Quasi-Associative Algebras Overview

• Quasi-associative algebras are defined by a deformed product A×B = λAB + (1-λ)BA that weakens classical associativity while ensuring power-associativity.
• They bridge historical frameworks like Jordan’s quasi-multiplication with modern models such as quasi-Jordan algebras, dual quasi-bialgebras, and quasi-twilled associative algebras.
• Contemporary approaches employ reassociators and deformation maps to control nonassociativity, offering versatile methods for algebraic and physical applications.

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 (Liebmann et al., 2019). 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 (Bremner, 2010, Ardizzoni et al., 2015, Das et al., 2024).

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 (Liebmann et al., 2019). 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 (Liebmann et al., 2019).

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 (Liebmann et al., 2019).

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 (Liebmann et al., 2019).

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 (Liebmann et al., 2019).

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 $A\times B=\lambda AB+(1-\lambda)BA.$0 with operations $A\times B=\lambda AB+(1-\lambda)BA.$1 and $A\times B=\lambda AB+(1-\lambda)BA.$2, the right quasi-Jordan product is

$A\times B=\lambda AB+(1-\lambda)BA.$3

or, in Bremner’s notation,

$A\times B=\lambda AB+(1-\lambda)BA.$4

This construction is the dialgebraic analogue of Jordan symmetrization, but it is one-sided rather than fully symmetric (Bremner, 2010).

The defining low-degree identities are correspondingly one-sided. The first is right commutativity,

$A\times B=\lambda AB+(1-\lambda)BA.$5

The second is the right quasi-Jordan identity,

$A\times B=\lambda AB+(1-\lambda)BA.$6

The third, found in degree $A\times B=\lambda AB+(1-\lambda)BA.$7, is the associator-derivation identity

$A\times B=\lambda AB+(1-\lambda)BA.$8

Bremner proves that over characteristic $A\times B=\lambda AB+(1-\lambda)BA.$9, every polynomial identity of degree $a\times b=\lambda ab+\mu ba,$0 satisfied by the quasi-Jordan product is a consequence of right commutativity, and every polynomial identity of degree $a\times b=\lambda ab+\mu ba,$1 is a consequence of right commutativity together with the multilinear identities $a\times b=\lambda ab+\mu ba,$2 and $a\times b=\lambda ab+\mu ba,$3, where $a\times b=\lambda ab+\mu ba,$4 is the linearization of the right quasi-Jordan identity and $a\times b=\lambda ab+\mu ba,$5 is equivalent to the associator-derivation law (Bremner, 2010).

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\times b=\lambda ab+\mu ba,$6

It then shows that every identity for the dialgebra-induced product in degree $a\times b=\lambda ab+\mu ba,$7 is a consequence of these identities in degree $a\times b=\lambda ab+\mu ba,$8, but that six new irreducible identities appear in degree $a\times b=\lambda ab+\mu ba,$9; some but not all are noncommutative preimages of the Glennie identity (Bremner et al., 2010). 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 $A\times B=\lambda AB+(1-\lambda)BA,$0.

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 $A\times B=\lambda AB+(1-\lambda)BA,$1: the coalgebra structure remains strictly coassociative, while multiplication is weakened by a convolution-invertible reassociator

$A\times B=\lambda AB+(1-\lambda)BA,$2

The central quasi-associative law is

$A\times B=\lambda AB+(1-\lambda)BA,$3

together with the pentagon-type identity

$A\times B=\lambda AB+(1-\lambda)BA,$4

and the normalization condition

$A\times B=\lambda AB+(1-\lambda)BA,$5

(Ardizzoni et al., 2015).

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 (Ardizzoni et al., 2015). 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 $A\times B=\lambda AB+(1-\lambda)BA,$6 whose underlying vector space decomposes as

$A\times B=\lambda AB+(1-\lambda)BA,$7

such that $A\times B=\lambda AB+(1-\lambda)BA,$8 is a subalgebra (Das et al., 2024). Equivalently, the multiplication on $A\times B=\lambda AB+(1-\lambda)BA,$9 is encoded by

$[a,b,c]=(ab)c-a(bc),$0

with

$[a,b,c]=(ab)c-a(bc),$1

Here $[a,b,c]=(ab)c-a(bc),$2 fails to be a subalgebra exactly by the $[a,b,c]=(ab)c-a(bc),$3-valued defect $[a,b,c]=(ab)c-a(bc),$4 (Das et al., 2024).

The Maurer–Cartan formulation is central. If

$[a,b,c]=(ab)c-a(bc),$5

then the data define a quasi-twilled associative algebra precisely when

$[a,b,c]=(ab)c-a(bc),$6

Within this framework, a strong deformation map $[a,b,c]=(ab)c-a(bc),$7 is defined by the condition that its graph $[a,b,c]=(ab)c-a(bc),$8 is a subalgebra; such maps generalize associative algebra homomorphisms, derivations, crossed homomorphisms, and the associative analogue of modified $[a,b,c]=(ab)c-a(bc),$9-matrices. Dually, a weak deformation map $(a,b,c)=(ab)c-a(bc).$0 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 (Das et al., 2024).

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 $(a,b,c)=(ab)c-a(bc).$1, derivations, homomorphisms, and crossed homomorphisms, while left deformation maps unify relative Rota-Baxter operators of weight $(a,b,c)=(ab)c-a(bc).$2, twisted Rota-Baxter operators, Reynolds operators, and deformation maps of matched pairs of associative algebras (Liu et al., 2024). Both papers construct controlling algebras via derived brackets and associated cohomologies, so the deformation equations become Maurer–Cartan equations in curved $(a,b,c)=(ab)c-a(bc).$3- or $(a,b,c)=(ab)c-a(bc).$4-algebras (Das et al., 2024, Liu et al., 2024). 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: $(a,b,c)=(ab)c-a(bc).$5 while distributivity is weakened to order inequalities,

$(a,b,c)=(ab)c-a(bc).$6

Its quasi-algebras are therefore associative ordered subdistributive algebra-like objects, not algebras satisfying a weakened associativity law (Dehghanizade et al., 2020).

Shift associative algebras provide a different nearby notion. They satisfy the permutation identity

$(a,b,c)=(ab)c-a(bc).$7

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 $(a,b,c)=(ab)c-a(bc).$8 (Abdelwahab et al., 2024). 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 (Ladkani, 2014). “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 (Zhao, 2010). Likewise, work on quasi-centroids and quasi-derivations studies quasi-operator identities on associative algebras rather than quasi-associative multiplication laws (Fiidow et al., 2023). Finally, compatible associative algebras are vector spaces with two associative multiplication operations satisfying a natural compatibility condition (Damdji et al., 8 May 2025). 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,b,c)=(ab)c-a(bc).$9, which is non-associative in general but power-associative (Liebmann et al., 2019). 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 (Bremner, 2010, Ardizzoni et al., 2015, Das et al., 2024).