---
title: Quasi-Associative Algebras Overview
url: https://www.emergentmind.com/topics/quasi-associative-algebras
type: topic
---

# Quasi-Associative Algebras Overview

Searching arXiv for the cited papers to ground the response in current arXiv records.
Searching arXiv for historical and modern uses of “quasi-associative” and related structures.
Quasi-associative algebras, in the narrow historical sense explicitly used in the literature, are algebras obtained from an associative algebra by replacing the original product \(AB\) with the deformed multiplication
\[
A\times B=\lambda AB+(1-\lambda)BA.
\]
In the historical survey of Jordan’s program, these algebras are non-associative in general but power-associative, and the term belongs to a broader landscape of attempts to weaken associativity while retaining enough structure for algebraic or physical applications [1909.04027]. Modern arXiv literature also places beside this classical notion several related frameworks—most notably quasi-Jordan algebras, dual quasi-bialgebras, and quasi-twilled associative algebras—in which the failure of ordinary associativity is one-sided, cocycle-controlled, or encoded by graph-subalgebra conditions rather than by a single universal identity [1008.2009] [1507.02402] [2409.00443].

## 1. Historical definition and Jordan’s quasi-multiplication

The historical point of departure is Jordan’s quasi-multiplication
\[
a\times b=\lambda ab+\mu ba,
\]
together with the special form
\[
A\times B=\lambda AB+(1-\lambda)BA,
\]
which the historical survey identifies as the defining construction of “quasi-associative-algebras” in the literature [1909.04027]. This product is distributive over addition, but generally neither commutative nor associative. The same source records two basic features that delimit the class sharply: Jordan thought quasi-associative algebras would probably not be useful for generalizing quantum mechanics, and all quasi-associative algebras are power-associative [1909.04027].

The surrounding context is Jordan’s early program of constructing algebras “with weakened associativity.” In that program, quasi-associative algebras form one branch among several others, including Jordan algebras, flexible algebras, alternative algebras, and later non-power-associative systems. The historical significance of the quasi-associative case is therefore not that it became the dominant replacement for associative algebras, but that it provided an early and explicit model of a product obtained by interpolating between left and right multiplication inside an associative algebra [1909.04027].

## 2. Associativity defects, power-associativity, and adjacent weak laws

The common language for these structures is the associator
\[
[a,b,c]=(ab)c-a(bc),
\]
or, in some noncommutative Jordan-type settings,
\[
(a,b,c)=(ab)c-a(bc).
\]
Within the historical classification, quasi-associative algebras remain on the power-associative side of the spectrum: powers are unambiguous, and the survey states this as
\[
a^{n+m}=a^n a^m.
\]
That property sharply separates them from later non-power-associative proposals such as Lie-triple and elementary or almost-Jordan algebras, as well as from higher octonionic matrix constructions where even power-associativity can fail [1909.04027].

The same historical account situates quasi-associative algebras beside several neighboring weakenings of associativity. Flexibility is expressed by
\[
(ab)a=a(ba),
\]
alternativity by
\[
[x,x,y]=[x,y,x]=[y,x,x]=0,
\]
and the Jordan identity by
\[
[a,b,a^2]=0.
\]
These identities are not equivalent to quasi-associativity. They encode distinct ways in which associativity may fail while preserving different fragments of the ordinary associative calculus. This suggests that “quasi-associative algebra” is best treated as one historically specific member of a wider family of controlled associativity weakenings, rather than as the generic name for all such systems [1909.04027].

## 3. One-sided quasi-associativity: quasi-Jordan algebras

A modern and technically important descendant of the nearly associative viewpoint is the theory of quasi-Jordan algebras. In an associative dialgebra \(D\) with operations \(\dashv\) and \(\vdash\), the right quasi-Jordan product is
\[
a\triangleleft b=\frac12(a\dashv b+b\vdash a),
\]
or, in Bremner’s notation,
\[
ab=a\dashv b+b\vdash a.
\]
This construction is the dialgebraic analogue of Jordan symmetrization, but it is one-sided rather than fully symmetric [1008.2009].

