Questions tagged [nonassociative-algebras]

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1answer
45 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 ...
0
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0answers
12 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
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1answer
21 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 ...
0
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0answers
12 views

How may one compute the proportion of isomorphisms among the total quantity of different combinations of element sequences and associative groupings?

If I have n elements in each possible sequence with each possible Tamari lattice grouping of those sequences with respect to a non-associative commutative operation, how may one compute the proportion ...
0
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0answers
9 views

What is the preferred convention for denoting Tamari lattice groupings?

Is there a common method or standard for denoting Tamari lattice/associative groupings with a character length less than that of the sum of the quantity of elements and parenthesis to be denoted? I ...
1
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0answers
30 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
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2answers
68 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
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0answers
34 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
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1answer
143 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
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2answers
27 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
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1answer
38 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 ...
5
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1answer
111 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+...
2
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0answers
44 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
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3answers
115 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 $b=c$?
2
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1answer
97 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,...