# Tagged Questions

Questions about non-associative algebras other than Lie algebras.

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### 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 ...
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Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product $A\... 1answer 245 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 ... 0answers 136 views ### 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. ... 2answers 269 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)? 1answer 217 views ### 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 ... 1answer 60 views ### 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-... 0answers 187 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 ... 2answers 134 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? 0answers 76 views ### Is the generated subalgebra of a subset of pairwise operator-commuting element in a JB-algebra associative? In a Jordan algebra elements$a$and$b$are said to operator-commute, whenever$a \circ (b \circ x) = b \circ (a \circ x)$for every other element$x$. (That is:$T_aT_b = T_bT_a$, writing$T_x(y) = ...
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 ...
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 $|M:Z(M)|\geq 4$, ...