. Our objective is to analyse the point scheme of these algebras and the coordinate ring of the point scheme. In particular, we prove that the point scheme determines the defining relations and that its coordinate ring is a Frobenius algebra.

This paper begins with an introduction to β-Frobenius structure on a finite-dimensional Hopf subalgebra pair. In Section 2 a study is made of a generalization of Frobenius bimodules and β-Frobenius extensions. Also a special type of twisted Frobenius bimodule which gives an endomorphism ring theorem and converse is studied. Section 3 brings together material on separable bimodules, the dual definitions of split and separable extension, and a theorem of Sugano on endomorphism rings of separable bimodules. In Section 4, separable twisted Frobenius bimodules are characterized in terms of data that generalizes a Frobenius homomorphism and a dual base. In the style of duality, two corollaries characterizing split β-Frobenius and separable β-Frobenius extensions are proven. Sugano"s theorem is extended to β-Frobenius extensions and their endomorphism rings. In Section 5, the problem of when separable extensions are Frobenius extensions is discussed. A Hopf algebra example and a matrix example are given of finite rank free separable β-Frobenius extensions which are not Frobenius in the ordinary sense.

J. A. Green proved a theorem which is the converse to a theorem of R. Brauer (Proc. Camb. Philos. Soc.51(1955), 237–239). In this article we give another proof of the theorem concretely without using Frobenius's formula for induced characters and we also state some comments on Brauer's induction theorem.

Resume In his theory of D † X -modules, P. Berthelot has proved a general theorem of Frobenius descent. On the other hand, Christol and others have also proved various statements about weak Frobenius structures on an annulus of the rigid projective line. It is the aim of this paper to give some explicit and global formulas for Frobenius descent for D † X -modules. It requires us to build a new kind of differential operator which represents Dwork's ψ operator. The construction uses mainly Taylor series properties. It is possible to derive from this some new proofs of Christol's theorems; it is also useful for computations in characteristicplike those arising in the construction of Cartier's isomorphism or Cartier's operator.

In fact, if X is any vector space on which the primitive Banach algebra A acts faithfully and irreducibly, then X can be converted in a Banach space in such a way that the requirements in the theorem are satisfied and even the inclusion A ↪→ BL(X) is contractive. Roughly speaking, the aim of this paper is to prove the appropriate Jordan variant of the above theorem. The notion of primitiveness for Jordan algebras was introduced and developed in 1981 by L. Hogben and K. McCrimmon [10]. Primitive Jordan algebras are relevant particular types of prime nondegenerate Jordan algebras but, although the celebrated Zel’manov prime theorem ([19], 1983) gave a precise description of these last algebras, it has happened only very recently that the appropriate variant of Zel’manov’s theorem for primitive Jordan algebras has been obtained (see [3] and [17]). Also very recently several particular normed versions of Zel’manov’s theorem have been provided (see [8], [6], [16], and [7]). Nevertheless, to obtain a Zel’manov type theorem for primitive Jordan-Banach algebras has remained an open problem in the last years [15]. In fact we have been able to prove such a theorem but only passing through a general normed version of the Zel’manov prime theorem (see Theorem 1) which is in our opinion one of the most important novelties in the paper. Since Theorem 1 will probably have applications different from that in the paper, we have included in its statement and proof some details not strictly needed for our main purpose. The same comment should be made concerning Theorem 2, which is nothing but a fine improvement of Theorem 1 under the additional assumption of completeness. From Theorem 2 and the main results in [3], [18], and [5], the desired Jordan variant of Theorem 0 (Theorem 3) follows easily. Again roughly speaking, it asserts that primitive complex Jordan-Banach algebras, different from the simple exceptional 27-dimensional one and the simple Jordan algebras of a continuous symmetric bilinear form on a complex Banach space, can be continuously regarded as Jordan algebras of bounded linear operators ”acting irreducibly” on a suitable complex Banach space.

Double Frobenius algebras (or dF-algebras) were recently introduced by the author. The concept generalizes finite-dimensional Hopf algebras, adjacency algebras of (non-commutative) association schemes, and C-algebras (or character algebras). In this paper we define a dualization construction of a dF-algebra, the so-called linear dual. We show that in the case of a Hopf algebra the linear dual gives the usual dual Hopf algebra and in the case of a C-algebra it essentially gives the usual Kawada’s dual.

We show that a theorem of Kisyński on the generation of Banach-algebra homomorphisms of certain convolution algebras is equivalent to the Hille-Yosida theorem on the generation of operator-valued one-parameter semigroups.

Throughout this paper A is an algebra over a field k. We say that A is a connected k-algebra if there is a vector space decomposition A = $n>oAn, such that AiAj C and A0 = k. The augmentation ideal is the unique maximal graded ideal m := $i>oAi. We call Alm the trivial module, and usually denote it by k. A finite dimensional k-algebra A is F'robenius if Homk(A, k) is isomorphic to A as a left and as a right A-module. A Frobenius algebra is quasi-F'robenius, meaning that it is left and right artinian and left and right self-injective. For trivial reasons, a quasi-Frobenius algebra need not be Frobenius: take a division algebra infinite dimensional over its center k. Also, a self-injective ring need not be quasi-Frobenius: there is a local, commutative and self-injective ring which is neither artinian nor noetherian [2, page 2141. Frobenius and quasi-Frobenius algebras have been studied in great detail, but rarely under the additional hypothesis that they are connected graded. Our interest in connected graded Frobenius algebras arises from our interest in various non-commutative analogues of polynomial rings because these analogues are often Koszul algebras whose dual algebra is a connected graded Frobenius algebra of finite dimension (see (51 for details). We will prove the following result.

