# Questions tagged [non-associative-algebras]

Questions about non-associative algebras other than Lie algebras.

26 questions
Filter by
Sorted by
Tagged with
50 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 ...
18 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 ...
347 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 ...
243 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 ...
220 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 ...
79 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 $$(a+\lambda)(b+\mu):=ab+\lambda b +\mu a +\lambda \mu$$ ...
54 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 ...
78 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 ...
2k 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 ...
602 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 ...
274 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. ...
110 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 ...
103 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 ...
122 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
502 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. ...
158 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-...
1k views

121 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 $|M:Z(M)|\geq 4$, ...
156 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 ...
315 views

### Is there a cohomology for magmas?

Is there a cohomology theory for magmas? Or cohomology theories for any class of non-associative algebras (other than Lie and maybe Jordan)?
260 views

### Homotopes of simple Lie algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra with bracket $[x,y]$. For which $z\in \mathfrak{g}$ does the formula $$\mu(x,y)=ad (z)([x,y])=[z,[x,y]]$$ define another Lie bracket on the same ...
221 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 ...
307 views

### What is flexible about flexible algebras?

A possibly non-associative algebra is flexible if it satisfies the identity $$(xy)x=x(yx).$$ This is clearly a very weak form of associativity —and obviously an associative algebra is flexible— but it ...
We can define the algebra of quaternions $\mathbb H$ over any field $k$, and depending on the arithmetic of $k$ it is either a division algebra or a matrix algebra. We can also define the algebra of ...