Questions tagged [nonassociative-algebras]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
0
votes
0answers
24 views

When does dim(Aut) = dim(Der) in non-associative algebra? [closed]

When does dim(Aut) = dim(Der) in non-associative algebra? I read this is true for Lie algebras, but is this true in general? Where can I find a proof?
1
vote
0answers
28 views

"Algebraic dimension" for finite-dimensional (non-associative) algebras?

Let $V$ be some finite-dimensional vector space (over some field $\mathbb{K}$), then a (possibly non-associative) algebra $A$ on $V$ corresponds to a bilinear map $V \times V \to V$. I prefer answers ...
4
votes
1answer
78 views

Do the Moufang identities *themselves* imply diassociativity / Moufang's theorem / Artin's theorem?

A Moufang loop is a loop satisfying the Moufang identities. Famously, these are diassociative -- the subloop generated by any two elements is associative (is a group) -- and more generally, they ...
3
votes
1answer
41 views

How many triples of mutually conjugate linear maps are there?

I’m looking for finite-dimensional complex vector spaces $V$ with three invertible linear maps $r,s,t:V\to V$, satisfying the following “mutual-conjugation” condition: $rsr^{-1}=t$ $sts^{-1}=r$ $trt^{...
0
votes
0answers
41 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
votes
0answers
41 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 ...
0
votes
0answers
38 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 ...
0
votes
0answers
25 views

$C(f)$ is the associative enveloping algebra of $B(f)$

Let $B(f) = F \cdot 1 + M$, where $f:M \times M \to F$ is a symmetric bilinear form. The multiplication in $B(f)$ is $$(\alpha \cdot 1 + x) (\beta \cdot 1 + y)= (\alpha \beta + f(x,y)) \cdot 1 + (\...
3
votes
0answers
141 views

Nonassociative algebra's closed under $\sqrt{}$?

Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a Cayley table such that elements are generated with real number coefficients $(a_0, \dots, ...
0
votes
0answers
44 views

Power associative basis implies not nilpotent?

Consider a non-associative commutative unital algebra of finite dimension where the product is defined by a cayley table such that elements are generated with real number coefficients $(a_0, \dots, ...
0
votes
0answers
31 views

If $A$ is a composition algebra, is $H(A_n)$ a simple algebra?

Question I was trying to prove that if $C$ is a Cayley-Dickson algebra over a field of characteristic $\neq 2$, then $H(C_3)$ is a simple algebra (I already know it is Jordan although $C$ is not ...
0
votes
1answer
58 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
vote
0answers
35 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?
6
votes
1answer
96 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 ...
2
votes
1answer
107 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 ...
1
vote
0answers
16 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, ...
1
vote
0answers
27 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 ...
18
votes
1answer
335 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 ...
4
votes
1answer
144 views

Zorn vector-matrix description of octonion multiplication

Zorn's vector-matrices are a way to describe split octonions by treating them as matrices $$ \begin{bmatrix}a & \mathbf v\\ \mathbf w & b\end{bmatrix} $$ where $a,b \in \mathbb{R}$ and $\...
2
votes
2answers
105 views

Term for an additive abelian group equipped with a multiplication which is distributive over the addition, but not necessarily associative?

What is the official name of an additive abelian group with a biadditive multiplication (left and right distributivity of multiplication over addition and no other assumptions)?
4
votes
2answers
107 views

How many ways to write a commutative non-associative product of $n$ terms?

The Catalan numbers give the number of ways to write a non-commutative non-associative product of $n$ terms, as $C_{n-1}\cdot n!=\frac{(2n-2)!}{(n-1)!}$. For example, there are $C_{3-1}\cdot3!=12$ ...
3
votes
1answer
95 views

Does this three-dimensional commutative non-associative algebra satisfy any identities?

The algebra has this multiplication table: $$\begin{array}{c|ccc}\odot&E_1&E_2&E_3\\\hline E_1&0&E_3&-E_2\\E_2&E_3&0&E_1\\E_3&-E_2&E_1&0\end{array}$$ ...
1
vote
0answers
21 views

Polynomial alternative algebra

Let $F[x]$ be an polynomial alternative algebra over a field $F$. How I compute the universal multiplication envelope of $F[x]$?
1
vote
0answers
65 views

Associativity of infinite matrix product.

