# Tagged Questions

Questions about non-associative algebras other than Lie algebras.

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### Determinants of octonionic hermitian matrices

For quaternionic hermitian matrices (i.e. quaternionic square matrices $(a_{ij})$ satisfying $a_{ji}=\bar a_{ij}$) there is a nice notion of (Moore) determinant which can be defined as follows. ...
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### Does the Cayley-Dickson construction preserve isomorphism of quaternion algebras?

I posted this on math.stackexchange to no avail, so I hope it's appropriate to post here despite that it might not be research-level. I expect the answer to this is well-known to people studying non-...
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### Homotopes of simple Lie algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra with bracket $[x,y]$. For which $z\in \mathfrak{g}$ does the formula $$\mu(x,y)=ad (z)([x,y])=[z,[x,y]]$$ define another Lie bracket on the same ...
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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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### 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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### 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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### 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 ...
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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 ...
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 ...