>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*).

GL_{n}(**R**) = GL(*n*, **R**)
:= GL(**R**^{n}).

GL_{n}(**C**) = GL(*n*, **C**)
:= GL(**C**^{n}).

GL_{n}(**H**) = GL(*n*, **H**)
:= GL(**H**^{n}).

SL_{n}(**R**) = SL(*n*, **R**)
:= SL(**R**^{n}).

SL_{n}(**C**) = SL(*n*, **C**)
:= SL(**C**^{n}).

PSL_{n}(**R**) = PSL(*n*, **R**)
:= PSL(**R**^{n}).

PGL_{n}(**C**) = PGL(*n*, **C**)
:= PSL(**C**^{n}).

Theorem: PGL_{n}(**C**) = PGL(*n*, **C**) is
GL_{n}(**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*(*T***v**, *T***w**)
= *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*).

Sp_{n}(**R**) = Sp(*n*, **R**)
:= Sp(**R**^{n}, standard alternating form).

Theorem: The centre of Sp(*n*, **R**) is {1, -1}.

Sp_{n}(**C**) = Sp(*n*, **C**)
:= Sp(**C**^{n}, standard alternating form).

Theorem: The centre of Sp(*n*, **C**) is {1, -1}.

PSp_{n}(**R**) = PSp(*n*, **R**)
:= PSp(**R**^{n}, standard alternating form).

PSp_{n}(**C**) = PSp(*n*, **C**)
:= PSp(**C**^{n}, 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*(*T***v**, *T***w**)
= *Q*(**v**, **w**),

SO(*V*, *Q*) is
the group of *T* in SL(*V*) such that
*Q*(*T***v**, *T***w**)
= *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*).

O_{n}(**R**) = O(*n*, **R**) = O(*n*)
:= O(**R**^{n}, standard dot product).

Theorem: O(*n*) has 2 connected components
if *n* > 0.

O_{p, q}(**R**)
= O(*p*, *q*, **R**) = O(*p*, *q*)
:= O(**R**^{p+q},
*Q* has *p* positive eigenvalues and *q* negative ones).

Theorem: O(*p*, *q*)
has 4 connected components if *p*, *q* > 0.

O_{n}(**C**) = O(*n*, **C**)
:= O(**C**^{n}, bilinear dot product).

SO_{n}(**R**)
= SO(*n*, **R**) = SO(*n*)
:= SO(**R**^{n}, 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.

SO_{p, q}(**R**)
= SO(*p*, *q*, **R**) = SO(*p*, *q*)
:= SO(**R**^{p+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.

SO_{n}(**C**) = SO(*n*, **C**)
:= SO(**C**^{n}, bilinear dot product).

Theorem: The centre of SO(*n*, **C**) is
{1, -1} if *n* is even and {1} if *n* is odd.

PSO_{n}(**R**)
= PSO(*n*, **R**) = PSO(*n*)
:= PSO(**R**^{n}, standard dot product).

PSO_{n}(**C**) = PSO(*n*, **C**)
:= PSO(**C**^{n}, bilinear dot product).

Pin_{n}(**R**)
= Pin(*n*, **R**) = Pin(*n*)
:= Pin(**R**^{n}, standard dot product).

Pin_{p, q}(**R**)
= Pin(*p*, *q*,**R**) = Pin(*p*, *q*)
:= Pin(**R**^{p+q},
*Q* has *p* positive eigenvalues and *q* negative ones).

Pin_{n}(**C**) = Pin(*n*, **C**)
:= Pin(**C**^{n}, bilinear dot product).

Spin_{n}(**R**)
= Spin(*n*, **R**) = Spin(*n*)
:= Spin(**R**^{n}, standard dot product).

Spin_{p, q}(**R**)
= Spin(*p*, *q*,**R**) = Spin(*p*, *q*)
:= Spin(**R**^{p+q},
*Q* has *p* positive eigenvalues and *q* negative ones).

Spin_{n}(**C**) = Spin(*n*, **C**)
:= Spin(**C**^{n}, 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*(*T***v**, *T***w**)
= *Q*(**v**, **w**),

SU(*V*, *Q*) is
the group of *T* in SL(*V*) such that
*Q*(*T***v**, *T***w**)
= *Q*(**v**, **w**).

U_{n}(**C**)
= U(*n*, **C**) = U(*n*)
:= U(**C**^{n}, standard dot product).

Theorem: U(*n*)
= O(2*n*) intersect Sp(2*n*, **R**).

U_{p, q}(**C**)
= U(*p*, *q*, **C**) = U(*p*, *q*)
:= U(**C**^{p+q},
*Q* has *p* positive eigenvalues and *q* negative ones).

U_{n}(**H**)
= U(*n*, **H**) = Sp(*n*)
:= U(**H**^{n}, standard dot product).

Theorem (*): Sp(*n*)
= U(2*n*) intersect Sp(2*n*, **C**).

SU_{n}(**C**)
= SU(*n*, **C**) = SU(*n*)
:= SU(**C**^{n}, standard dot product).

SU_{p, q}(**C**)
= SU(*p*, *q*, **C**) = SU(*p*, *q*)
:= SU(**C**^{p+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
Sp_{n}(**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 papers.

This web page was written between 1999 and 2002 by Toby Bartels. Toby reserves no legal rights to it.