All Questions
Tagged with nonassociative-algebra or non-associative-algebras
29 questions with no upvoted or accepted answers
12
votes
0
answers
224
views
Do compact inverse-property loops (or just compact Moufang loops) have invariant uniformities and bi-invariant Haar measure?
So, the overall question is in the title: Does a compact topological loop with the inverse property have a Haar measure that is simultaneously left and right invariant? (And we can restrict to ...
- 2,585
8
votes
0
answers
112
views
Identity for the associator involving a third root of unity
This is a reference request. I came across the class of nonassociative algebras satisfying the following identity:
$$
(a,b,c)+\omega(b,c,a)+\omega^2(c,a,b)=0.
$$
Here:
by an "algebra" I mean a ...
- 16.9k
6
votes
0
answers
68
views
Vector algebra in a Tarski space
By a Tarski space I understand a mathematical structure $(X,B,E)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and a 4-ary equidistance relation $E\subseteq X^2\times X^2$ ...
- 41.8k
5
votes
0
answers
79
views
Reference request: associative subalgebras of Cayley algebras are at most 4-dimensional
By a Cayley algebra I mean an 8-dimensional algebra (over an arbitrary field) formed in the Cayley-Dickson process. (They are also called octonion algebras, but I prefer to reserve the term octonion ...
- 2,493
5
votes
0
answers
142
views
Is there a notion of octonionic structure on a Lie algebra? In the same way as there is one for complex and quaternionic
Is there a notion of octonionic structure on a Lie group? In the same way as there is one for complex and quaternionic
- 59
5
votes
0
answers
274
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 ...
- 133
4
votes
0
answers
75
views
Conceptual meaning of the Dickson construction
The Cayley-Dickson construction (a.k.a. Dickson construction or Dickson doubling) constructs a new *-algebra $A'$ out of a given *-algebra $A$. As a vector space $A' = A \oplus A$, and the ...
- 2,040
4
votes
0
answers
122
views
Is the average associator over a finite subloop of octonions necessarily zero?
For any three octonions $a,b,c$, their associator is defined as \begin{equation*} [a,b,c]=a(bc)-(ab)c \end{equation*} and measures their non-associativity so to speak.
Now suppose that $L$ is a finite ...
- 439
3
votes
0
answers
106
views
Finite dimensional real division algebra up to isotopy
Finite dimensional real division (non necessarily associative) algebras exist in dimensions 1, 2, 4, and 8. The standard example is a Hurwitz algebra $(A,*)$: reals, complexes, quaternions, octonions. ...
- 12.3k
3
votes
0
answers
89
views
Non-associative algebras and determinant over 3 by 3 matrices
I have a non associative algebra $A$ that is not unital. And I have three by three matrices $X$ with coefficients over $A$ with a matrix product $*$. I'd like to define something like the determinant ...
- 295
3
votes
0
answers
93
views
Nonassociative quaternion algebras
I'm interested in nonassociative central simple algebras; I've found Lee and Waterhouse's article 'Maximal Orders in Nonassociative Quaternion Algebras', and this cites a previous article by ...
- 323
3
votes
0
answers
139
views
Is there a characterisation of Cayley–Dickson Algebras?
The Cayley–Dickson construction takes an algebra with involution and produces another algebra with involution of twice the dimension.
Starting from the reals (with trivial involution), we ...
- 2,369
3
votes
0
answers
37
views
Is there a unique way to define an Euclidean Jordan Algebra as the product of two simple Euclidean Jordan Algebras?
Let $A$ and $B$ be two simple Euclidean Jordan Algebras. It is well known that $A\times B$ can be made into a Jordan Algebra in a unique way by defining the operation component-wise. Is it true that ...
- 1,638
3
votes
0
answers
134
views
Language representation problem regarding non-commutative, non-associative algebras
Consider a sentence as a series of words with an associated set of labels that tell one how information is passed through the sentence - examples include combinatory categorical grammars or Lambek ...
- 149
2
votes
0
answers
43
views
An elementary proof of the equivalence of the Moufang identities
By a well-known result of Bol (1937) and Bruck (1958), for any loop the following two identities are equivalent:
B: $x(y(xz))=((xy)x)z$
M: $(xy)(zx)=(x(yz))x$.
A proof of the equivalence (B)$\...
- 41.8k
2
votes
0
answers
46
views
Are two notions of power-associativity equivalent for loops?
According to Groupprops, a magma $X$ is called power-associative if for every element $x\in X$ there exists a sequence $(x^n)_{n\in\mathbb N}$ of elements of $X$ such that $x^1=x$ and $x^m\cdot x^n=x^{...
- 41.8k
2
votes
0
answers
26
views
Functor from Leibniz algebra category to Lie-Yamaguti algebra category
Is there any functor from $\operatorname{Leib}_{\mathbb{K}}$ (Leibniz algebra category) to $\operatorname{LYA}_{\mathbb{K}}$ (Lie-Yamaguti algebra category)?
From Kinyon and Weinstein's paper I saw ...
- 229
2
votes
0
answers
99
views
Combinatorics on non-associative words
In my P.h.d research, I deal (among other things) with non-associative words, which we call monomials, and we need to consider two types of operations with these monomials.
The first one is simply ...
2
votes
0
answers
147
views
Non-associative Clifford algebra
Let $V$ be a finite-dimensional $\mathbb{R}$ vector space equipped with a symmetric, bilinear form $b : V \times V \to \mathbb{R}$.
My question is if there exists an analog of a Clifford algebra in ...
- 179
2
votes
0
answers
58
views
Gelfand-Kirillov dimension for non-associative algebras
Let $A$ be any finitely generated algebra - non necessarely unital neither associative - over a base field $k$. Let us denote the product $*$. Suppose $A$ is finitely generated by $S$, and introduce $...
- 3,318
2
votes
0
answers
171
views
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 ...
- 407
1
vote
0
answers
76
views
What is the operator norm of the sedenions and beyond?
Suppose that $K$ is a field. Then for all $n$, define a bilinear operation $*$ (or $*_{n,K}$ in case there may be ambiguity) on $K^{2^n}$ along with a conjugation operation $^*$ on $K^{2^n}$ by ...
1
vote
0
answers
35
views
An algebraic characterization of dual translation projective planes
It is well-known that translation projective planes are coordinatized by quasifields. More precisely, a projective plane is translation if and only if it has a ternary-ring $R$ which is linear, the ...
- 41.8k
1
vote
0
answers
32
views
Is the (left or right) Bol property Isotopy-invariant?
It is well known that a loop satisfies both the left Bol property $(x(yx))z = x(y(xz))$ and the right Bol property $((zx)y)x = z((xy)x)$ if and only if it is a Moufang loop. It is also well known that ...
- 1,947
1
vote
0
answers
166
views
Nonassociative algebras 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, \dotsc, ...
- 763
1
vote
0
answers
50
views
Separable nonassociative algebras
In his paper "Structure and Representation of Nonassociative Algebras", Schafer notes that an arbitrary nonassociative algebra over a field is separable "if and only if the ...
- 323
0
votes
0
answers
46
views
Nonassociativity in Cayley-Algebras
Let $(E,s)$ be a Cayley algebra over a unital commutative ring $A$ with unit element $e$ and $s$ an antiautomorphism (i.e. $s(uv) = s(v)s(u)$, $u,v \in E$) of $E$ such that $u + s(u) \in Ae$ and $N(u) ...
- 2,101
0
votes
0
answers
172
views
When does this commutative non-associative algebra have nilpotent elements?
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, \dotsc, ...
- 763
0
votes
0
answers
125
views
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 $|M:Z(M)|\geq 4$, ...
- 123