# Tagged Questions

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### Notion of trace in a Jordan algebra

Let A be a Jordan algebra (with identity). If x is in A let A[x] be the subalgebra generated by x and the identity. An element x is regular if the dimension of A[x] is maximal. For x regular, denote ...
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### 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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### 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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### 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. $U_{p'}$). ...
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### 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 ...
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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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### 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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### 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 ...
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 ...