Questions tagged [nonassociative-algebras]
70 questions
18
votes
1 answer
Can octonions be represented by infinite matrices?
It is sometimes possible to multiply matrices of countably-infinite dimension. (Matrix multiplication is defined in the usual way, with rows and columns multiplied termwise and summed.) However, it turns out the associative property fails in general…
anon
- 155,259
7
votes
1 answer
Does Distributivity Imply Power Associativity?
Say we have an algebra $(A, +, \cdot)$, where $(A, +)$ is an Abelian Group. All we know about $\cdot$ is that it is both left and right distributive over addition. So, $\forall a,b,c \in A, a \cdot (b+c) = (a \cdot b) + (a \cdot c)$ and $(b+c) \cdot…
RothX
- 1,731
- 14
- 21
6
votes
1 answer
Are inverse unique in unital algebra?
Let $A$ be a unital algebra over a field $F$ with unity $1$. If $a,b,c\in A$ such that $ab=ba=1=ac=ca$, does this imply that $b=c$?
We have $c(ab)=c$. However, algebras are not necessarily associative so we can't conclude $c=c(ab)=(ca)b=b$.
Are…
user815463
5
votes
1 answer
Do the Moufang identities *themselves* imply diassociativity / Moufang's theorem / Artin's theorem?
A Moufang loop is a loop satisfying the Moufang identities. Famously, these are diassociative -- the subloop generated by any two elements is associative (is a group) -- and more generally, they satisfy Moufang's theorem, that if any three elements…
Harry Altman
- 4,993
5
votes
1 answer
Zorn vector-matrix description of octonion multiplication
Zorn's vector-matrices are a way to describe split octonions by treating them as matrices
$$ \begin{bmatrix}a & \mathbf v\\ \mathbf w & b\end{bmatrix} $$
where $a,b \in \mathbb{R}$ and $\mathbf{v}, \mathbf{w} \in \mathbb{R}^3$. The multiplication…
John C. Baez
- 1,902
- 15
- 31
5
votes
1 answer
Classification of subalgebras of composition algebras
Let $F$ be an algebraically closed field. It is known that the only composition algebras over $F$ are $F$ itself, the direct sum $F\oplus F$ (also called split-complexes), the algebra of $2\times 2$ matrices (or split-quaternions) $M_2(F)$, and…
pregunton
- 6,358
4
votes
1 answer
Is this algebra simple?
I have this algebra $A$, defined on the elements of the basis $ \left\{ L_{k}\right\} _{k\in\mathbb{Z}} $ with a bilinear product such that
$$L_{a}\circ L_{b}=\frac{1}{2}\left(L_{\phi(a,b)+2a-b}+L_{-\phi(a,b)+2b-a}\right),$$
where…
Dac0
- 9,504
4
votes
1 answer
Easy examples of right alternative rings that are not alternative
By a ring I mean a $\mathbb{Z}$-module $A$ equipped with a binary operation $*:A\times A\rightarrow A$ that is $\mathbb{Z}$-bilinear and denoted by $(x,y)\mapsto xy$.
Let $A$ be a ring.
$A$ is said to be right alternative if $(xy)y=x(yy)$ for any…
Daniel Kawai
- 1,027
- 6
- 19
4
votes
2 answers
How many ways to write a commutative non-associative product of $n$ terms?
The Catalan numbers give the number of ways to write a non-commutative non-associative product of $n$ terms, as $C_{n-1}\cdot n!=\frac{(2n-2)!}{(n-1)!}$. For example, there are $C_{3-1}\cdot3!=12$ ways to write a product of $3$…
mr_e_man
- 5,986
3
votes
1 answer
About 2 operators and $A*(B+C) =(A+B)*(A+C)$
Consider $2$ binary operators defined for a finite set with $n$ elements.
Operator $*$ behaves like a commutative latin quandle :
$$x*x = x$$
$$a*b=b*a$$
$$a*(b*c)=(a*b)*(a*c)$$
And forms a latin square.
Commutative operator $+$ behaves…
mick
- 17,886
3
votes
1 answer
Examples of Non-Associative Algebras with (Non-Associative) Division and with Unity without Classical Division
Let $\mathbb{K}$ be a field. By an $\mathbb{K}$-algebra I mean a $\mathbb{K}$-module $A$ equipped with a binary operation $∗:A\times A\rightarrow A$ that is $\mathbb{K}$-bilinear and denoted by $(x,y)\mapsto xy$.
Let $A$ be an algebra.
$A$ is said…
Daniel Kawai
- 1,027
- 6
- 19
3
votes
1 answer
How many triples of mutually conjugate linear maps are there?
I’m looking for finite-dimensional complex vector spaces $V$ with three invertible linear maps $r,s,t:V\to V$, satisfying the following “mutual-conjugation” condition:
$rsr^{-1}=t$
$sts^{-1}=r$
$trt^{-1}=s$
For lack of a better term, I’m calling…
Alvaro Martinez
- 1,967
- 8
- 22
3
votes
0 answers
Investigating and Generalizing Octonionic Nonassociativity
A possible multiplication convention for the octonions is given by the following 7 sets of integers from 1 to 7.
{1,2,3},{1,4,5},{1,6,7},{2,4,6},{2,5,7},{3,4,7},{3,5,6}.
Notice that each of the seven sets of integers above has exactly 3 elements,…
Michael Riberdy
- 127
3
votes
0 answers
Nonassociative algebra's 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, \dots, a_n) $
for a basis
$\{1, i_1, \dots, i_n \}$
and we…
mick
- 17,886
3
votes
1 answer
Does this three-dimensional commutative non-associative algebra satisfy any identities?
The algebra has this multiplication table:
$$\begin{array}{c|ccc}\odot&E_1&E_2&E_3\\\hline E_1&0&E_3&-E_2\\E_2&E_3&0&E_1\\E_3&-E_2&E_1&0\end{array}$$
Equivalently, any vector $aE_1+bE_2+cE_3=\begin{bmatrix}a\\b\\c\end{bmatrix}$ acts on other vectors…
mr_e_man
- 5,986