Properties of higher characters are developed and applied to symmetric products and Frobenius algebras. A `constructive' proof of the Gel'fand-Kolmogorov theorem is given. Generalisations of that theorem and the Nullstellensatz to symmetric products are discussed.Applications to the theory of multi-symmetric functions are also discussed. It is proved that the first three characters determine the Jordan algebra associated to a Frobenius algebra and as a corollary one obtains the theorem of Hoehnke and Johnson that a finite group is determined by the first three characters of its regular representation.
In order to study graded Frobenius algebras from a ring theoretical perspective, we introduce graded quasi-Frobenius rings, graded Frobenius rings and a shift-version of the latter ones, and we investigate the structure and representations of such objects. We need to revisit graded simple graded left Artinian rings, graded semisimple rings, and to provide graded versions of certain results concerning the Jacobson radical, the singular radical, and their connection to finiteness conditions and injectivity. We prove a structure result for (shift-)graded Frobenius rings.
The classes of FP-injective and weakly quasi-Frobenius rings are investigated. The properties for both classes of rings are closely linked with embedding of finitely presented modules in fp-flat and free modules respectively. Using these properties, we describe the classes of coherent CF and FGF-rings. Moreover, it is proved that the group ring R(G) is FP-injective (resp. weakly quasi-Frobenius) if and only if the ring R is FP-injective (resp. weakly quasi-Frobenius) and the group G is locally finite.
We introduce the notion of the Frobenius-Perron dimension of an integral $\Bbb Z_+$-ring and give some applications of this notion to classification of finite dimensional quasi-Hopf algebras with a unique nontrivial simple module, and of quasi-Hopf and Hopf algebras of prime dimension $p$.
MacWilliams proved that every finite field has the extension property for Hamming weight which was later extended in a seminal work by Wood who characterized finite Frobenius rings as precisely those rings which satisfy the MacWilliams extension property. In this paper, the question of when is a MacWilliams ring quasi-Frobenius is addressed. It is proved that a right or left noetherian left 1-MacWilliams ring is quasi-Frobenius thus answering the different questions asked in [M. C. Iovanov, On infinite MacWilliams rings and minimal injectivity conditions, Proc. Amer. Math. Soc., DOI: 10.1090/p…
We provide new characterizations of pseudo-Frobenius and quasi-Frobenius rings in terms of tight modules. In the process, we also provide fresh perspectives on FGF and CF conjectures. In particular, we propose new natural extensions of these conjectures which connect them with the classical theory of PF rings. Our techniques are mainly based on set-theoretic counting arguments initiated by Osofsky. Several corollaries and examples to illustrate their applications are given.
Let $i: A\to R$ be a ring morphism, and $χ: R\to A$ a right $R$-linear map with $χ(χ(r)s)=χ(rs)$ and $χ(1_R)=1_A$. If $R$ is a Frobenius $A$-ring, then we can define a trace map $\tr: A\to A^R$. If there exists an element of trace 1 in $A$, then $A$ is right FBN if and only if $A^R$ is right FBN and $A$ is right noetherian. The result can be generalized to the case where $R$ is an $I$-Frobenius $A$-ring, and the condition on the trace can be replaced by a weaker condition. We recover results of Garc\'ıa and del R\'ıo and by Dǎscǎlescu, Kelarev and Torrecillas on actions of group and Hopf algeb…
Let M be an (A,B)-bi-module over left and right respectively Artinian rings. Let C be a multiplication ring of bi-module M, i.e. the ring C is generated by images of a and B in End(M). Let Z be a center of C. Bi-module M over rings C and Z we call a bi-module of translation of (A,B)-bi-module M. We investigate a relationship between the fact that (A,B)-bi-module M is quasi-Frobenius and the fact that (C,Z)-bi-module M is quasi-Frobenius.
