Questions tagged [jordan-algebras]

This tag is for questions related to Jordan algebras. They constitute one of the first classes of non-associative algebras.

Filter by
Sorted by
Tagged with
0
votes
0answers
26 views

Properties of the Jordan basis of a matrix.

Suppose $A$ is a square matrix $d\times d$ and that it's a linear opetaror over the albgeraicly closed field $\mathbb{C}$. In particular, $A$ has $n$ different eigen values, each one with multiplicity ...
1
vote
1answer
17 views

Proving that a Jordan basis is not uniquely determined. [duplicate]

It seems that a Jordan basis, for, say, an operator $φ:V→V$ with $V$ a vector space can have several Jordan bases. However I don't see how this is true and I didn't achieve to prove it. Indeed I tried ...
0
votes
0answers
22 views

$C(f)$ is the associative enveloping algebra of $B(f)$

Let $B(f) = F \cdot 1 + M$, where $f:M \times M \to F$ is a symmetric bilinear form. The multiplication in $B(f)$ is $$(\alpha \cdot 1 + x) (\beta \cdot 1 + y)= (\alpha \beta + f(x,y)) \cdot 1 + (\...
3
votes
0answers
134 views

Nonassociative algebra's 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, \dots, ...
0
votes
0answers
34 views

Power associative basis implies not nilpotent?

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, \dots, ...
0
votes
0answers
20 views

If $A$ is a composition algebra, is $H(A_n)$ a simple algebra?

Question I was trying to prove that if $C$ is a Cayley-Dickson algebra over a field of characteristic $\neq 2$, then $H(C_3)$ is a simple algebra (I already know it is Jordan although $C$ is not ...
0
votes
1answer
43 views

Let $A \in \mathbb{C}^{3 \times 3}$ with $A^2(A^2 - 4 I) = 0$. What are possible minimal polynomials of $A^3$?

Let $A \in \mathbb{C}^{3 \times 3}$ with $A^2(A^2 - 4 I) = 0$. What are possible minimal polynomials of $A^3$? Give examples of $A$ with their corresponding $A^3$ matrices. At the moment I know that $...
2
votes
0answers
74 views

What does this: $\{a\}\doteq\emptyset$ mean to you? Is this notation acceptable in any sense?

I'm kind of embarassed for asking this question, but... The problem: I'm having a hard time trying to establish a consistent notation for the spectral decomposition of the elements of a second-order ...
1
vote
1answer
26 views

Calculate the Jordan matrix of a matrix $C$ where $C$ exists of the unity matrix and a matrix $A$ which is diagonalisable.

Let $A \in M^{3 \times 3}(\mathbb{C})$ be diagonalisable matrix with 3 different eigenvalues $\lambda_1, \lambda_2, \lambda_3$ and their corresponding eigenvalues $v_1, v_2, v_3$. Consider the matrix $...
3
votes
2answers
56 views

Ring homomorphism $f: \Bbb{Z}_p[X] \to \Bbb{Z}_p^{n\times n}$. Prove it is never bijective and find the amount of elements in Im$(f)$.

Let $f: \mathbb{Z}_p[X] \to \mathbb{Z}_p^{n \times n}$ with $n \geq 2$ be a ring homomorphism with $f(1) = I_n, f(X) = M$. a) Prove it is never injective and never surjective. b) Find the minimal ...
0
votes
1answer
33 views

Calculate the value of $\sum_{(x,y,z) \in S} g(x,y,z).$

Let S be a triplet set of integers $(x, y, z)$ satisfying the equation $∣x∣ + ∣y∣ + ∣z∣ = 2021$. If we define $g (x, y, z) = x + y + z$ for every real number $x, y$, and $z$, calculate the value of $\...
2
votes
1answer
88 views

Is this proof about Jordan algebra's correct?

Let $A$ be a unital commutative non-associative alternative algebra. Let $J$ be a unital Jordan algebra. Notice Jordan algebra's are commutative and non-associative by definition. Conjecture : The set ...
0
votes
0answers
23 views

Free product of algebras

I am studing Free Product of Algebras in a variety. I would like a good book to study this, and an exemples os free product. Thank you.
1
vote
0answers
14 views

Coordinatization Theorem Exemple

I am studying Coordinatization Theorems, in Jacobson's book. But it is so hard to understand, I would like an easy exemple of direct application of the Coordinatization Theorem, with an easy algebra, ...
2
votes
1answer
55 views

Is there a relationship between associators and commutators?

Let $A$ a algebra over a field $K$. We define the associator of elements $a,b,c \in A$ as $$(a,b,c) = (ab)c - a(bc).$$ We define the commutator too by $$[a,b] = ab - ba.$$ I'm looking for some ...
0
votes
1answer
28 views

Proof of the formula for the number of Jordan blocks

here's what I need help with: Let A$\in M_{n\times n}(F)$ be a Jordanizable matrix. Let n$_k(\lambda)$ be the number of J$_k$($\lambda$) blocks. I'm trying to prove this formula: n$_k$($\lambda$)=rank(...
1
vote
1answer
49 views

$\mathfrak{J}_{ij}^{\;\;.2} \cdot e_i$ is an ideal of $\mathfrak{J}_{ii}$

Let $\mathfrak{J}$ be a Jordan algebra, and $\mathfrak{J} = \sum \mathfrak{J}_{ij}$ the Peirce decomposition of $\mathfrak{J}$ relative to orthogonal idempotents $e_i$ with sum $1$. Prove that $\...
-2
votes
1answer
58 views

Matrix and eigenvalue matrix? [closed]

If we have the eigenvalues of a 6*6 matrix with values such as = [1 1 1 2 2 3], how to write different matrix based on the same eigenvalue matrix. I really need you all to answer this question. THANK ...
1
vote
1answer
89 views

$(x,yz,t) = (x,y,t)z + (x,z,t)y$ holds in every Jordan algebra.

