All Questions
Tagged with nonassociative-algebra or non-associative-algebras
59 questions
2
votes
0
answers
42
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
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 ...
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
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
4
votes
1
answer
685
views
Who and when proved Artin's Theorem on alternative rings?
I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings).
Question. When has Artin proved this theorem and where was it published for the first ...
- 41.8k
5
votes
1
answer
117
views
An isomorphic classification of non-associative division octonion algebras
A division octonion algebra over a field $F$ is a $8$-dimensional unital non-associative algebra $A$ over the field $F$, endowed with a quadratic form $N:A\times A\to F$ such that $N(xy)=N(x)N(y)$ and ...
- 41.8k
0
votes
0
answers
46
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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
3
votes
1
answer
279
views
Wedderburn–Artin like theorem for infinite dimensional Lie algebras?
The Wedderburn–Artin Theorem is one of the cornerstones of the structure theory of (associative) rings.
Wedderburn–Artin Theorem : Let $R$ be a left Artinian ring with zero Jacobson radical. Then $R$ ...
- 3,318
9
votes
2
answers
383
views
Does the Affine Pappus Axiom imply the Affine Desargues Axiom in affine planes?
I am interested in the affine version of the well-known Hessenberg's Theorem (saying that Pappian projective planes are Desarguesian).
First I introduce all necessary definitions.
Definition L. A ...
- 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
6
votes
1
answer
384
views
What is the cardinality of liners of rank 4? Is it always equal 27?
Definition 1. A binary operation $\cdot:X\times X\to X$, $\cdot:(xy)\mapsto xy$, on a set $X$ will be called a line operation if
$$xx=x,\quad xy=yx,\quad (xy)x=y$$
for every $x,y\in X$.
Remark 1. ...
- 41.8k
6
votes
2
answers
407
views
Chirality of octonion algebras
Octonion multiplication can be defined with respect to a set of triads. A set of such triads can be represented by a directed Fano plane diagram such as the following two diagrams.
This depicts two ...
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
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 ...
10
votes
2
answers
218
views
Degree 8 multilinear operations on Jordan algebras
I am interested in the dimension, or, even better, in the $S_8$-module structure of the space of degree 8 multilinear operations on Jordan algebras.
Recall that a Jordan algebra is a commutative but ...
- 16.9k
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
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
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
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
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
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
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
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
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
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
16
votes
1
answer
1k
views
Why is the automorphism group of the octonions $G_2$ instead of $SO(7)$
I can calculate the derivation of the octonions and I clearly find the 14 generators that form the algebra of $G_2$. However, when I do the same calculation for the quaternions, I end up with the ...
- 369
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
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
1
answer
260
views
Left- (right-) multiplications of an algebra that are derivations
Let us say that $A$ is a (finite-dimensional) algebra over a field of characteristic zero. We can assume commutativity
but not associativity, if that makes it easier. Indeed, I am mostly interested in ...
- 4,729
10
votes
1
answer
807
views
How many Lie and associative algebras over a finite field are there?
This question is related to the following general question:
Given a variety of (non-associative) algebras $\mathcal V$, a finite field $\mathbb{F}_q$, with $q$ elements, and a positive integer $n$, ...
- 398
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
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
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
2
votes
1
answer
186
views
Principal ideal of a non-associative magma
The definitions of an ideal of an algebraic structure $A$ (as a substructure $I$ such that the product of $A$ and $I$ is a subset of $I$) do not involve associativity.
However, the definitions of a ...
- 133
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
2
votes
2
answers
600
views
Good reference on the algebraic geometry of non-associative rings
I am looking for a good reference about the algebraic geometry of non-associative rings. I am in particular interested in derivation algebras.
Preferrably an online resource or a book that is ...
- 1,805
5
votes
1
answer
351
views
"Non-associative" standard polynomials
I saw somewhere (I appreciate if anyone has any references to proof of this fact) that if $A$ is a finite dimensional associative algebra such that $\textrm{dim}(A)<n$, then $A$ satisfies the ...
- 353
7
votes
2
answers
386
views
Non-associative deformation quantization
Several physicists consider non-Poisson bivectors but still apply Kontsevich formality in order to get deformation quantization type results: see e.g. Szabos's review An introduction to nonassociative ...
- 3,880
3
votes
1
answer
129
views
Is there a way to adjoin a counit to a non counital coalgebra?
Let $k$ be a field.
If $A$ is a non unital $k$-algebra, there is a simple way to make it unital by taking $\tilde{A}:=A\oplus k$ and setting
$$ (a+\lambda)(b+\mu):=ab+\lambda b +\mu a +\lambda \mu$$
...
- 131
1
vote
2
answers
191
views
Determinants in Jordan algebras of Euclidean type
As far as I heard (I am not sure about the precise statement) there is a classification of simple Jordan algebras of Euclidean type. Each (some?) of such algebras admits a cone of positive definite ...
- 21.8k
6
votes
1
answer
214
views
Weak associativity
Let $(V,*)$ be an algebra and denote $A_*\in \text{Hom}(V^{\otimes 3},V)$ the associator of the binary product $*\in \text{Hom}(V^{\otimes 2},V)$ defined as $A_*(a,b,c):=(a*b)*c-a*(b*c)$.
The ...
32
votes
1
answer
3k
views
Have you ever seen this bizarre commutative algebra?
I have encountered very strange commutative nonassociative algebras without unit, over a characteristic zero field, and I cannot figure out where do they belong. Has anybody seen these animals in any ...
- 18.4k
24
votes
2
answers
710
views
What's the maximum probability of associativity for triples in a nonassociative loop?
In a finite nonabelian group, the probability that two randomly chosen elements commute cannot exceed 5/8. One easy proof also makes it easy to find the smallest groups that attain this bound, namely ...
- 22.3k
-2
votes
1
answer
560
views
Non-associative module theory [closed]
I'm looking for a reference that treats basic module theory over non-associative rings, the isomorphism theorems and so on. I imagine the theory is known, but have not been able to find a reference.
...
- 1
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
1
vote
1
answer
119
views
Multiplication on cubic hypersurfaces and partially defined groups
Let $V$ be a general smooth projective cubic hypersurface. Doing literally as in case of cubic curves we define a relation on $V\times V\times V$: $(x,y,z)$ satisfy it iff $x+y+z$ is an intersection ...
- 2,934
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
11
votes
1
answer
740
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.
...
- 21.8k
6
votes
1
answer
230
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-...
- 2,436
15
votes
4
answers
2k
views
Applications of Jordan algebras
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\...
- 21.8k