We continue a study of separable equivalence from Hokkaido Mathematical Journal 24 (1995), 527-549. We prove that symmetric separable equivalent rings $A$ and $B$ are linked by a Frobenius bimodule ${}_AP_B$ such that $A$ is $P$-separable over $B$. Separably equivalent rings are linked by a biseparable bimodule $P$. In addition, the ring extension $A \rightarrow$ End $P_B$ is split, separable Frobenius. It is observed that left and right finite projective bimodules over symmetric algebras are Frobenius bimodules; twisted by the Nakayama automorphisms if over Frobenius algebras.
We investigate adjoint and Frobenius pairs between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings, which leads to a reasonable notion of Frobenius coring extension. When applied to corings stemming from entwining structures, we obtain new results in this setting and in graded ring theory.
We study the existence of a Batalin-Vilkovisky differential on the complete cohomology ring of a Frobenius algebra.We construct a Batalin-Vilkovisky differential on the complete cohomology ring in the case of Frobenius algebras with diagonalizable Nakayama automorphisms.
The Faith-Menal conjecture says that every strongly right $Johns$ ring is $QF$. The conjecture is also equivalent to say every right noetherian left $FP$-injective ring is $QF$. In this short article, we show that the conjecture is true under the condition(a proper generalization of left $CS$ condition)that every nonzero complement left ideal is not small(a left ideal $I$ is called small if for every left ideal $K$, $K$+$I$=$R$ implies $K$=$R$). It is also proved that (1) $R$ is $QF$ if and only if $R$ is a left and right mininjective ring with $ACC$ on right annihilators in which $S_{r}\subse…
We give a sufficient condition for a bi-invariant weight on a Frobenius bimodule to satisfy the extension property. This condition applies to bi-invariant weights on a finite Frobenius ring as a special case. The complex-valued functions on a Frobenius bimodule are viewed as a module over the semigroup ring of the multiplicative semigroup of the coefficient ring.
This expository article presents a unified ring theoretic approach, based on the theory of Frobenius algebras, to a variety of results on Hopf algebras. These include a theorem of S. Zhu on the degrees of irreducible representations, the so-called class equation, the determination of the semisimplicity locus of the Grothendieck ring, the spectrum of the adjoint class and a non-vanishing result for the adjoint character.
For a finite group $G$, let $\tilde{\mathbb{Z}}$ be the semilocalization of $\mathbb{Z}$ at the prime divisors of $|G|$. If $G$ is a Frobenius group with Frobenius kernel $K$, it is shown that each torsion unit in the group ring $\tilde{\mathbb{Z}} G$ which maps to the identity under the natural ring homomorphism $\tilde{\mathbb{Z}} G \rightarrow \tilde{\mathbb{Z}} G/K$ is conjugate to an element of $G$ by a unit in $\tilde{\mathbb{Z}} G$.
The values of the homogeneous weight are determined for finite Frobenius rings that are a direct product of local Frobenius rings. This is used to investigate the partition induced by this weight and its dual partition under character-theoretic dualization. A characterization is given of those rings for which the induced partition is reflexive or even self-dual.
A pair of adjoint functors $(F,G)$ is called a Frobenius pair of the second type if $G$ is a left adjoint of $βFα$ for some category equivalences $α$ and $β$. Frobenius ring extensions of the second kind provide examples of Frobenius pairs of the second kind. We study Frobenius pairs of the second kind between categories of modules, comodules, and comodules over a coring. We also show that a finitely generated projective Hopf algebra over a commutative ring is always a Frobenius extension of the second kind, and prove that the integral spaces of the Hopf algebra and its dual are isomorphic.
