Questions about non-associative algebras other than Lie algebras.

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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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0answers
111 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 ...
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2answers
235 views

Is there a cohomology for magmas?

Is there a cohomology theory for magmas? Or cohomology theories for any class of non-associative algebras (other than Lie and maybe Jordan)?
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Homotopes of simple Lie algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra with bracket $[x,y]$. For which $z\in \mathfrak{g}$ defines $$ \mu(x,y)=ad (z)([x,y])=[z,[x,y]] $$ another Lie bracket on the same vector space ? For ...
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153 views

Reference request: The relationship between norm and trace forms on an Albert algebra

I am interested in either a nice reference, or some clarification. Overview: I am considering $J_3(\mathbb{O})$, the Jordan algebra of $3\times 3$ self adjoint octonionic matrices. This algebra is a ...
11
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1answer
217 views

What is flexible about flexible algebras?

A possibly non-associative algebra is flexible if it satisfies the identity $$(xy)x=x(yx).$$ This is clearly a very weak form of associativity —and obviously an associative algebra is flexible— but it ...
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2answers
118 views

Any results or concise introduction about nonassociative algebra that even does not satisify Power associativity?

Any results or concise introduction about nonassociative algebra that even does not satisify Power associativity?
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1answer
706 views

The octonions on a bad day

We can define the algebra of quaternions $\mathbb H$ over any field $k$, and depending on the arithmetic of $k$ it is either a division algebra or a matrix algebra. We can also define the algebra of ...