The identity $$(x,yz,t) = (x,y,t)z + (x,z,t)y$$ holds in every Jordan algebra. Remember that a Jordan algebra satisfies $xy=yx$ and $(x^2,y,x) = 0$ for all $x,y$. Here $(a,b,c) = (ab)c - a(bc)$ is ...
1
vote
0answers
37 views

In free special Jordan algebra is valid $T(x,y,z,t) = \frac{1}{4}([x,z] \circ [t,y] + [x,t] \circ [z,y])$.

In the free special Jordan algebra $SJ[X]$ is valid the equality $$T(x,y,z,t) = \frac{1}{4}([x,z] \circ [t,y] + [x,t] \circ [z,y]),$$ where $T(x,y,z,t) = (xy,z,t) - x(y,z,t) - y(x,z,t)$. Here $[x,y] ...
0
votes
0answers
21 views

Proving that Jordan's lemma can be used in integral

I have the next integral: $\int_{-\infty }^{\infty }\frac{e^{ik(x-x')}}{m^{2}+k^{2}}dk$. I want to use Jordan's lemma to solve this integral so I started with proving that I can use it. I know that ...
1
vote
1answer
33 views

Operator commutation in Jordan algebras

Let $(A,*,1)$ be a Jordan algebra and for an element $a\in A$ write $T_a:A\rightarrow A$ for the Jordan product map $T_a(b) = a*b$. We say that $a,b\in A$ operator commute when $T_aT_b = T_bT_a$. The ...
1
vote
1answer
48 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 ...
1
vote
1answer
53 views

There exists an algebra that for every linear map is derivation? [closed]

From the formal definition of Derivation (differential algebra) . I want know if there exists an non-trivial algebra that for every linear map on it, is a derivation.
1
vote
1answer
79 views

Recommended books on JB-algebra

I want to find a self-contained book on JB-algebras, just like Murphy's book on C*-algebra. Any suggestion?
2
votes
1answer
47 views

Analogy of Exponential Map for Jordan Algebras

Today I gave a talk about this paper that constructs a Jordan algebra (more precisely, a JB algebra) to model (bounded) physical observables. It cites this paper, that proves that every JB algebra $A$ ...
3
votes
0answers
40 views

When is a homogeneous cone a Jordan Banach algebra?

A (closed) positive cone $C$ in a vector space $V$ is called homogeneous if for for all $a$ and $b$ in the interior of $C$ there exists an order isomorphism $\Phi: V\rightarrow V$ (i.e. a linear ...
2
votes
1answer
42 views

Mirror symmetries of the Albert algebra

A simple Euclidean Jordan algebra (i.e. a factor) is either a spin-factor, the matrices over the reals/complex-numbers/quaternions or the exceptional Albert algebra of 3x3 octonian matrices. My ...
0
votes
1answer
77 views

Is the quadratic form of a Euclidean Jordan Algebra positive?

A Jordan Algebra $(V,*)$ is an algebra with a commutative (not associative) multiplication operator $*$ that satisfies the Jordan identity: $(x*y)*(x*x) = x*(y*(x*x))$. The positive cone consists of ...
2
votes
2answers
59 views

Complex integration using a suitable contour

$$\int_{-\infty}^{\infty} \frac{x\sin x}{x^2 +4} \ dx$$ Can someone show me how to evaluate this integral by integrating around a suitable contour. I've seen similar questions however I think you ...
2
votes
1answer
211 views

Algebra Symbols $\mathfrak h$ and $\mathfrak{so}$

What do these symbols mean in algebra? I found them as follows: $$\mathfrak h_3(\Bbb O(\Bbb Z_p))$$ $$\mathfrak{so}(\Bbb O)\oplus\Bbb O^3$$
-2
votes
1answer
50 views

What is meant by $A^{+}$ and $A^{-}$ in algebra? [closed]

What is meant by $A^{+}$ and $A^{-}$ in algebra? I read it in Jordan Algebra $A^+$
2
votes
2answers
153 views

Jordan form from the minimal polynomial $m_A$

Let the matrix \begin{equation} A=\begin{bmatrix} 1 & 0 & -1 \\ 4 & 3 & 2 \\\ 2 & 1 & 1 \end{bmatrix}. \end{equation} So far I found the characteristic polynomial $C_A(x)=(x-...
0
votes
0answers
27 views

Proof-Explanation, why does this imply 1-1

Theorem : Let T be a linear operator on a vector space V and let $\lambda$ be an eigenvalue of T. Then : For any scalar $\mu \not = \lambda$, the restriction of T - $\mu$I to $K_\lambda$ is one - to -...