# Questions tagged [non-associative-algebras]

Questions about non-associative algebras other than Lie algebras.

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### An algebraic characterization of dual translation projective planes

It is well-known that translation projective planes are coordinatized by quasifields. More precisely, a projective plane is translation if and only if it has a ternary-ring $R$ which is linear, the ...

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### Who and when proved Artin's Theorem on alternative rings?

I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings).
Question. When has Artin proved this theorem and where was it published for the first ...

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### An isomorphic classification of non-associative division octonion algebras

A division octonion algebra over a field $F$ is a $8$-dimensional unital non-associative algebra $A$ over the field $F$, endowed with a quadratic form $N:A\times A\to F$ such that $N(xy)=N(x)N(y)$ and ...

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### Nonassociativity in Cayley-Algebras

Let $(E,s)$ be a Cayley algebra over a unital commutative ring $A$ with unit element $e$ and $s$ an antiautomorphism (i.e. $s(uv) = s(v)s(u)$, $u,v \in E$) of $E$ such that $u + s(u) \in Ae$ and $N(u) ...

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### Wedderburn–Artin like theorem for infinite dimensional Lie algebras?

The Wedderburn–Artin Theorem is one of the cornerstones of the structure theory of (associative) rings.
Wedderburn–Artin Theorem : Let $R$ be a left Artinian ring with zero Jacobson radical. Then $R$ ...

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### Does the Affine Pappus Axiom imply the Affine Desargues Axiom in affine planes?

I am interested in the affine version of the well-known Hessenberg's Theorem (saying that Pappian projective planes are Desarguesian).
First I introduce all necessary definitions.
Definition L. A ...

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### Functor from Leibniz algebra category to Lie-Yamaguti algebra category

Is there any functor from $\operatorname{Leib}_{\mathbb{K}}$ (Leibniz algebra category) to $\operatorname{LYA}_{\mathbb{K}}$ (Lie-Yamaguti algebra category)?
From Kinyon and Weinstein's paper I saw ...

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### What is the cardinality of liners of rank 4? Is it always equal 27?

Definition 1. A binary operation $\cdot:X\times X\to X$, $\cdot:(xy)\mapsto xy$, on a set $X$ will be called a line operation if
$$xx=x,\quad xy=yx,\quad (xy)x=y$$
for every $x,y\in X$.
Remark 1. ...

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### Chirality of octonion algebras

Octonion multiplication can be defined with respect to a set of triads. A set of such triads can be represented by a directed Fano plane diagram such as the following two diagrams.
This depicts two ...

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### Finite dimensional real division algebra up to isotopy

Finite dimensional real division (non necessarily associative) algebras exist in dimensions 1, 2, 4, and 8. The standard example is a Hurwitz algebra $(A,*)$: reals, complexes, quaternions, octonions. ...

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### Combinatorics on non-associative words

In my P.h.d research, I deal (among other things) with non-associative words, which we call monomials, and we need to consider two types of operations with these monomials.
The first one is simply ...

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### Degree 8 multilinear operations on Jordan algebras

I am interested in the dimension, or, even better, in the $S_8$-module structure of the space of degree 8 multilinear operations on Jordan algebras.
Recall that a Jordan algebra is a commutative but ...

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### When does this commutative non-associative algebra have nilpotent elements?

Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients
$(a_0, \dotsc, ...

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### Vector algebra in a Tarski space

By a Tarski space I understand a mathematical structure $(X,B,E)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and a 4-ary equidistance relation $E\subseteq X^2\times X^2$ ...

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### Non-associative algebras and determinant over 3 by 3 matrices

I have a non associative algebra $A$ that is not unital. And I have three by three matrices $X$ with coefficients over $A$ with a matrix product $*$. I'd like to define something like the determinant ...

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### Identity for the associator involving a third root of unity

This is a reference request. I came across the class of nonassociative algebras satisfying the following identity:
$$
(a,b,c)+\omega(b,c,a)+\omega^2(c,a,b)=0.
$$
Here:
by an "algebra" I mean a ...

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### Do compact inverse-property loops (or just compact Moufang loops) have invariant uniformities and bi-invariant Haar measure?

