>I really hope someone who actually knows this stuff weighs in on this. For my
>part, one of these days I'm going to have to buckle down and really delve
>into a good, modern notation book on all this [S]O(n), [S]Pin(n), [S]U(n),
>Sp(n) ... compact(?) Lie group theory. I would appreciate any recommendations
>of good books in this area.

Here are some definitions from lectures 7 and 20 of Fulton & Harris, Representation Theory: a First Course. All my vector spaces are over either the real numbers R, the complex numbers C, or the quaternions H.

Given a vector space V,
GL(V) is the group of invertible linear transformations of V,
SL(V) is the group of linear transformations of V of determinant 1 (in which case V must be over R or C),
PSL(V) is SL(V) modulo its centre.
Theorem: SL(V) is a subgroup of GL(V).
GLn(R) = GL(n, R) := GL(Rn).
GLn(C) = GL(n, C) := GL(Cn).
GLn(H) = GL(n, H) := GL(Hn).
SLn(R) = SL(n, R) := SL(Rn).
SLn(C) = SL(n, C) := SL(Cn).
PSLn(R) = PSL(n, R) := PSL(Rn).
PGLn(C) = PGL(n, C) := PSL(Cn).
Theorem: PGLn(C) = PGL(n, C) is GLn(C) = GL(n, C) modulo its centre (hence the notation).

Given a vector space V and a nondegenerate alternating bilinear form Q (in which case V must have even dimension),
Sp(V, Q) is the group of T in GL(V) such that Q(Tv, Tw) = Q(v, w),
PSp(V, Q) is Sp(V, Q) modulo its centre.
Theorem: If V is over R or C, then Sp(V, Q) is a subgroup of SL(V).
Spn(R) = Sp(n, R) := Sp(Rn, standard alternating form).
Theorem: The centre of Sp(n, R) is {1, -1}.
Spn(C) = Sp(n, C) := Sp(Cn, standard alternating form).
Theorem: The centre of Sp(n, C) is {1, -1}.
PSpn(R) = PSp(n, R) := PSp(Rn, standard alternating form).
PSpn(C) = PSp(n, C) := PSp(Cn, standard alternating form).

Note that any degenerate alternating bilinear form is just a nondegenerate alternating bilinear form on some subspace. So, all alternating bilinear forms are covered.

Given a vector space V and a nondegenerate symmetric bilinear form Q,
O(V, Q) is the group of T in GL(V) such that Q(Tv, Tw) = Q(v, w),
SO(V, Q) is the group of T in SL(V) such that Q(Tv, Tw) = Q(v, w),
PSO(V, Q) is SO(V, Q) modulo its centre,
Pin(V, Q) is the double cover of O(V, Q),
Spin(V, Q) is the double cover of SO(V, Q).
Theorem: Spin(V, Q) is a subgroup of Pin(V, Q).
On(R) = O(n, R) = O(n) := O(Rn, standard dot product).
Theorem: O(n) has 2 connected components if n > 0.
Op, q(R) = O(p, q, R) = O(p, q) := O(Rp+q, Q has p positive eigenvalues and q negative ones).
Theorem: O(p, q) has 4 connected components if p, q > 0.
On(C) = O(n, C) := O(Cn, bilinear dot product).
SOn(R) = SO(n, R) = SO(n) := SO(Rn, standard dot product).
Theorem: SO(n) is a connected component of O(n).
Theorem: The centre of SO(n) is {1, -1} if n is even and {1} if n is odd.
SOp, q(R) = SO(p, q, R) = SO(p, q) := SO(Rp+q, Q has p positive eigenvalues and q negative ones).
Theorem: SO(p, q) is 2 connected components of O(p, q) if p, q > 0.
SO+(p, q) is the connected component of SO(p, q) containing the identity.
Note: SO+(3, 1) is the group of Lorentz transformations.
SOn(C) = SO(n, C) := SO(Cn, bilinear dot product).
Theorem: The centre of SO(n, C) is {1, -1} if n is even and {1} if n is odd.
PSOn(R) = PSO(n, R) = PSO(n) := PSO(Rn, standard dot product).
PSOn(C) = PSO(n, C) := PSO(Cn, bilinear dot product).
Pinn(R) = Pin(n, R) = Pin(n) := Pin(Rn, standard dot product).
Pinp, q(R) = Pin(p, q,R) = Pin(p, q) := Pin(Rp+q, Q has p positive eigenvalues and q negative ones).
Pinn(C) = Pin(n, C) := Pin(Cn, bilinear dot product).
Spinn(R) = Spin(n, R) = Spin(n) := Spin(Rn, standard dot product).
Spinp, q(R) = Spin(p, q,R) = Spin(p, q) := Spin(Rp+q, Q has p positive eigenvalues and q negative ones).
Spinn(C) = Spin(n, C) := Spin(Cn, bilinear dot product).

Note that any degenerate symmetric bilinear form is just a nondegenerate symmetric bilinear form on some subspace. So, all symmetric bilinear forms are covered.

Note that any bilinear form is the sum of a symmetric bilinear form and an alternating bilinear form. So, all bilinear forms are covered (except for the spin business).

Given a vector space V and a nondegenerate Hermitian form Q (in which case V must be over C or H),
U(V, Q) is the group of T in GL(V) such that Q(Tv, Tw) = Q(v, w),
SU(V, Q) is the group of T in SL(V) such that Q(Tv, Tw) = Q(v, w).
Un(C) = U(n, C) = U(n) := U(Cn, standard dot product).
Theorem: U(n) = O(2n) intersect Sp(2n, R).
Up, q(C) = U(p, q, C) = U(p, q) := U(Cp+q, Q has p positive eigenvalues and q negative ones).
Un(H) = U(n, H) = Sp(n) := U(Hn, standard dot product).
Theorem (*): Sp(n) = U(2n) intersect Sp(2n, C).
SUn(C) = SU(n, C) = SU(n) := SU(Cn, standard dot product).
SUp, q(C) = SU(p, q, C) = SU(p, q) := SU(Cp+q, Q has p positive eigenvalues and q negative ones).

Note that any degenerate Hermitian form is just a nondegenerate Hermitian form on some subspace. So, all Hermitian forms are covered.

Names: "G" stands for "general", "L" for "linear", "S" for "special", "P" for "projective", "Sp" for "symplectic", "O" for "orthogonal", and "U" for "unitary".

Warning: Don't confuse the "compact symplectic" groups Sp(n) with the "algebraic symplectic" groups Spn(R/C) = Sp(n, R/C). They are unrelated, except through the theorem (*). Also, since n is even in the algebraic symplectic groups, some people use n/2 instead, so (*) looks simpler. But the notation for the compact symplectic groups, where n can be even or odd, is the same.

Go back to my expository notes.
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