Questions tagged [nonassociative-algebras]
The nonassociative-algebras tag has no usage guidance.
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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 ...
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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
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$(ab)c + a(bc) = 2 b (ac) \implies^? x(yz) = (xy)z$? [closed]
Consider some unital commutative algebra $A$ such that for all its elements we have
$$(ab)c + a(bc) = 2 b (ac) $$
Does this imply the algebra is associative ?
or in symbols :
$$(ab)c + a(bc) = 2 b (ac)...
- 16.4k
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Do we ever reason about a non-associative algebra without embedding it in an associative algebra?
This question most certainly contains some errors in phrasing. It is on the subject of the philosophy of mathematics, and it is hard to stay precise when reaching towards the fundamentals of math.
...
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Does $(x^2)(x^3) \neq 0$ imply $(x)(x^3) \neq 0$?
Consider a commutative unital algebra $A$ of finite dimension $n>3$ over the reals.
The product is defined such that elements are generated with real number coefficients
$(a_0, \dotsc, a_n) $
for ...
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About algebras such that $ j_a j_b \in \{ -1,0, +1,j_c,-j_c \}$ for all $a,b$
Consider a commutative unital algebra $A$ of finite dimension $n>3$ over the reals.
The product is defined such that elements are generated with real number coefficients
$(a_0, \dotsc, a_n) $
for ...
- 16.4k
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Representation of Lie triple systems
While reading the definition of a representation of a Lie triple system, I have a few doubts which I have stated at the end.
Let $(T, \{\cdot,\cdot,\cdot\})$ be a Lie triple system and $V$ be a vector ...
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Algebra : Conjecture about basis for algebra.
Consider a non-associative commutative unital algebra of finite dimension such that elements are generated with real number coefficients
$(a_0, \dotsc, a_n) $
for a basis $\{1, i_1, \dotsc, i_n \}$.
...
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Given an alternative algebra A, is the algebra generated by two commutative, associative subalgebras B and C always associative?
The algebras are over a field.
The answer to the question is yes when B and C are generated by 1 element each. Over the octonions, the answer to the question is then yes. What about in general?
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Jordan algebras over $\mathbb{F}_2$ are power-associative?
By Jordan algebra I mean a finite-dimensional vector space $A$ endowed with a bilinear product satisfying $xy=yx$ and $(x^2,y,x)=0$ for every $x,y\in A$.
I was studying the paper "Power-...
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A problem about a group-like structure
I was sitting in an algebra course a year ago and while commutativity and associativity were discussed I wondered:
Does there exist a set $S$ with an operation $\cdot$ which is commutative, has an ...
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Representations of (opposite) non-associative algebras on dual spaces
I want to better understand the relationship between representations on dual vector spaces and opposite algebras when the algebra being represented is nonassociative. Specifically, my question is: ...
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Proving an algebra is nilpotent
I recently have started studyng about free algebras and I want to know what kind of methods or approaches are there to prove that free algebra $A$ is nilpotent with nilpotency index $n.$
Let me remind ...
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Is every solvable algebras are nilpotent?
An algebra $A$ is called nilpotent if $A^n=0$ for some positive integer $n$. Also, we call algebra $A$ is solvable if $A^{(n)}=0$, with solvable index $n$, where $A^{(n)} = A^{(n-1)}A^{(n-1)}$.
Can we ...
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What kind of algebra and how can I learn its properties
I try to find any information on the following algebra defined as:
$$(a,b)+(c,d)=(a+c,b+d)$$
$$(a,b)*(c,d)=(ac,ac-bd)$$
It is non-associative, but commutative and distributive.
Can it be classified ...
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