We show that the Nakayama automorphism of a Frobenius algebra $R$ over a field $k$ is independent of the field (Theorem 4). Consequently, the $k$-dual functor on left $R$-modules and the bimodule isomorphism type of the $k$-dual of $R$, and hence the question of whether $R$ is a symmetric $k$-algebra, are independent of $k$. We give a purely ring-theoretic condition that is necessary and sufficient for a finite-dimensional algebra over an infinite field to be a symmetric algebra (Theorem 7). Key words: Nakayama automorphism, Frobenius algebra, Frobenius ring, symmetric algebra, dual module, du…
With a small suitable modification, dropping the projectivity condition, we extend the notion of a Frobenius algebra to grant that a Frobenius algebra over a Frobenius commutative ring is itself a Frobenius ring. The modification introduced here also allows Frobenius finite rings to be precisely those rings which are Frobenius finite algebras over their characteristic subrings. From the perspective of linear codes, our work expands one's options to construct new finite Frobenius rings from old ones. We close with a discussion of generalized versions of the MacWilliams identities that may be ob…
Analogous to a recent result of N. Kowalzig and U. Krähmer for twisted Calabi-Yau algebras, we show that the Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra, thus generalizing a result of T.Tradler for finite dimensional symmetric algebras. We give a criterion to determine when a Frobenius algebra given by quiver with relations has semisimple Nakayama automorphism and apply it to some known classes of tame Frobenius algebras. We also provide ample examples including quantum complete intersections, finite dimensional Hopf a…
In this paper, we define and discuss higher-dimensional and absolute versions of symmetric, Frobenius, and quasi-Frobenius algebras. In particular, we compare these with the relative notions defined by Scheja and Storch. We also prove the validity of codimension two-argument for modules over a coherent sheaf of algebras with a $2$-canonical module, generalizing a result of the author.
Let $\Cc$ and $\Dd$ be two corings over a ring $A$ and $\Cc\stackrelλ{\longrightarrow}\Dd$ be a morphism of corings. We investigate the situation when the associated induced ("corestriction of scalars") functor $\Mm^\Cc\longrightarrow \Mm^\Dd$ is a Frobenius functor, and call these morphisms Frobenius extensions of corings. The characterization theorem generalizes notions such as Frobenius corings and is applied to several situations; in particular, provided some (general enough) flatness conditions hold, the notion proves to be dual to that of Frobenius extensions of rings (algebras). Several…
In this paper we consider stabilizer codes over local Frobenius rings. First, we study the relative minimum distances of a stabilizer code and its reduction onto the residue field. We show that for various scenarios, a free stabilizer code over the ring does not underperform the according stabilizer code over the field. This leads us to conjecture that the same is true for all free stabilizer codes. Secondly, we focus on the isometries of stabilizer codes. We present some preliminary results and introduce some interesting open problems.
In this paper, we study the Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between the Poisson homology and the Poisson cohomology, similar to the duality between the Hochschild homology and the Hochschild cohomology of a Frobenius algebra. Using the non-degenerated bilinear form on a Frobenius algebra we construct a Batalin-Vilkovisky structure on the Poisson cohomology ring of a class of Frobenius Poisson algebras.
We consider a finite acyclic quiver $\mathcal{Q}$ and a quasi-Frobenius ring $R$. We endow the category of quiver representations over $R$ with a model structure, whose homotopy category is equivalent to the stable category of Gorenstein-projective modules over the path algebra $R\mathcal{Q}$. As an application, we then characterize Gorenstein-projective $R\mathcal{Q}$-modules in terms of the corresponding quiver $R$-representations; this generalizes a result obtained by Luo-Zhang to the case of not necessarily finitely generated $R\mathcal{Q}$-modules, and partially recover results due to Eno…
Let $K$ be a field of characteristic $p>0$. It is proved that each automorphism $\s \in \Aut_K(\CDPn)$ of the ring $\CDPn$ of differential operators on a polynomial algebra $P_n= K[x_1, ..., x_n]$ is {\em uniquely} determined by the elements $\s (x_1), ... ,\s (x_n)$, and the set $\Frob (\CDPn)$ of all the extensions of the Frobenius from certain maximal commutative polynomial subalgebras of $\CDPn$, like $P_n$, is equal to $\Aut_K(\CDPn) \cdot \CF$ where $\CF$ is the set of all the extensions of the Frobenius from $P_n$ to $\CDPn$ that leave invariant the subalgebra of scalar differential ope…
We prove that for a Frobenius extension, a module over the extension ring is Gorenstein projective if and only if its underlying module over the base ring is Gorenstein projective. For a separable Frobenius extension between Artin algebras, we obtain that the extension algebra is CM-finite (resp. CM-free) if and only if so is the base algebra. Furthermore, we prove that the reprensentation dimension of Artin algebras is invariant under separable Forbenius extensions.
