Move \leftmult (eja) to \leftreg (linear algebra)

And make the argument to \quadrepr pseudo-optional in the process.

diff --git a/examples.tex b/examples.tex
index f8165b680f295f815108900a8af12b01d9e1c2e1..31acc21cfb89f1e3bff39f4a9932f626c1c28428 100644 (file)
--- a/examples.tex
+++ b/examples.tex
@@ -167,10 +167,10 @@
   \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
-    $\JAut{V}$. 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
-    $\leftmult{x}$. The Jordan-algebraic trace is available either as
+    $\JAut{V}$. One important operator in an EJA is its quadratic
+    representation. The quadratic representation operator itself is
+    $\quadrepr{}$ and the quadratic representation of $x$ is
+    $\quadrepr{x}$. The Jordan-algebraic trace is available either as
     a function $\tr{x}$, or in its operator form $\tr{}$.
 
     The one EJA that fits better here than anywhere else is the Jordan
@@ -256,10 +256,12 @@
     \normalsize
 
     If $V$ is an algebra, then $\Der{V}$ is the space of all (linear)
-    derivations on $V$. We also have the group of isometries on $V$,
-    if $V$ has a metric: $\Isom{V}$. More generally, if $V$ and $W$
-    are both metric spaces, then we can represent the isometries from
-    one to the other by $\Isom[W]{V}$.
+    derivations on $V$. The left regular representation of $x \in V$
+    is $\leftreg{x}$, and the representation operator itself is
+    $\leftreg{} : V \to V$. We also have the group of isometries on
+    $V$, if $V$ has a metric: $\Isom{V}$. More generally, if $V$ and
+    $W$ are both metric spaces, then we can represent the isometries
+    from one to the other by $\Isom[W]{V}$.
   \end{section}
 
   \begin{section}{Listing}

diff --git a/mjo-eja.tex b/mjo-eja.tex
index 1d245facc4cba714b3148a93bee535cf532151d2..79ab1770ed78a1f17b31d81ff415a115d61e4cd5 100644 (file)
--- a/mjo-eja.tex
+++ b/mjo-eja.tex
@@ -41,12 +41,7 @@
 % 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}}
+\newcommand*{\quadrepr}[1]{P\if\relax\detokenize{#1}\relax\else_{#1}\fi}
 
 % The ``Jordan automorphism group of'' operator. Using
 % \Aut{} is too ambiguous sometimes.

diff --git a/mjo-linear_algebra.tex b/mjo-linear_algebra.tex
index 32d4d6cde083c9d6f2a5ce8fec8260eef2297234..2b43317ef0c3945255b9b6107d563fdd99d13453 100644 (file)
--- a/mjo-linear_algebra.tex
+++ b/mjo-linear_algebra.tex
@@ -40,6 +40,11 @@
 % The adjoint of a linear operator.
 \newcommand*{\adjoint}[1]{ #1^{*} }
 
+% The "left regular representation" of its argument, i.e. the "left
+% multiplication by" operator. For the linear representation operator
+% itself, use a blank argument.
+\newcommand*{\leftreg}[1]{L\if\relax\detokenize{#1}\relax\else_{#1}\fi}
+
 % The ``transpose'' of a linear operator; namely, the adjoint, but
 % specialized to real matrices.
 \newcommand*{\transpose}[1]{ #1^{T} }