Questions about non-associative algebras other than Lie algebras.

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### 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 ...

**7**

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**2**answers

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### 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 ...

**5**

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### 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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**1**answer

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### 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 ...

**2**

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**2**answers

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### 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?

**13**

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**1**answer

617 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 ...