So, the overall question is in the title: Does a compact topological loop with the inverse property have a Haar measure that is simultaneously left and right invariant? (And we can restrict to ...

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### Non-associative Clifford algebra

Let $V$ be a finite-dimensional $\mathbb{R}$ vector space equipped with a symmetric, bilinear form $b : V \times V \to \mathbb{R}$.
My question is if there exists an analog of a Clifford algebra in ...

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### Nonassociative quaternion algebras

I'm interested in nonassociative central simple algebras; I've found Lee and Waterhouse's article 'Maximal Orders in Nonassociative Quaternion Algebras', and this cites a previous article by ...

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### Is the (left or right) Bol property Isotopy-invariant?

It is well known that a loop satisfies both the left Bol property $(x(yx))z = x(y(xz))$ and the right Bol property $((zx)y)x = z((xy)x)$ if and only if it is a Moufang loop. It is also well known that ...

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### Is there a characterisation of Cayley–Dickson Algebras?

The Cayley–Dickson construction takes an algebra with involution and produces another algebra with involution of twice the dimension.
Starting from the reals (with trivial involution), we ...

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### Nonassociative algebras closed under $\sqrt{\ }$?

Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients
$(a_0, \dotsc, ...

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### Why is the automorphism group of the octonions $G_2$ instead of $SO(7)$

I can calculate the derivation of the octonions and I clearly find the 14 generators that form the algebra of $G_2$. However, when I do the same calculation for the quaternions, I end up with the ...

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### Separable nonassociative algebras

In his paper "Structure and Representation of Nonassociative Algebras", Schafer notes that an arbitrary nonassociative algebra over a field is separable "if and only if the ...

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### Conceptual meaning of the Dickson construction

The Cayley-Dickson construction (a.k.a. Dickson construction or Dickson doubling) constructs a new *-algebra $A'$ out of a given *-algebra $A$. As a vector space $A' = A \oplus A$, and the ...

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### Left- (right-) multiplications of an algebra that are derivations

Let us say that $A$ is a (finite-dimensional) algebra over a field of characteristic zero. We can assume commutativity
but not associativity, if that makes it easier. Indeed, I am mostly interested in ...

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### How many Lie and associative algebras over a finite field are there?

This question is related to the following general question:
Given a variety of (non-associative) algebras $\mathcal V$, a finite field $\mathbb{F}_q$, with $q$ elements, and a positive integer $n$, ...

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### Gelfand-Kirillov dimension for non-associative algebras

Let $A$ be any finitely generated algebra - non necessarely unital neither associative - over a base field $k$. Let us denote the product $*$. Suppose $A$ is finitely generated by $S$, and introduce $...

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### Reference request: associative subalgebras of Cayley algebras are at most 4-dimensional

By a Cayley algebra I mean an 8-dimensional algebra (over an arbitrary field) formed in the Cayley-Dickson process. (They are also called octonion algebras, but I prefer to reserve the term octonion ...

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### Is the average associator over a finite subloop of octonions necessarily zero?

For any three octonions $a,b,c$, their associator is defined as \begin{equation*} [a,b,c]=a(bc)-(ab)c \end{equation*} and measures their non-associativity so to speak.
Now suppose that $L$ is a finite ...

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### Principal ideal of a non-associative magma

The definitions of an ideal of an algebraic structure $A$ (as a substructure $I$ such that the product of $A$ and $I$ is a subset of $I$) do not involve associativity.
However, the definitions of a ...

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### Is there a unique way to define an Euclidean Jordan Algebra as the product of two simple Euclidean Jordan Algebras?

Let $A$ and $B$ be two simple Euclidean Jordan Algebras. It is well known that $A\times B$ can be made into a Jordan Algebra in a unique way by defining the operation component-wise. Is it true that ...

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### Good reference on the algebraic geometry of non-associative rings

I am looking for a good reference about the algebraic geometry of non-associative rings. I am in particular interested in derivation algebras.
Preferrably an online resource or a book that is ...

