Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0103001

An Approach to Hopf Algebras via Frobenius Coordinates I

Lars Kadison·A. A. Stolin
View original source ↗Read source document ↗Export metadata ↓

Abstract

In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of an FH-algebra H \cite{Par72} and extend two recent theorems in \cite{EG}. We obtain two Radford formulas for the antipode in H and generalize in Section 7 the results on its order in \cite{FMS}. We study the Frobenius structure on an FH-subalgebra pair in Sections 5 and 6. In Section 8 we show that the quantum double of H is symmetric and unimodular.

Explore related subjects

math.RAmath.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lars Kadison, A. A. Stolin. 2001-03-05. An Approach to Hopf Algebras via Frobenius Coordinates I. https://arxiv.org/abs/math/0103001

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Directed partial orders on the complex number field

We construct a class of positive cones that make $\mathbb{C}$ into a directed partially ordered ring. The positive cones are defined using integral closures of local rings, associated with a transcendence basis and a chosen real generator. A localization criterion also yields such orders on every transcendental extension of $\mathbb{Q}$. For a fixed coefficient field, we prove that two real generators define the same cone if and only if they differ by an affine map with positive real-algebraic slope and translation algebraic over that field. The nonzero positive elements are closed under inversion.

math.RA↗
arXiv2026

Anti-successors and anti-replicators, Manin products and Koszul duality

We develop an operadic theory of anti-splitting and anti-replicating for binary operads. For a binary operad $\opd$, we define its anti-bisuccessor $\ABSu(\opd)$ and anti-trisuccessor $\ATSu(\opd)$. Each generating operation is recovered as the negative sum of two or three split operations, yielding canonical operad morphisms $\opd\to\ABSu(\opd)$ and $\opd\to\ATSu(\opd)$. For binary quadratic operads, these constructions are described by Manin black products: \[ \ABSu(\opd)\cong\antiprelie\bullet\opd, \qquad \ATSu(\opd)\cong\antipostlie\bullet\opd. \] Dually, we also introduce the canonical quadratic anti-duplicator and anti-triplicator, together with their combinatorial counterparts defined by labeled planar binary trees. We show that they coincide precisely under a full-support condition. As applications, we obtain anti-dupli-com and anti-dupli-Lie algebras. We also display an example in which the two constructions differ. Furthermore, we establish the Koszul duality relations \[ \widehat{\ADup}(\opd^!) \cong\ABSu(\opd)^!, \qquad \widehat{\ATrip}(\opd^!) \cong\ATSu(\opd)^!, \] providing an effective description of the canonical quadratic anti-duplicator in terms of the Manin white product. The framework recovers known constructions, including anti-dendriform, anti-pre-Lie and anti-pre-Poisson algebras, and produces further examples and Koszul dual pairs.

math.RA↗
arXiv2026

On the Connected Components of Structure Graphs of Associative Rings

The article studies the notion of the structure graph of an associative ring with unity. It is known that the structural graph of a ring with $n$ elements has at least $n$ connected components. The article investigates when this lower bound is attained. A hypothesis is stated that the only rings of order $n$ having exactly $n$ connected components are the rings $\mathbb{Z}_2$, $\mathbb{Z}_3$, $\mathbb{Z}_7$, $GF(4)$, and their direct sums. The hypothesis is proved for all rings except residue rings modulo a prime of the form $7+12k$, $k\geq181$, and Galois fields of characteristic~2 with extension degree $\geq17$.

math.RA↗