From: Michael Orlitzky <michael@orlitzky.com>
Date: Wed, 15 Mar 2023 22:33:02 +0000 (-0400)
Subject: mjo-eja.tex: add \quadrepr and \leftmult.
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=5b800f8b14e519dc313fbcd2db2553080587d9c8;p=mjotex.git

mjo-eja.tex: add \quadrepr and \leftmult.
---

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.