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 ...
- 2,139
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 ...
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27
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 ...
- 108k
16
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
$$q^{\frac{2}{27} n^3 + O(n^{8/3})...
- 118k
9
votes
Accepted
Who and when proved Artin's Theorem on alternative rings?
Artin published [1] on the related Wedderburn theorem, but Zorn does not cite a publication on the theorem he attributes to Artin in his 1930 paper [2]. Moufang [3] also cites Zorn in her 1935 ...
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9
votes
Accepted
Left- (right-) multiplications of an algebra that are derivations
An algebra whose (left) multiplications are derivations is referred to as a (left) Leibniz algebra (or Loday algebra). There is a large literature about this class of non-associative algebras. See e....
- 5,878
6
votes
Accepted
Chirality of octonion algebras
Perhaps this is more a question about the Fano plane than about the (real) octonions. Notice that the automorphism group of the Fano plane is the simple group $\operatorname{GL}(3, \mathbb{F}_2) \cong ...
- 6,614
5
votes
What's the maximum probability of associativity for triples in a nonassociative loop?
This is a bit too long for a comment, so it's an answer. The Loops package for Gap, by Gabor Nagy and Petr Vojtechovsky contains implementations of all the nonassociative Moufang loops of order $\leq ...
- 3,768
4
votes
Accepted
Non-associative module theory
A Google search of this term brings a number of references, did you try it?
In any case, a very obvious relevant reference is the old paper of Osborn called Modules over nonassociative rings.
More ...
- 16.9k
3
votes
Determinants in Jordan algebras of Euclidean type
We denote by $J_{3}(\mathbb{O})$ be the space:
$$ J_{3}(\mathbb{O}) = \left\{ \begin{pmatrix} \lambda_1 & a_1 & \overline{a_2} \\ \overline{a_1} & \lambda_2 & a_3 \\ a_2 & \...
- 7,300
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 ...
- 8,385
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:
$$ \Delta'(c + x)= \Delta(c) + c \otimes 1 + 1 \...
- 42.4k
2
votes
Accepted
Is the generated subalgebra of a subset of pairwise operator-commuting element in a JB-algebra associative?
Yes, this is true. I couldn't find any proof of the statement you quote in the article, and even after emailing the authors I didn't get any wiser, so I decided to work out the details myself, see my ...
- 136
2
votes
The octonions on a bad day
@Marty has already answered this question; this answer is to provide additional context. Marty’s answer addresses the case where $k$ is a field of characteristic different from 2. But the results ...
- 2,178
2
votes
Accepted
Multiplication on cubic hypersurfaces and partially defined groups
Here is an explicit example over the rationals.
Consider the diagonal Clebsch cubic surface given by $\sum_{i=0}^4 X_i = 0$ and $\sum_{i=0}^4 X_i^3 = 0$. Let me take the point $u := (0:0:0:1:-1)$ so ...
- 32.5k
2
votes
Determinants in Jordan algebras of Euclidean type
Makt wrote:
As far as I heard (I am not sure about the precise statement) there is a classification of simple Jordan algebras of Euclidean type.
Yes, in this paper
Pascual Jordan, John von Neumann ...
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2
votes
Accepted
Weak associativity
Let me assume that the characteristic of the ground field is different from two. Let me start by replacing your identity by something where the existing symmetries are a bit more apparent. I claim ...
- 16.9k
2
votes
Accepted
Principal ideal of a non-associative magma
In a magma $M$, one can describe the 2-sided ideal generated by a subset $Y$ as follows: define by induction
$$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\...
- 63.9k
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. ...
- 3,318
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 ...
- 501
2
votes
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 ...
- 16.2k
2
votes
Accepted
An isomorphic classification of non-associative division octonion algebras
Yes, the number of isomorphism classes can be greater than 1 and it can be infinite.
One good reference on octonions is the book [1] by Springer-Veldkamp. It gives the following example on page 22:
...
- 2,178
2
votes
Accepted
Wedderburn–Artin like theorem for infinite dimensional Lie algebras?
I had the opportunity to chat with an expert in nonassociative algebras, and I report here what he told me.
The short answer is: as yet, there is no Wedderburn–Artin theory for Lie algebras. The ...
- 3,318
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:
- 41.8k
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 ...
- 16.9k
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): ...
- 16.9k
1
vote
Determinants of octonionic hermitian matrices
May I point out the paper of J. Liao, J. Wang, and X. Li, "The all-associativity of octonions and its applications",
Anal. Theory Appl. Vol. 26, No. 4 (2010), 326-338. The abstract, in part, reads "a ...
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