mjo-eja.tex

   1 %
   2 % Euclidean Jordan algebras.
   3 %
   4 \ifx\havemjoeja\undefined
   5 \def\havemjoeja{1}
   6
   7
   8 % The jordan product of its two arguments.
   9 %
  10 % The choice of \circ here (as opposed to, say, \bullet) was a tough
  11 % one. On the one hand, the \circ operator already means function
  12 % composition, and function composition does appear in the context of
  13 % (Euclidean) Jordan algebras. On the other hand, the meaning would
  14 % be unambiguous to a mathematician. Perhaps more importantly -- the
  15 % \circ is much easier to write on the board if one intends to teach
  16 % this material. Finally, the \circ is used by Koecher, Alizadeh,
  17 % Baes, Gowda, et cetera and seems to be more popular. Full
  18 % disclosure: the \bullet is used by McCrimmon.
  19 %
  20 % It is for this reason that we have segregated the Jordan product
  21 % into its own LaTeX file; we probably wouldn't want to use \circ as
  22 % a (bilinear) algebra multiplication in any other context.
  23 \newcommand*{\jp}[2]{{#1} \circ {#2}}
  24
  25 % The "quadratic representation" of the ambient space applied to its
  26 % argument. We have standardized on the "P" used by Faraut and Korányi
  27 % rather than the "U" made popular by Jacobson.
  28 \newcommand*{\quadrepr}[1]{P_{#1}}
  29
  30 % The "left multiplication by" operator. Takes one argument, the thing
  31 % to multiply on the left by. This has meaning more generally than in
  32 % an EJA, but an EJA is where I use it.
  33 \newcommand*{\leftmult}[1]{L_{#1}}
  34
  35 % The ``Jordan automorphism group of'' operator. Using
  36 % \Aut{} is too ambiguous sometimes.
  37 \newcommand*{\JAut}[1]{ \operatorname{JAut}\of{{#1}} }
  38
  39
  40 \fi