Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.09884

Induced structures of averaging commutative and cocommutative infinitesimal bialgebras via a new splitting of perm algebras

Chengming Bai·Li Guo·Guilai Liu·Quan Zhao
View original source ↗Read source document ↗Export metadata ↓

Abstract

It is well-known that an averaging operator on a commutative associative algebra gives rise to a perm algebra. This paper lifts this process to the level of bialgebras. For this purpose, we first give an infinitesimal bialgebra structure for averaging commutative associative algebras and characterize it by double constructions of averaging Frobenius commutative algebras. To find the bialgebra counterpart of perm algebras that is induced by such averaging bialgebras, we need a new two-part splitting of the multiplication in a perm algebra, which differs from the usual splitting of the perm algebra (into the pre-perm algebra) by the characterized representation. This gives rise to the notion of an averaging-pre-perm algebra, or simply an apre-perm algebra. Furthermore, the notion of special apre-perm algebras which are apre-perm algebras with the second multiplications being commutative is introduced as the underlying algebra structure of perm algebras with nondegenerate symmetric left-invariant bilinear forms. The latter are also the induced structures of symmetric Frobenius commutative algebras with averaging operators. Consequently, a double construction of averaging Frobenius commutative algebra gives rise to a Manin triple of special apre-perm algebras. In terms of bialgebra structures, this means that an averaging commutative and cocommutative infinitesimal bialgebra gives rise to a special apre-perm bialgebra.

Explore related subjects

math.RAmath.QAmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Chengming Bai, Li Guo, Guilai Liu, Quan Zhao. 2025-09-11. Induced structures of averaging commutative and cocommutative infinitesimal bialgebras via a new splitting of perm algebras. https://arxiv.org/abs/2509.09884

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

KEEP EXPLORING

Related papers

arXiv2026

Maximal tails,character fibres and induced modules for pullback Kumjian-Pask algebras

Let $f:\N^{k}\to\N^{\ell}$ be a surjective monoid homomorphism and let $Γ$ be a row-finite $\ell$-graph with no sources and finitely many vertices. We give an explicit graded isomorphism from the Kumjian--Pask algebra of the pullback $f^{*}Γ$ onto the tensor product of $\KP_{\K}(Γ)$ and the group algebra of the kernel of the group completion of $f$. When $Γ$ is strongly aperiodic, but need not be cofinal, every maximal tail $T$ and every maximal ideal $\mathfrak m$ of the kernel group algebra determine an explicit primitive ideal and primitive quotient. If, in addition, $\K$ is uncountable and algebraically closed, these ideals exhaust the primitive spectrum. We prove that the resulting parametrisation is a homeomorphism for the product of the maximal-tail and Zariski topologies. Each primitive ideal is realised as the annihilator of a simple module induced from the isotropy of a path which is cofinal in $T$, and the character fibres are algebraic tori. Two examples exhibit, respectively, a single character fibre and the non-Hausdorff gluing of two such fibres.

math.RA↗
arXiv2026

A classification of group gradings on incidence algebras over commutative rings

Let $R$ be a commutative ring with 1, $P$ a locally finite partially ordered set, and $G$ a group. We derive necessary and sufficient conditions for an $R$-algebra isomorphism between the incidence algebra $I(P,R)$ and the group algebra $RG$. Then, for an indecomposable ring $R$, a finite poset $P$ and an arbitrary group $G$, we classify the $G$-gradings of $I(P,R)$ up to graded isomorphism. The classification rests on a complete set of primitive orthogonal homogeneous idempotents. The corner algebras are split group algebras of finite abelian subgroups of $G$, and the off-diagonal Peirce blocks are multiplicity-free sums of bimodules induced from characters of double coset stabilizers. Graded isomorphisms are shown to have a rigid form, and a grading is determined up to graded isomorphism by the poset of idempotents, the corner groups, the types of the atomic bimodules and the structure constants of their multiplication. The data which occur are characterized by polynomial conditions, and over an algebraically closed field of characteristic zero only finitely many graded isomorphism classes share given partial invariants. An example shows that the structure constants cannot be omitted. Some previous results are extended and enhanced, while providing alternative proofs for some known facts.

math.RA↗