In this paper we study initial topological properties of the (non-)finitely-generated locus of Frobenius Algebra coming from Stanley-Reisner rings defined through face ideals. More specifically, we will give a partial answer to a conjecture made by M. Katzman about the openness of the finitely generated locus of such Frobenius algebras. This conjecture can be formulated in a precise manner: Let us define \begin{equation*} U=\{P\in \text{Spec}(R)~:~\mathcal{F}(E_{R_{p}})~\text{is a finitely generated}~R_{p}\text{-algebra}\} \end{equation*} where $\mathcal{F}$ denotes the Frobenius functor and $…
Gendo-Frobenius algebras are a common generalisation of Frobenius algebras and of gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius and on gendo-symmetric algebras. In addition, Frobenius algebras are shown to be precisely those gendo-Frobenius algebras that have a counit compatible with this comultiplication. Moreover, a new characterisation of gendo-Frobenius algebras is given. This new characterisation is a key for constructing the comultiplication of gendo-Frobenius algebras.
Comparing self-dual corings, Frobenius corings and Ringel self-duality
Swansea University（斯旺西大学）；University of Białystok（比亚韦斯托克大学）；University of Bielefeld（比勒费尔德大学）；TRR 358, University of Bielefeld（比勒费尔德大学 TRR 358）；University of Stuttgart（斯图加特大学）；Uppsala University（乌普萨拉大学）
We calculate Frobenius-Schur indicator values for some fusion categories obtained from inclusions of finite groups $H\subset G$, where more concretely $G$ is symmetric or alternating, and $H$ is a symmetric, alternating or cyclic group. Our work is strongly related to earlier results by Kashina-Mason-Montgomery, Jedwab-Montgomery, and Timmer for bismash product Hopf algebras obtained from exact factorizations of groups. We can generalize some of their results, settle some open questions and offer shorter proofs; this already pertains to the Hopf algebra case, while our results also cover fusio…
We prove some cases of a conjecture of Lewis, Reiner and Stanton regarding Hilbert series corresponding to the action of $Gl_n(\mathbb{F}_q)$ on a polynomial ring modulo Frobenius powers. We also give a few conjectures about the invariant ring for certain cases that we don't prove completely.
The representation theory of finite groups began with Frobenius's factorization of Dedekind's group determinant. In this paper, we consider the case of the semigroup determinant. The semigroup determinant is nonzero if and only if the complex semigroup algebra is Frobenius, and so our results include applications to the study of Frobenius semigroup algebras. We explicitly factor the semigroup determinant for commutative semigroups and inverse semigroups. We recover the Wilf-Lindström factorization of the semigroup determinant of a meet semilattice and Wood's factorization for a finite commutat…
We study cup product and cap product in Tate-Hochschild theory for a finite dimensional Frobenius algebra. We show that Tate-Hochschild cohomology ring equipped with cup product is isomorphic to singular Hochschild cohomology ring introduced by Wang. An application of cap product occurs in Tate-Hochschild duality; as in Tate (co)homology of a finite group, the cap product with the fundamental class of a finite dimensional Frobenius algebra provides certain duality result between Tate-Hochschild cohomology and homology groups. Moreover, we characterize minimal complete resolutions over a finite…
We study the K-group K_1 of the group ring of a finite group over a coefficient ring which is p-adically complete and admits a lift of Frobenius. In this paper, we consider the image of K_1 under the determinant map; the central tool is the group logarithm which we can define using the Frobenius lift. Using this we prove a fixed point theorem for the determinantal image of K_1.