Many texts reads "It is well known that for infinite matrices multiplication is non-associative". A treatise on this can be found in On the associativity of infinite matrix multiplication. However, ...
2
votes
1answer
51 views

Principal ideal of a non-associative magma

The definitions of a left, right, and two-sided ideal of an algebra do not involve associativity (R.D. Schafer "An Introduction To Nonassociative Algebras"). The same we can say about the definitions ...
0
votes
0answers
50 views

Universal multiplication enveloping algebra

I don’t understood what is a universal multiplication enveloping algebra, I studied in Jacobson book, but the concept is very formal and difficult to understand. I wound like exemples of universal ...
2
votes
0answers
61 views

General formula for permutation of modes in Virasoro algebra/ Explicit formula for general OPE structure constants

Let $Vir_c$ denote the state space of the Virasoro VOA with central charge $c$. Let $n$ be a positive integer(i.e. $\geq 0$) and let $(n_k,...,n_2)$ be a sequence of positive integers. I am interested ...
1
vote
1answer
65 views

Proving inverse image of ideal is an ideal

Let $L_1,L_2$ two Lie algebras. For a Lie morphism $\varphi : L_1 \to L_2$, let $I\subset L_2$ an ideal. How we prove $$\varphi^{-1}(I) \subset L_1$$ is an ideal? My try: $$[\varphi^{−1}(I),L_1]=[\...
0
votes
0answers
27 views

Several doubts and questions related with definitions on Leibniz algebras

Here are my questions for a (left) Leibniz ${\mathbb{F}}$-algebra $L$ and $M$ a Leibniz module: In $L$ when you define the Leibniz Kernel as $\mathfrak{I}≔𝑠𝑝𝑎𝑛_{\mathbb{F}}\{[𝑥,𝑥] :𝑥∈L\}$, ...
0
votes
0answers
27 views

About a characterization of solvability for Leibniz algebras

Let $\mathcal{L}$ be a Leibniz algebra. Then the derived series of $\mathcal{L}$ is the series $$\cdots \subset \mathcal{L}^{(2)} \subset \mathcal{L}^{(1)} \subset \mathcal{L}$$ being $$\mathcal{L}^{(...
1
vote
2answers
51 views

Why Leibniz algebras are called on this way?

I´m interested to know the reason to call the Leibniz algebras on this way. I know that previously they were known as $D$-algebras Why Loday chose this name? Did Leibniz study something related? ...
1
vote
0answers
30 views

Is there an ideal $J$ with $Ann(J)=0$ such that $J\subseteq I_1\cap\cdots\cap I_n$?

Let $L$ be a Lie algebra and $I_1,\dots,I_n$ ideals in $L$ with $Ann(I_k)=0$ for all $k=1,\dots,n$. Of course, $I_1\cap\cdots\cap I_n\neq0$. I would like to know if there exists an ideal $J\subseteq ...
1
vote
1answer
90 views

$(x,yz,t) = (x,y,t)z + (x,z,t)y$ holds in every Jordan algebra.

The identity $$(x,yz,t) = (x,y,t)z + (x,z,t)y$$ holds in every Jordan algebra. Remember that a Jordan algebra satisfies $xy=yx$ and $(x^2,y,x) = 0$ for all $x,y$. Here $(a,b,c) = (ab)c - a(bc)$ is ...
2
votes
1answer
73 views

An octonion generalization of the usual quantum angular momentum operators.

I'm working through, developing the formalism for an octonion generalization of angular momentum operators, but the whole time I'm thinking that this has probably been done elsewhere. My searches ...
1
vote
1answer
111 views

About augmented algebra

An augmented algebra is equipped with a morphism of algebras $ \epsilon: A \to \mathbb{K}$. In this case $ A \equiv \mathbb{K} \oplus \ker(\epsilon)$. May you please clarify what the meaning of $ A \...
2
votes
1answer
78 views

Does there exist an anti-associative structure on a set with three elements?

