Questions tagged [nonassociative-algebras]

The tag has no usage guidance.

67 questions
Filter by
Sorted by
Tagged with
1 vote
43 views

Can we construct a free structure on a non associative algebraic structure.

For any set we can construct a free group on it. Also for non associative structures like Lie algebra, Lie ring we may construct free structures, but these are non associative structures and having ...
• 1,958
34 views

Non associativity of geometric algebra??

I can't resolve this bizarreness. First consider this: $$\tilde{R}\gamma_\mu R = e_\mu$$ where $\gamma_\mu$ is a gamma matrix, R is a rotor (exponential of a bivector) and $e_\mu$ is a rotated basis ...
• 2,589
73 views

• 9,254
294 views

• 1,913
97 views

Are there aspects of non-associative algebras which cannot be captured using Category Theory?

Category theory uses morphisms, which are associative by definition. This means that non-associative operations cannot be trivially mapped into morphisms. One must find clever ways to deal with this....
• 3,397
55 views

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 ...
134 views

Alternative algebra with division is a division ring

In the book Rings that are Nearly Associative, has the follow definitions: $1)$ An algebra $A$ is called an $\textit{algebra with division}$ if, for any elements $a, b \in A$, with $a \neq 0$, each of ...
• 104
150 views

• 16.4k
189 views

What is the definition of derivation of tensor algebra?

Derivation $d$ of a Lie algebra $L$ is defined as $d([x,y])=[d(x),y]+[x,d(y)]$ for all $x,y \in L$. Let consider $T(L)$ be the tensor algebra and $S(L)$ be the symmetric algebra of $L$. How a ...
• 1,316
1 vote
41 views

Looking for Nonassociative Algebra notes

I'm reading Shafer's book on nonassociative algebras, but it contains no exercises. Could you share with me some other references, especially course's notes?
• 952
169 views

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 ...
253 views

Is this proof about Jordan algebra's correct?

Let $A$ be a unital commutative non-associative alternative algebra. Let $J$ be a unital Jordan algebra. Notice Jordan algebra's are commutative and non-associative by definition. Conjecture : The set ...
• 16.4k
1 vote
21 views

Coordinatization Theorem Exemple

I am studying Coordinatization Theorems, in Jacobson's book. But it is so hard to understand, I would like an easy exemple of direct application of the Coordinatization Theorem, with an easy algebra, ...
• 104
1 vote
32 views

Universal multiplicative envelope $\mathfrak{U}(A) = \dfrac{T(A \oplus A_1)}{\mathfrak{R}}$

Let $\Phi[x, y]$ be the polynomial algebra over the field $\Phi$. Then $\Phi [x, y]$ is an alternative algebra too. Let $A = \Phi [x,y]$ and $A_1 = \Phi[w, z]$, where $A_1$ is a vector space ...
• 104
485 views

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 ...
• 153k