Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Suppose $f$ is a nondegenerate, symmetric bilinear form on a vector space $V$ over a field $F$. Then $J = V + F\cdot 1$ is a unital Jordan algebra (known as the "spin factor" Jordan algebra when $F = \mathbb{R}$ and $V = \mathbb{R}^n$). We can construct the Lie algebra $TKK(J) = L = L_{-1} + L_0 + L_1$ via the Tits-Kantor-Koecher construction where $L_1 \cong L_{-1} \cong J$. I have seen several papers reference the fact that $TKK(J) \cong K(R, *)$, the Lie algebra of skew-symmetric elements in an associative algebra $R$ with involution *. Can someone direct me to a proof of this fact? In particular, I would like to know what is $R$ and I'd like a proof when $V$ is not necessarily finite dimensional.

share|improve this question
    
Is $R$ the Clifford algebra of the form you start with? –  Bruce Westbury Apr 27 '13 at 6:58

1 Answer 1

This is contained in Jacobson's Blue Book (Structure and Representations of Jordan Algebras, AMS Colloquium Publications, 1968), as Exercise 1 on p. 342, for arbitrary fields, and with no assumptions on the dimension of $V$.

More precisely, let $W$ be the $3$-dimensional vector space with bilinear form $g$ given by the matrix $\left( \begin{smallmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{smallmatrix} \right)$, and consider the orthogonal sum $h := f \perp g$ on $U := V \oplus W$. Then $TKK(J)$ is isomorphic to the Lie algebra of linear transformations of $U$ that are skew w.r.t. $h$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.