To any Frobenius superalgebra $A$ we associate towers of Frobenius nilCoxeter algebras and Frobenius nilHecke algebras. These act naturally, via Frobenius divided difference operators, on Frobenius polynomial algebras. When $A$ is the ground ring, our algebras recover the classical nilCoxeter and nilHecke algebras. When $A$ is the two-dimensional Clifford algebra, they are Morita equivalent to the odd nilCoxeter and odd nilHecke algebras.
Let R > S be finite Frobenius rings for which there exists a trace map T from R onto S as left S modules. Let C:= {x -> T(ax + bf(x)) : a,b in R}. Then C is an S-linear subring-subcode of a left linear code over R. We consider functions f for which the homogeneous weight distribution of C can be computed. In particular, we give constructions of codes over integer modular rings and commutative local Frobenius that have small spectra.
This article is divided into two parts. In the first part we work over a field $\mathbb{k}$ and prove that the Frobenius space associated to a Frobenius algebra is generated as left A-module by the Frobenius coproduct. In particular, we prove that the Frobenius dimension coincides with the dimension of the algebra. In the second part we work with a commutative ring $k$. We introduce the concept of nearly Frobenius algebras in this context and construct solutions of the Yang-Baxter equation starting from elements in the Frobenius space. Also, we give a list of equivalent characterizations of ne…
We study quasi-Hopf algebras and their subobjects over certain commutative rings from the point of view of Frobenius algebras. We introduce a type of Radford formula involving an anti-automorphism and the Nakayama automorphism of a Frobenius algebra, then view several results in quantum algebras from this vantage-point. In addition, separability and strong separability of quasi-Hopf algebras are studied as Frobenius algebras.
A finite ring R and a weight w on R satisfy the Extension Property if every R-linear w-isometry between two R-linear codes in R^n extends to a monomial transformation of R^n that preserves w. MacWilliams proved that finite fields with the Hamming weight satisfy the Extension Property. It is known that finite Frobenius rings with either the Hamming weight or the homogeneous weight satisfy the Extension Property. Conversely, if a finite ring with the Hamming or homogeneous weight satisfies the Extension Property, then the ring is Frobenius. This paper addresses the question of a characterization…
Finite Frobenius rings have been characterized as precisely those finite rings satisfying the MacWilliams extension property, by work of Wood. In the present note we offer a generalization of this remarkable result to the realm of Artinian rings. Namely, we prove that a left Artinian ring has the left MacWilliams property if and only if it is left pseudo-injective and its finitary left socle embeds into the semisimple quotient. Providing a topological perspective on the MacWilliams property, we also show that the finitary left socle of a left Artinian ring embeds into the semisimple quotient i…
It is shown in this paper that, if $R$ is a Frobenius ring, then the quaternion ring $\mathcal{H}_{a,b}(R)$ is a Frobenius ring for all units $a,b \in R$. In particular, if $q$ is an odd prime power then $\mathcal{H}_{a,b}(\mathbb{F}_q)$ is the semisimple non-commutative matrix ring $M_2(\mathbb{F}_q)$. Consequently, a homogeneous weight that depends on the field size $q$ is obtained. On the other hand, the homogeneous weight of a finite Frobenius ring with a unique minimal ideal is derived in terms of the size of the ideal. This is illustrated by the quaternions over the Galois ring $GR(2^r,m…
Let $R$ be the homogeneous coordinate ring of the Grassmannian $\mathbb{G}=Gr(2,n)$ defined over an algebraically closed field $k$ of characteristic $p \geq \max\{n-2,3\}$. In this paper we give a description of the decomposition of $R$, considered as graded $R^{p^r}$-module, for $r \geq 2$. This is a companion paper to our earlier paper, where the case $r=1$ was treated, and taken together, our results imply that $R$ has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring f…
