From 5b800f8b14e519dc313fbcd2db2553080587d9c8 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Wed, 15 Mar 2023 18:33:02 -0400 Subject: [PATCH] mjo-eja.tex: add \quadrepr and \leftmult. --- examples.tex | 8 ++++++-- mjo-eja.tex | 9 +++++++++ 2 files changed, 15 insertions(+), 2 deletions(-) diff --git a/examples.tex b/examples.tex index d59975e..740dd02 100644 --- a/examples.tex +++ b/examples.tex @@ -161,8 +161,12 @@ \end{section} \begin{section}{Euclidean Jordan algebras} - The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra $V$ - is $\jp{x}{y}$. The Jordan-automorphism group of $V$ is $\JAut{V}$. + The Jordan product of $x$ and $y$ in some Euclidean Jordan algebra + $V$ is $\jp{x}{y}$. The Jordan-automorphism group of $V$ is + $\JAut{V}$. Two popular operators in an EJA are its quadratic + representation and ``left multiplication by'' operator. For a + given $x$, they are, respectively, $\quadrepr{x}$ and + $\leftmult{x}$. \end{section} \begin{section}{Font} diff --git a/mjo-eja.tex b/mjo-eja.tex index 8b4efbf..efa68de 100644 --- a/mjo-eja.tex +++ b/mjo-eja.tex @@ -22,6 +22,15 @@ % a (bilinear) algebra multiplication in any other context. \newcommand*{\jp}[2]{{#1} \circ {#2}} +% The "quadratic representation" of the ambient space applied to its +% argument. We have standardized on the "P" used by Faraut and Korányi +% rather than the "U" made popular by Jacobson. +\newcommand*{\quadrepr}[1]{P_{#1}} + +% The "left multiplication by" operator. Takes one argument, the thing +% to multiply on the left by. This has meaning more generally than in +% an EJA, but an EJA is where I use it. +\newcommand*{\leftmult}[1]{L_{#1}} % The ``Jordan automorphism group of'' operator. Using % \Aut{} is too ambiguous sometimes. -- 2.43.2