The defining low-degree identities are correspondingly one-sided. The first is right commutativity,
\[
a(bc)=a(cb).
\]
The second is the right quasi-Jordan identity,
\[
(ba)a^2=(ba^2)a.
\]
The third, found in degree \(4\), is the associator-derivation identity
\[
(b,a^2,c)=2(b,a,c)a.
\]
Bremner proves that over characteristic \(0\), every polynomial identity of degree \(\le 3\) satisfied by the quasi-Jordan product is a consequence of right commutativity, and every polynomial identity of degree \(\le 4\) is a consequence of right commutativity together with the multilinear identities \(J\) and \(K\), where \(J\) is the linearization of the right quasi-Jordan identity and \(K\) is equivalent to the associator-derivation law [1008.2009].

The later paper on special identities extends this picture. It states that semispecial quasi-Jordan algebras, also called Jordan dialgebras, are defined by the three identities
\[
a(bc)=a(cb),\qquad (ba)a^2=(ba^2)a,\qquad (b,a^2,c)=2(b,a,c)a.
\]
It then shows that every identity for the dialgebra-induced product in degree \(\le 7\) is a consequence of these identities in degree \(\le 4\), but that six new irreducible identities appear in degree \(8\); some but not all are noncommutative preimages of the Glennie identity [1008.2723]. In this sense quasi-Jordan algebras furnish a concrete one-sided model of controlled nonassociativity inherited from associative dialgebras, rather than from a direct deformation \(AB\mapsto \lambda AB+(1-\lambda)BA\).

## 4. Categorical quasi-associativity and reassociators

A different modern formalization appears in the theory of dual quasi-bialgebras. The relevant paper develops dual quasi-bialgebras as objects in \(\mathrm{NAlg}(\mathrm{Coalg}_k)\): the coalgebra structure remains strictly coassociative, while multiplication is weakened by a convolution-invertible reassociator
\[
\omega:U\otimes U\otimes U\to k.
\]
The central quasi-associative law is
\[
\sum \omega(x_1\otimes y_1\otimes z_1)\,(x_2y_2)z_2
=
\sum x_1(y_1z_1)\,\omega(x_2\otimes y_2\otimes z_2),
\]
together with the pentagon-type identity
\[
(\omega\circ (U\otimes U\otimes m)) * (\omega\circ (m\otimes U\otimes U))
=
(\varepsilon\otimes \omega) * (\omega\circ (U\otimes m\otimes U)) * (\omega\otimes \varepsilon)
\]
and the normalization condition
\[
\omega(x\otimes y\otimes 1)=\omega(x\otimes 1\otimes y)=\omega(1\otimes x\otimes y)=\varepsilon(x)\varepsilon(y)
\]
[1507.02402].

This framework is not historical quasi-associativity in Jordan’s sense, but it gives an exact algebraic model of associativity weakened in a coherent, cocycle-controlled manner. The same paper shows that finite duality transports quasi-coassociativity on quasi-bialgebras into quasi-associativity on dual quasi-bialgebras, and establishes a contravariant adjunction between quasi-bialgebras and split dual quasi-bialgebras [1507.02402]. A plausible implication is that the modern categorical use of “quasi-associative” is best understood not as a single identity but as associativity governed by a reassociator satisfying its own coherence equations.

## 5. Quasi-twilled associative algebras and deformation maps

Another modern development keeps the total algebra strictly associative while relaxing the behavior of a decomposition. A quasi-twilled associative algebra is an associative algebra \(\mathbb A\) whose underlying vector space decomposes as
\[
\mathbb A=A\oplus B
\]
such that \(B\subset \mathbb A\) is a subalgebra [2409.00443]. Equivalently, the multiplication on \(\mathbb A\) is encoded by
\[
(A_\mu,B_\nu,\triangleright,\triangleleft,\rightharpoonup,\leftharpoonup,\phi),
\]
with
\[
(a,x)\cdot_{\mathbb A}(b,y)
=
\bigl(
a\cdot_A b+a\leftharpoonup y+x\rightharpoonup b,\;
x\cdot_B y+a\triangleright y+x\triangleleft b+\phi(a,b)
\bigr).
\]
Here \(A\) fails to be a subalgebra exactly by the \(B\)-valued defect \(\phi:A\times A\to B\) [2409.00443].

