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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 ...
Taras Banakh's user avatar
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4 votes
1 answer
657 views

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 ...
Taras Banakh's user avatar
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5 votes
1 answer
105 views

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 ...
Taras Banakh's user avatar
  • 41.1k
0 votes
0 answers
41 views

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) ...
Dieter Kadelka's user avatar
3 votes
1 answer
244 views

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 $R$ ...
jg1896's user avatar
  • 3,104
9 votes
2 answers
376 views

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 ...
Taras Banakh's user avatar
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2 votes
0 answers
21 views

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 ...
Saikat's user avatar
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6 votes
1 answer
382 views

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. ...
Taras Banakh's user avatar
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3 votes
2 answers
326 views

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 ...
John Wayland Bales's user avatar
3 votes
0 answers
101 views

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. ...
Bugs Bunny's user avatar
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2 votes
0 answers
96 views

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 ...
José Victor Gomes's user avatar
10 votes
2 answers
211 views

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 ...
Vladimir Dotsenko's user avatar
0 votes
0 answers
161 views

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, ...
mick's user avatar
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6 votes
0 answers
66 views

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 $E\subseteq X^2\times X^2$ ...
Taras Banakh's user avatar
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3 votes
0 answers
87 views

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 ...
Dac0's user avatar
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8 votes
0 answers
110 views

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 ...
Vladimir Dotsenko's user avatar
12 votes
0 answers
222 views

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 ...
Harry Altman's user avatar
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2 votes
0 answers
138 views

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 ...
Julian Seipel's user avatar
3 votes
0 answers
91 views

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 ...
a196884's user avatar
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1 vote
0 answers
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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 ...
saolof's user avatar
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3 votes
0 answers
131 views

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 ...
Mozibur Ullah's user avatar
1 vote
0 answers
163 views

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, ...
mick's user avatar
  • 763
16 votes
1 answer
940 views

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 ...
p6majo's user avatar
  • 355
1 vote
0 answers
49 views

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 ...
a196884's user avatar
  • 323
3 votes
0 answers
71 views

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 ...
Jakob's user avatar
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4 votes
1 answer
249 views

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 ...
Claudio Gorodski's user avatar
10 votes
1 answer
796 views

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 $n$, ...
Thiago's user avatar
  • 398
2 votes
0 answers
57 views

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 $...
jg1896's user avatar
  • 3,104
5 votes
0 answers
76 views

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 ...
Vincent's user avatar
  • 2,467
4 votes
0 answers
122 views

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 ...
Bob's user avatar
  • 439
2 votes
1 answer
178 views

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 ...
Alex C's user avatar
  • 133
3 votes
0 answers
36 views

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 ...
Shake Baby's user avatar
  • 1,638
2 votes
2 answers
591 views

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 ...
The Thin Whistler's user avatar
5 votes
1 answer
340 views

"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 ...
cl4y70n____'s user avatar
7 votes
2 answers
377 views

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 ...
Jim Stasheff's user avatar
  • 3,860
3 votes
1 answer
127 views

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 $$ (a+\lambda)(b+\mu):=ab+\lambda b +\mu a +\lambda \mu$$ ...
Operadbeginner's user avatar
1 vote
2 answers
180 views

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 ...
asv's user avatar
  • 21.3k
6 votes
1 answer
210 views

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 ...
thingsthatmighthavebeen's user avatar
32 votes
1 answer
3k views

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 ...
მამუკა ჯიბლაძე's user avatar
24 votes
2 answers
699 views

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 ...
John Baez's user avatar
  • 21.8k
-2 votes
1 answer
532 views

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. ...
Per's user avatar
  • 1
3 votes
0 answers
134 views

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 ...
East's user avatar
  • 149
1 vote
1 answer
115 views

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 ...
Dmitrii Korshunov's user avatar
5 votes
0 answers
141 views

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
Rony Kaplan's user avatar
11 votes
1 answer
726 views

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. ...
asv's user avatar
  • 21.3k
6 votes
1 answer
226 views

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-...
j0equ1nn's user avatar
  • 2,468
15 votes
4 answers
2k views

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\...
asv's user avatar
  • 21.3k
5 votes
1 answer
149 views

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) = ...
westerbaan's user avatar
0 votes
0 answers
124 views

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 $|M:Z(M)|\geq 4$, ...
Ahmadi's user avatar
  • 123
2 votes
0 answers
169 views

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 ...
A.B.'s user avatar
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