>
> Who was that algebraist, I think in the 1920s, who wrote a book about
> groups which had besides letters from 3 or 4 alphabets (if I remember
> correctly), also used musical notations such as clef signs, signs for flats
> and sharps, etc.?
Another common use for sharps and flats is in the duality of vector spaces, such
as in Riemannian/symplectic geometry.
A (pseudo-)Riemannian structure gives you an (in)definite non-degenerate
symmetirc bilinear form (G, say -- descended from the graviatational constant
via General Relativity) on tangent spaces. A symplectic structure (as in
Hamiltonian mechanics) gives you a non-degenerate skew-symmetric bilinear form
(often called J) on tangent spaces to the phase space.
Given any bilinear form B on any vector space V there is a mapping from
V (thought of as a space full of vectors, thought of as pointy arrows) to its
dual
V* (the collection of linear functionals on V, thought of geometrically
as dividing V into level sets -- a hyperplane for each level),
B_flat : v |--> (the linear mapping w |--> B(v,w) ).
The name is suggested by the mapping's replacing sharp pointy things by wide
flat things.
If B is non-degenerate and V is finite-dimensional
(or if we throw in some topology, in the infinite-dimensional version),
this mapping is invertible, and its inverse is written (naturally) as B_sharp.
Other notations are common for the values of these maps (for instance, Dirac
wrote a typical v as < \psi | and its image as | \psi >, calling them "bras"
and "kets": some people write v and v*, and indexed-quantity enthusiasts write
v^i and v_i), but by suppressing B they create confusion if more than one B is
in play. The sharp and flat notation is a very useful scheme for directly
naming the maps involved, and makes discussion of adjoints, etc., much easier.
I hesitated to mention indexed quantities above, as this could bring up the
ongoing feud between those who see them as among the triumphs of the power of
notation and those who mutter about the "debauch of indices". On some
notations, the jury is still out.
Tim Poston