The Maurer–Cartan formulation is central. If
\[
\Omega=\widetilde\phi+\widehat\mu+\widetilde\nu,
\]
then the data define a quasi-twilled associative algebra precisely when
\[
[\Omega,\Omega]=0.
\]
Within this framework, a strong deformation map \(D:A\to B\) is defined by the condition that its graph \(\mathrm{Gr}(D)\subset \mathbb A\) is a subalgebra; such maps generalize associative algebra homomorphisms, derivations, crossed homomorphisms, and the associative analogue of modified \({\sf r}\)-matrices. Dually, a weak deformation map \(r:B\to A\) is defined by the same graph-subalgebra condition and generalizes relative Rota-Baxter operators of any weight, twisted Rota-Baxter operators, Reynolds operators, left-averaging operators, and right-averaging operators [2409.00443].

The companion paper reformulates these constructions as right and left deformation maps of quasi-twilled associative algebras. Right deformation maps unify modified Rota-Baxter operators of weight \(\lambda\), derivations, homomorphisms, and crossed homomorphisms, while left deformation maps unify relative Rota-Baxter operators of weight \(0\), twisted Rota-Baxter operators, Reynolds operators, and deformation maps of matched pairs of associative algebras [2409.02651]. Both papers construct controlling algebras via derived brackets and associated cohomologies, so the deformation equations become Maurer–Cartan equations in curved \(L_\infty\)- or \(L_\infty\)-algebras [2409.00443] [2409.02651]. This suggests a modern shift in emphasis: instead of asking only for a single quasi-associative identity, one studies how associative structure is split, curved, and deformed by maps whose graphs remain multiplicatively stable.

## 6. Related modern usages and terminological distinctions

Several arXiv papers use “quasi” in ways that are adjacent to, but not identical with, quasi-associativity. The paper on quasi-algebras is explicit that associativity is kept exactly:
\[
x*(y*z)=(x*y)*z,
\]
while distributivity is weakened to order inequalities,
\[
x*(y+z)\le x*y+x*z,\qquad (x+y)*z\le x*z+y*z.
\]
Its quasi-algebras are therefore associative ordered subdistributive algebra-like objects, not algebras satisfying a weakened associativity law [2010.08724].

Shift associative algebras provide a different nearby notion. They satisfy the permutation identity
\[
(xy)z=y(zx),
\]
which the paper describes as a permutation form of associativity. It proves, among other structural results, that these algebras are power-associative, that every unital shift associative algebra is commutative associative, and that the first non-associative shift associative algebra appears only in dimension \(5\) [2408.07078]. Although the term “quasi-associative” is not used there, the identity belongs to the same general landscape of generalized associativity laws.

Other papers explicitly warn against terminological confusion. “Algebras of quasi-quaternion type” are finite-dimensional associative algebras; the “quasi” modifies quaternion type, not associativity [1404.6834]. “Quasi-stable algebras” in the theory of Mathieu subspaces are also associative algebras; the term refers to the abundance of Mathieu subspaces, not to any weakened associativity law [1005.4260]. Likewise, work on quasi-centroids and quasi-derivations studies quasi-operator identities on associative algebras rather than quasi-associative multiplication laws [2306.14331]. Finally, compatible associative algebras are vector spaces with two associative multiplication operations satisfying a natural compatibility condition [2505.05529]. A plausible implication is that these compatible pairs furnish an exact-associative comparison class for quasi-associative questions, since they exhibit nontrivial interaction of two products without introducing a nonzero associator.

In contemporary usage, therefore, “quasi-associative algebra” has a dual status. In the strict historical sense it refers to Jordan’s deformed product \(A\times B=\lambda AB+(1-\lambda)BA\), which is non-associative in general but power-associative [1909.04027]. In a broader modern sense, it points toward a family of structures in which associativity is one-sided, reassociator-controlled, permutation-modified, or encoded through quasi-twilled decompositions and deformation maps rather than abandoned outright [1008.2009] [1507.02402] [2409.00443].

Source: https://www.emergentmind.com/topics/quasi-associative-algebras