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### "Non-associative" standard polynomials

I saw somewhere (I appreciate if anyone has any references to proof of this fact) that if $A$ is a finite dimensional associative algebra such that $\textrm{dim}(A)<n$, then $A$ satisfies the ...

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### Non-associative deformation quantization

Several physicists consider non-Poisson bivectors but still apply Kontsevich formality in order to get deformation quantization type results: see e.g. Szabos's review An introduction to nonassociative ...

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### Is there a way to adjoin a counit to a non counital coalgebra?

Let $k$ be a field.
If $A$ is a non unital $k$-algebra, there is a simple way to make it unital by taking $\tilde{A}:=A\oplus k$ and setting
$$ (a+\lambda)(b+\mu):=ab+\lambda b +\mu a +\lambda \mu$$
...

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### Determinants in Jordan algebras of Euclidean type

As far as I heard (I am not sure about the precise statement) there is a classification of simple Jordan algebras of Euclidean type. Each (some?) of such algebras admits a cone of positive definite ...

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### Weak associativity

Let $(V,*)$ be an algebra and denote $A_*\in \text{Hom}(V^{\otimes 3},V)$ the associator of the binary product $*\in \text{Hom}(V^{\otimes 2},V)$ defined as $A_*(a,b,c):=(a*b)*c-a*(b*c)$.
The ...

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### Have you ever seen this bizarre commutative algebra?

I have encountered very strange commutative nonassociative algebras without unit, over a characteristic zero field, and I cannot figure out where do they belong. Has anybody seen these animals in any ...

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### What's the maximum probability of associativity for triples in a nonassociative loop?

In a finite nonabelian group, the probability that two randomly chosen elements commute cannot exceed 5/8. One easy proof also makes it easy to find the smallest groups that attain this bound, namely ...

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### Non-associative module theory [closed]

I'm looking for a reference that treats basic module theory over non-associative rings, the isomorphism theorems and so on. I imagine the theory is known, but have not been able to find a reference.
...

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### Language representation problem regarding non-commutative, non-associative algebras

Consider a sentence as a series of words with an associated set of labels that tell one how information is passed through the sentence - examples include combinatory categorical grammars or Lambek ...

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### Multiplication on cubic hypersurfaces and partially defined groups

Let $V$ be a general smooth projective cubic hypersurface. Doing literally as in case of cubic curves we define a relation on $V\times V\times V$: $(x,y,z)$ satisfy it iff $x+y+z$ is an intersection ...

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### Is there a notion of octonionic structure on a Lie algebra? In the same way as there is one for complex and quaternionic

Is there a notion of octonionic structure on a Lie group? In the same way as there is one for complex and quaternionic

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### Determinants of octonionic hermitian matrices

For quaternionic hermitian matrices (i.e. quaternionic square matrices $(a_{ij})$ satisfying
$a_{ji}=\bar a_{ij}$) there is a nice notion of (Moore) determinant which can be defined as follows.
...

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### Does the Cayley-Dickson construction preserve isomorphism of quaternion algebras?

I posted this on math.stackexchange to no avail, so I hope it's appropriate to post here despite that it might not be research-level. I expect the answer to this is well-known to people studying non-...

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### Applications of Jordan algebras

Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product $A\...

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### Is the generated subalgebra of a subset of pairwise operator-commuting element in a JB-algebra associative?

In a Jordan algebra elements $a$ and $b$ are said to operator-commute, whenever $a \circ (b \circ x) = b \circ (a \circ x)$ for every other element $x$. (That is: $T_aT_b = T_bT_a$, writing $T_x(y) = ...

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### A question about index of the commutant in a Moufang loop

Let $M$ be a non-commutative Moufang loop and $C(M)$ be its commutant. I can prove that the index of $C(M)$ in $M$, $|M:C(M)|$, is greater than or equal to 4. Also, I can show that $|M:Z(M)|\geq 4$, ...

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### Is there a system of quasigroup equations implying non-associativity?

I have read that if 4 quasigroup operations, $\cdot,\circ,\star,\square$, on a set $S$ respect the following equation:
$$x\cdot (y\circ z) = (x \star y) \square z$$
for all $x,y,z\in S$, then all 4 ...