47 votes
Accepted

Have you ever seen this bizarre commutative algebra?

This algebra is defined on the permutation module of the symmetric group. It was studied by K. Harada and R. Griess in the 1970s and a proof that its automorphism group is the symmetric group can ...
Dan Fox's user avatar
  • 2,140
28 votes

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

I found the following example due to J. Jezek and T. Kepka from "Notes on the number of associative triples" Acta Universitatis Carolinae 31 (1990), 15-19 (Example 2.1): Suppose $Q(+)$ is ...
Gjergji Zaimi's user avatar
26 votes
Accepted

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

The quaternions are generated by any two imaginary elements $x$ and $y$ that are orthonormal, i.e., $\bigl(1,\, x,\, y,\, xy\bigr)$ is an orthonormal basis of the quaternions. Moreover, the ...
Robert Bryant's user avatar
15 votes

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

Bjorn Poonen addresses this question for commutative (associative, unital) algebras in The moduli space of commutative algebras of finite rank; asymptotically we have $$q^{\frac{2}{27} n^3 + O(n^{8/3})...
Qiaochu Yuan's user avatar
13 votes

Applications of Jordan algebras

I would like to elaborate on the link to "associative" problems (such as the Zelmanov's solution of the restricted Burnside problems mentioned by Tom De Medts) - mainly because by saying that Jordan ...
Vladimir Dotsenko's user avatar
12 votes

Applications of Jordan algebras

They turn up quite often in the study of (exceptional) linear algebraic groups. The most famous instance of this is the fact that algebraic groups of type $F_4$ are precisely the automorphism groups ...
Tom De Medts's user avatar
  • 6,494
9 votes
Accepted

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

An algebra whose (left) multiplications are derivations is referred to as a (left) Leibniz algebra (or Loday algebra). There is a large literature about this class of non-associative algebras. See e....
Salvatore Siciliano's user avatar
7 votes

Applications of Jordan algebras

Jordan algebras were originally introduced by Pascual Jordan in a hope to generalize the orthodox formulation of quantum mechanics, but this program was not successful as far as the generalization of ...
Zurab Silagadze's user avatar
6 votes
Accepted

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

Suppose $K$ is a field, and $B$ your quaternion algebra $K$. The octonion algebra $C$ made from $B$ using the Cayley-Dickson construction depends on an auxiliary choice of an element $c \in K^{\times}...
user25514's user avatar
  • 203
5 votes
Accepted

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

Every liner is an involutory Latin quandle, and we have a characterization of involutory Latin quandles, so we may simply restrict this characterization to liners. A quandle is an algebra $(X,*,*^{-1})...
Joseph Van Name's user avatar
5 votes
Accepted

Chirality of octonion algebras

Perhaps this is more a question about the Fano plane than about the (real) octonions. Notice that the automorphism group of the Fano plane is the simple group $\operatorname{GL}(3, \mathbb{F}_2) \cong ...
Tom De Medts's user avatar
  • 6,494
5 votes

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

This is a bit too long for a comment, so it's an answer. The Loops package for Gap, by Gabor Nagy and Petr Vojtechovsky contains implementations of all the nonassociative Moufang loops of order $\leq ...
Robert Furber's user avatar
4 votes
Accepted

Non-associative module theory

A Google search of this term brings a number of references, did you try it? In any case, a very obvious relevant reference is the old paper of Osborn called Modules over nonassociative rings. More ...
Vladimir Dotsenko's user avatar
3 votes

Applications of Jordan algebras

Jordan algebras and more generally Jordan pairs and triple systems have applications in the theory of Riemannian symmetric spaces. Some relevant links: https://en.wikipedia.org/wiki/...
Vít Tuček's user avatar
  • 8,159
3 votes

Determinants in Jordan algebras of Euclidean type

We denote by $J_{3}(\mathbb{O})$ be the space: $$ J_{3}(\mathbb{O}) = \left\{ \begin{pmatrix} \lambda_1 & a_1 & \overline{a_2} \\ \overline{a_1} & \lambda_2 & a_3 \\ a_2 & \...
Libli's user avatar
  • 7,210
3 votes

Non-associative deformation quantization

This is probably not really an answer to this question, but there are two different context I know where deformation quantization produces something not exactly associative, but associative in a ...
DamienC's user avatar
  • 8,093
3 votes

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

Yes, it worked pretty much in the exact dual way: If $(C, \Delta)$ is a nonunital coalgebra, then $C \oplus k$ has a co-algebra structure given by: $$ \Delta'(c + x)= \Delta(c) + c \otimes 1 + 1 \...
Simon Henry's user avatar
  • 39.9k
2 votes

Determinants in Jordan algebras of Euclidean type

Makt wrote: As far as I heard (I am not sure about the precise statement) there is a classification of simple Jordan algebras of Euclidean type. Yes, in this paper Pascual Jordan, John von Neumann ...
John Baez's user avatar
  • 21.3k
2 votes
Accepted

Weak associativity

Let me assume that the characteristic of the ground field is different from two. Let me start by replacing your identity by something where the existing symmetries are a bit more apparent. I claim ...
Vladimir Dotsenko's user avatar
2 votes
Accepted

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

Yes, this is true. I couldn't find any proof of the statement you quote in the article, and even after emailing the authors I didn't get any wiser, so I decided to work out the details myself, see my ...
John's user avatar
  • 136
2 votes
Accepted

Multiplication on cubic hypersurfaces and partially defined groups

Here is an explicit example over the rationals. Consider the diagonal Clebsch cubic surface given by $\sum_{i=0}^4 X_i = 0$ and $\sum_{i=0}^4 X_i^3 = 0$. Let me take the point $u := (0:0:0:1:-1)$ so ...
Gro-Tsen's user avatar
  • 29.8k
2 votes
Accepted

Principal ideal of a non-associative magma

In a magma $M$, one can describe the 2-sided ideal generated by a subset $Y$ as follows: define by induction $$M_1=M,\;Y_1=Y,\; M_n=\bigcup_{p,q\ge 1,p+q=n}M_pM_q,\;Y_n=\bigcup_{p,q\ge 1,p+q=n}(M_pY_q\...
YCor's user avatar
  • 60.1k
2 votes

Good reference on the algebraic geometry of non-associative rings

As mentioned in the answer by user6976, there is the idea of development of algebraic geometry to (essentialy) any general algebraic system. This is carried out(following Plotkin's work) by E. ...
jg1896's user avatar
  • 2,683
2 votes
Accepted

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

There is German book Plane geometry (2007) which contains equivalent adapted Hessenberg Theorem for affine planes. It's too long to translate and adapt proof to your terminology, so here are ...
Ihromant's user avatar
  • 471
1 vote

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

It seems that the proof from the book posted by @ihromant follows the lines of the original Hessenberg's proof from his paper in Mathematische Annalen of 1905:
Taras Banakh's user avatar
  • 40.7k
1 vote

Chirality of octonion algebras

Negating the seven imaginary basis vectors of the octonions is equivalent to reversing every arrow in the oriented Fano plane. This operation swaps the two orientations, and displays the isomorphism ...
Dave Benson's user avatar
  • 11.6k
1 vote

Degree 8 multilinear operations on Jordan algebras

I managed to run Albert on a very powerful computer at work, and the computation of the desired dimension converged: it seems equal to 19089. I would very much like to confirm that it is correct (I am ...
Vladimir Dotsenko's user avatar
1 vote

Non-associative deformation quantization

I figured out that in full generality this problem has no chance of leading to a different algebraic structure for which the given one is a quasi-classical limit (like it is for associative/Poisson): ...
Vladimir Dotsenko's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible