# Questions tagged [jordan-algebras]

A Jordan algebra is an algebra with multiplication satisfying two axioms (J1) xy=yx (J2) (xxy)x=xx(yx). They were defined in 1934 by Jordan, von Neuman, and Wigner seeking a better formalism for quantum mechanics. In 1966 McCrimmon proposed to analyze instead the operator Ux(y)=xyx, which lead to a notion of quadratic Jordan algebras. Three axioms (Q1, Q2, Q3) of these objects can be found below.

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### What is about nonassociative geometry?

At the end of a conference given by Alain Connes in 2000 (here is a video in French), a member of the audience asked a question. I transcribed and translated it for you below: Audience: You showed ...
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### Are there nice isomorphisms $\operatorname{S}^2(k^n)\cong\Lambda^2(k^{n+1})$?

This might be forced to migrate to math.SE but let me still risk it. The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism ...
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### Algebraic axiomatization for AB+BA^T operation on matrices

Let us consider a matrix algebra $Mat_{n\times n}(K)$, where $K$ is a field, $char K \neq 2.$ It is well-known that the axiomatization of commutator operation $[A,B]=AB-BA$ on matrix algebra leads us ...
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### Jordan algebra identities

A Jordan algebra is a vector space with a commutative bilinear operation $\circ$ obeying an identity that's often written as $$(x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x)) .$$ ...
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### How do Jordan algebras help one understand representations of exceptional Lie algebras?

For this question I'm happy to take the complex numbers as the base field. I've been trying to learn a little bit about the exceptional Lie algebras and for a while they seemed inaccessible. I looked ...
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### Formally real non-Jordan algebras

Jordan, von Neumann and Wigner  showed that for any finite-dimensional real vector space $A$ with a bilinear commutative power-associative operation $\circ : A \times A \to A$, the formal reality ...
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### A characterisation of certain $C^*$-algebras

I was wondering if there is a characterisation for $C^*$-algebras (unital) for which the bidual does not have any central atoms. It is not sufficient for example to demand that the $C^*$-algebra does ...
269 views

### Looking for Severi varieties

Let $K$ be an algebraically closed field of characteristic $0$, and let $\mathbb{O}$ be the Cayley algebra over $K$. Let  \mathfrak{J}_{3}=\{A\in\mathcal{M}_{3}(\mathbb{O}):A\text{ is Hermitian}\}, ...
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### ABA-product of matrices and length of chains of principal inner ideals

Let $k$ be a field, $p,q$ positive integers, and let $R$ be the space of $(p \times q)$-matrices over $k$, and $S$ be the space of $(q \times p)$-matrices over $k$. For every matrix $A \in R$, we ...
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### Alfsen Shultz theorem-the space of states of $C^*$-algebra depends only on Jordan structure

According to the article on nLab the Alfsen Shultz theorem states that the space of states of a given $C^*$-algebra depends on somehow weaker structure namely on the so called Jordan algebra structure....
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### 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 ...
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### polarization/linearization as in jordan forms

I am new to this branch of math, so bear with me. This question started when reading Kevin McCrimmon's "A Taste of Jordan Algebras" It talks about polarization and gives a general description. ...
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### Lie Groups and Lie algebras related to Jordan algebras

Let $J$ be a Jordan algebra. I knew three relative Lie groups/Lie algebras to $J$. In the paper "The Capelli Identity, Tube Domains, and the Generalized Laplace Transform" Jacobson [J] has ...
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### Automorphism group of formally real Jordan algebras of hermitian matrices

It is well known that the automorphism group of exceptional Jordan algebra $\mathcal{h}_{3}(\mathbb{O})$ is the exceptional Lie group $F_{4}$. I am trying to understand the automorphism group of the ...
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### Is -1 a sum of Hermitian squares in associative *-envelopes of formally real Jordan algebras?

Let $J$ be an unital Jordan algebra (over $\mathbb{R}$) - recall that this means that $J$ is an unital $\mathbb{R}$-algebra (whose product we denote by $\bullet$) satisfying $x\bullet y=y\bullet x$ ...
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### Automorphism groups of symmetric cones

Indecomposable symmetric cones fall into five classes. The automorphism group of any symmetric cone $C$ is a real Lie group $Aut(C)$. What is the associated class of Lie groups $Aut(C)$ for each of ...
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### 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 ...
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### Relating classic spectral decomposition with Euclidean Jordan algebras

I'm currently getting into studying optimization problems over symmetric cones (NSCP) and I'm having some trouble to understand something. Let me first give some context, sorry if it is repetitive to ...
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### Good source for Jordan Fréchet algebras

Is there any good source for Jordan Fréchet (or more generally, Jordan locally convex) algebras? I'm looking for something on the level similar to the level of the book "Banach and Locally Convex ...
I have seen written that the space of $m\times m$-complex matrices $M_m(\mathbb C)$ endowed with the usual Jordan product is isomorphic, as a Jordan algebra to the complexification of the space \$Herm(...