# 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 Neumann, 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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### Symmetric cones and symmetric spaces

I start by stating what I think I understood on symmetric cones (https://en.wikipedia.org/wiki/Symmetric_cone). Let $\mathcal{C}$ be a symmetric cone in a vector space $V$. There are Riemannian ...
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### 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 ...
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### Is it true that any semisimple Jordan algebra has the unit element?

I found this theorem in Minnesota notes of Koecher [Thm. 9 p.70]: Thm. Any semisimple Jordan algebra has a unit element. During the text Koecher do not state that the Jordan algebra has to be finite ...
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### Do the Tits-Kantor-Koecher construction extend to non unital Jordan algebras?

I have an infinite dimensional Jordan algebra which is not-unital. I would like to study a Lie algebra related to it (if it exists). Natural choice would be to look at the KKT construction, but ...
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### Are the automorphism groups of simple symmetric cones algebraic groups?

This question arises when I tried to understand Chapter 2 of the celebrated book "Smooth compactification of locally symmetric varieties" by Ash–Mumford–Rapoport–Tai. The setting is as ...
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### Semisimple Lie algebra and convexity

There is any relation between semisimple lie algebras and symmetric cones? I'm saying this because the classification of the euclidean Jordan algebras, dual by Koecher-Vinberg theorem to homogeneous ...
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### Diagonalization of octonionic Hermitian matrices of size $2\times 2$

The group $Spin(9)$ is a subgroup of $SO(16)$ and acts transitively on the unit sphere $S^{15}$. $Spin(9)$ acts naturally on the space of octonionic Hermitian $2\times 2$-matrices (I do not define ...
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### Irreducible $G$-representations with unital algebra structure

Let us work over $\mathbb C$. Suppose that $G$ is a semisimple algebraic group and let $H \subset G$ be a maximal torus. Consider a dominant weight $\omega$, then one can associate a unique ...
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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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### 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 ...
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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 ...
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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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### 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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### 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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Suppose we have exceptional Jordan algebra, which is a $3\times3$ matrix $X=\left(\begin{matrix}x_1&\phi_1&\phi_2\\\bar{\phi_1}&x_2&\phi_3\\\bar{\phi_2}&\bar{\phi_3}&x_3\\\end{... 2 votes 0 answers 133 views ### Automorphism groups of symmetric cones Indecomposable symmetric cones fall into five classes. The automorphism group of any symmetric cone$C$is a real Lie group$Aut(C)$. What is the associated class of Lie groups$Aut(C)$for each of ... 1 vote 1 answer 86 views ### Square of Pierce 1/2 elements Let$p$be a nontrivial idempotent in a JB-algebra$A$with Pierce decomposition$A = A_1 \oplus A_{1/2} \oplus A_0$. Then the projection onto$A_1$(resp.$A_0$) is given by$U_p$(resp.$U_{p'}$). ... 5 votes 0 answers 260 views ### Reference request: The relationship between norm and trace forms on an Albert algebra I am interested in either a nice reference, or some clarification. Overview: I am considering$J_3(\mathbb{O})$, the Jordan algebra of$3\times 3$self adjoint octonionic matrices. This algebra is a ... 30 votes 1 answer 3k views ### 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 ... 2 votes 2 answers 607 views ### 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 ... 9 votes 3 answers 662 views ### Algebraic axiomatization for AB+BA^T operation on matrices Let us consider a matrix algebra$\operatorname{Mat}_{n\times n}(K)$, where$K$is a field,$\operatorname{char} K \neq 2$. It is well-known that the axiomatization of commutator operation$[A,B]=AB-...
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 ...