A friend and I discussed whether there exist operations that are never associative in the sense that $$x(yz)\neq (xy)z$$ for all x, y, z and after pondering I found a simple example on a set with 2 ...
1
vote
0answers
72 views

Subalgebras of the Real Octonions under Multiplication

I am interested to know what the subalgebras of the octonions look like. I tried searching for it but to no avail. What did show up was "Subalgebras of the Split Octonions," which was quite nice. Yet, ...
1
vote
1answer
54 views

Jordan Identity over $char\neq2$ implies power-associativity?

Let $A$ be non-comutative Algebra over a field of characteristic not 2 that satisfies $(xx)(xy)=x(y(xx))$. Can we say that $A$ is power-associative? My attempt: I'm trying to disprove the claim for ...
1
vote
0answers
50 views

Associator of a 2-cocycle is a 3-cocycle

Let $A$ be a $k$-algebra with underlying vector space $V$ and let $F_1:V\times V\to V$ be a bilinear map. Let $A:V\times V\times V\to V$ be the associator of $F_1$, i.e. $A(a,b,c)=F_1(F_1(a,b),c)-...
3
votes
2answers
140 views

What do we call the inverse of a right-multiplication when it's the left-multiplication of the inverse element?

I'm working with a non-commutative, non-associative, non-unital algebra with objects that are somewhat like matrices, but have the property that for almost all $X$ (except the "zero-like" elements), ...
2
votes
0answers
37 views

Number of possible solutions for this repeated commutative, non-associative operation?

If we define $x^n=\overbrace{x\star x\star\cdots\star x}^{n\text{ copies of }x},\; a\star b=-(a+b)$, where the final expression is evaluated with the usual definitions of $+$ and $-$, how many ...
2
votes
1answer
189 views

Is there a name for $x/x$ in a non-unital algebra?

I'm looking into an algebraic structure on $\mathbb{R}^3$ in which multiplication is defined (commutative), as well as division. That is, if $a\cdot b = c$, then $c/b=a$, if $c$ and the modulus of $b$ ...
0
votes
2answers
36 views

Powers of an element $x$ in a non-associative algebra

Consider an element $x\in A$ where $A$ is an arbitrary algebra. Denote by $*$ its product. Consider $X^5$ as the set of containing all the different ways to write the product of $x$ by itself five ...
2
votes
1answer
59 views

Identity in a composition algebra

Let $A$ be a real composition algebra ($A=\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$). I would like to prove that $$ |\lambda|=1 \implies(\lambda u) \overline{(\lambda v)}=u\overline{v}$$ In a ...
6
votes
1answer
239 views

Does Distributivity Imply Power Associativity?

Say we have an algebra $(A, +, \cdot)$, where $(A, +)$ is an Abelian Group. All we know about $\cdot$ is that it is both left and right distributive over addition. So, $\forall a,b,c \in A, a \cdot (b+...
3
votes
0answers
140 views

A question about the Dendriform algebra

Dendriform algebra is a well-known non-associative algebra defined by Loday and Ronco. More precisely, A Dendriform algebra is a $k$-module $D$ together with two binary operations $\{\prec, \succ\}$...
2
votes
3answers
712 views

Is an inverse element of binary operation unique? If yes then how?

I am trying to prove it but not getting any clue how to start it! $$a*b=b*a=e,$$ $$a*c=c*a=e$$ How to show $b=c$?
5
votes
1answer
143 views

Classification of subalgebras of composition algebras

Let $F$ be an algebraically closed field. It is known that the only composition algebras over $F$ are $F$ itself, the direct sum $F\oplus F$ (also called split-complexes), the algebra of $2\times 2$ ...
2
votes
1answer
190 views

The intersection of a maximal toral subalgebra with a simple ideal of a Lie algebra is a maximal toral subalgebra of the simple ideal.

I'm reading Humphreys' Introduction to Lie Algebras and Representation Theory and I have a question about Corollary 14.1, which reads: Humphreys Corollary 14.1. Let $L$ be a semisimple Lie algebra,...