It is shown that every Frobenius algebra over a commutative ring determines a class of solutions of the quantum Yang-Baxter equation, which forms a subbimodule of its tensor square. Moreover, this subbimodule is free of rank one as a left (right) submodule. An explicit form of a generator is given in terms of the Frobenius homomorphism. It turns out that the generator is invertible in the tensor square if and only if the algebra is Azumaya.

Let H be a finite dimensional Hopf algebra and A an H-module algebra. We prove that is Frobenius or quasi-Frobenius algebra if A is Frobenius or quasi-Frobenius as algebra and some other conditions are satisfied which are generalizations of theorems in group actions.

For a finite dimensional Frobenius cellular algebra, a sufficient and necessary condition for a simple cell module to be projective is given. A special case that dual bases of the cellular basis satisfying a certain condition is also considered. The result is similar to that in symmetric case., 10 pages

We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either “annihilator algebras” — algebras whose socle is a principal ideal — or field extensions. The relationship between two-dimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classification of the indecomposable two-dimensional topological quantum field theories.

We use addition theory on the Abelian group ( Z a 0 , +) to generalize Vitek's bound and obtain some estimated formulas for the Frobenius numbers.

The Diophantine Problem of Frobenius is to find a formula for the least integer not representable as a nonnegative linear form of positive integers. A reduction formula for the Diophantine Problem of Frobenius is presented. The formula can be applied whenever there are common divisors of the coefficients except for the whole set of them. The reduction formula is expressed in terms of the least common multiple of the coefficients. For some classes of coefficients this formula gives an exact answer for the problem of Frobenius, and these classes are fully characterized in the paper.

So far there has been elementary proof for Frobenius's theorem only in special cases: if the complement is solvable, see e.g. [3], if the complement is of even order, see e.g. [6]. In the first section we consider the case, when the order of the complement is odd. We define a graph the vertices of which are the set K# of elements of our Frobenius group with 0 fixed points. Two vertices are connected with an edge if and only if the corresponding elements commute. We prove with elementary methods that K is a normal subgroup in G if and only if there exists an element x in K# such that all elements of K# belonging to the connected component C of K# containing x are at most distance 2 from c and NG(C) is not a -group, where is the set of prime divisors of the Frobenius complement of G. In the second section we generalize the case when the order of the complement is even, proving that the Frobenius kernel is a normal subgroup, if a fixed element a of the complement, the order of which is a minimal prime diviso...

Frobenius showed that, if m divides the order of a finite group, then the number of m-torsion elements in that group is divisible by m. We provide a quick combinatorial proof.

Let G be a finite group and L e ( G ) = { x ∈ G ∣ x e = 1 } $L_e(G)=\lbrace x \in G \mid x^e=1\rbrace $ , where e is a positive integer dividing | G | $\vert G\vert $ . How do bounds on | L e ( G ) | $\vert L_e(G)\vert $ influence the structure of G? Meng and Shi [Arch. Math. (Basel) 96 (2011), 109–114] have answered this question for | L e ( G ) | ≤ 2 e $\vert L_e(G)\vert \le 2e$ . We generalize their contributions, considering the inequality | L e ( G ) | ≤ e 2 $\vert L_e(G)\vert \le e^2$ and finding a new class of groups of whose we study the structural properties.

In the first part we study nearly Frobenius algebras. The concept of nearly Frobenius algebras is a generalization of the concept of Frobenius algebras. Nearly Frobenius algebras do not have traces, nor they are self-dual. We prove that the known constructions: direct sums, tensor, quotient of nearly Frobenius algebras admit natural nearly Frobenius structures. In the second part we study algebras associated to some families of quivers and the nearly Frobenius structures that they admit. As a main theorem, we prove that an indecomposable algebra associated to a bound quiver (Q, I) with no monomial relations admits a non trivial nearly Frobenius structure if and only if the quiver Q is linearly oriented of type \(\overrightarrow {\mathbb {A}_{n}}\) and I = 0. We also present an algorithm that determines the number of independent nearly Frobenius structures for gentle algebras without oriented cycles.

For a Frobenius cellular algebra, we prove that if the left (right) dual basis of a cellular basis is again cellular, then the algebra is symmetric. Moreover, some ideals of the center are constructed by using the so-called Nakayama twisted center., Comment: 12 pages

A local existence and uniqueness theorem for ODEs in the special algebra of generalized functions is established, as well as versions including parameters and dependence on initial values in the generalized sense. Finally, a Frobenius theorem is proved. In all these results, composition of generalized functions is based on the notion of c-boundedness.