\documentclass[11pt]{amsart}
\usepackage[a4paper,scale=0.8,centering,nomarginpar]{geometry}
\usepackage{setspace}
\onehalfspacing
\newcommand{\bbR}{{\mathbb R}}
\newcommand{\bbG}{{\mathbb G}}
\newcommand{\bfi}{{\bf i}}
\newcommand{\bfj}{{\bf j}}
\newcommand{\bfk}{{\bf k}}
\DeclareMathOperator{\Spin}{Spin}
\usepackage{hyperref}
\begin{document}
\title{Lecture V\\ Linear Algebraic Groups}
\author{Kapil Hari Paranjape}
\maketitle
Other than addition and multiplication the first group that most
high school students get introduced to is the group whose
multiplication is defined by
\[ (x,y)\cdot(u,v) = (xu-yv,xv+yu) \]
Consider
the affine plane curve defined by the single equation
$X^2+Y^2=1$; when $(x,y)$ and $(u,v)$ are points of this curve then
the above rule defines a multiplication where $(1,0)$ acts as
identity. When the underlying field is $\bbR$, the field of real
numbers, there are real numbers $\theta$ and $\psi$ such that
\[
(x,y) = (\cos\theta,\sin\theta)
\text{~and~}
(u,v) = (\cos\psi,\sin\psi)
\]
In case you haven't already recognised it, the above group law
is recognisable as the addition law for sines and cosines. However,
the above law makes sense over \emph{any} field $k$. In fact, it works
under the weaker hypothesis that $X^2+Y^2$ is non-zero; in other words it
gives a group law on the points of the surface defined by
$Z(X^2+Y^2)=1$. This can be used to show that if $a$ and $b$ are elements
of the field that can each be written as the sum of two squares then
so can their product.
The above is also the first interesting example of a group scheme or
more specifically an affine algebraic group variety. We can write the
equations $(f_n(X,Y),g_n(X,Y))=(1,0)$ where $f_n$ and $g_n$ are polynomial
expressions for the $n$-fold product $(X,Y)\cdots(X,Y)$. Since the
original group law is commutative, these two equations also define an
interesting finite group scheme---the corners of a regular polygon.
There is another more common (garden variety) of group scheme which
is defined by the equation $XY=1$ in the plane. For this the
multiplication law is given by
\[ (x,y)\cdot (u,v) = (xu,yv) \]
This law points us in the direction in which we can look for
more general group schemes.
Let $A$ be a $k$-algebra (which is associative but not necessarily
commutative). Further assume that $A$ is finite dimensional as a $k$
vector space and let $e_1$, \dots, $e_n$ be a basis such that $e_1=1$
is the identity element of $A$. In terms of this basis the multiplication
law on $A$ can be written as $e_i \cdot e_j = \sum_k \gamma_{ij}^k
e_k$, or in other words
\[
(x_1,\dots,x_n)\cdot(y_1,\dots,y_n) =
\left(\sum_{ij} \gamma_{ij}^1 x_i x_j, \dots,
\sum_{ij} \gamma_{ij}^n x_i x_j \right)
\]
Consider the equations
\[
\sum_{ij} \gamma_{ij}^1 x_i x_j = 1 \text{~and~}
\sum_{ij} \gamma_{ij}^k x_i x_j = 0 \text{~for~} k>1
\]
The equation of the hyperbola can be thought of as the special case
of this equation for the case $A=k$. In general, this gives us an
algebraic group which we denote by $A^{\times}$ and call the (algebraic)
group of units of $A$.
In particular, we can think of the group of invertible $n\times n$
matrices. In this case the equations can be simplified a bit. We know
that a matrix is invertible if and only if its determinant is non-zero;
in fact, the inverse of a matrix $A$ is $(A^*)/(\det(A))$,
where $A^*$, the adjoint of $A$, is the matrix of $(n-1)\times(n-1)$
minors of $A$. Thus the above equation can be replaced with the
equation $\det(A)T=1$, which is an equation in the entries $A_{ij}$
of $A$ and the supplemental variable $T$.
It is thus natural to define the group of $2\times 2$ invertible
matrices as the locus of solutions of the equation $(XW-YZ)T=1$. To
simplify matters further, let us take matrices of determinant 1 which
are given by the equation $XW-YZ=1$. Multiplication is given in the
usual way
\[
\begin{pmatrix}
x & y \\
z & w
\end{pmatrix}
\cdot
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
=
\begin{pmatrix}
ax+cy & bx+dy \\
az+cw & bz+dw
\end{pmatrix}
\]
What is the sub-variety of this given by the equations $X=W$ and
$Z=-Y$? Is it a group? (Hint: Look around).
More generally, the various matrix groups that we study are defined
by algebraic equations; like $AA^{t}=I$ for the orthogonal group. The
study of these groups and algebraic \emph{group} homomorphisms among
them (which can be defined in an obvious way) goes under the name
``linear algebraic groups and their representations''. It is a theorem
that all affine algebraic groups are in fact linear algebraic groups.
Let us now look at a more exotic algebraic group
which (unfortunately) most of you may not have encountered before. It
was introduced by Hamilton to begin with but Clifford, Lipschitz,
Grassmann and others developed it further. Somehow it is not in our
mathematics syllabi but it is certainly in the physics syllabus!
Let us start again with the group of $2\times 2$ matrices of
determinant one given by the equation $XW-YZ=1$.
Over the field of complex numbers (or any field where the equation
$T^2+1=0$ has two distinct roots $\pm\iota$), this equation can
be transformed (reversibly) into the equation $A^2+B^2+C^2+D^2=1$ by means of
the substitutions
\[ X = A+\iota B ; W = A-\iota B ; Y = C + \iota D ; Z = -C +\iota D
\]
Now, in terms of these new co-ordinates, the multiplication is given
by
\[
(x,y,z,w)\cdot(a,b,c,d) =
(ax-by-cz-dw,ay+bx-cw+dz,az+bw+cx-dy,aw-bz+cy+dx)
\]
While this multiplication may look somewhat mysterious, it is not
really so. Regard $(x,y,z,w)$ as representing the quaternion
$x+y\bfi+z\bfj+w\bfk$, where $\bfi$, $\bfj$, $\bfk$ satisfy the rules
$\bfi^2=\bfj^2=\bfk^2=\bfi\bfj\bfk=-1$. We then see that the
multiplication given above is the linear extension of these
identities. It follows that the unit sphere of dimension 3
is also a group in a natural algebraic way. As before, the point
$(1,0,0,0)$ plays the role of identity. Moreover, we can again use this
product rule to show that if $a$ and $b$ are each a sum of 4 squares, then
so is the product $ab$.
Now consider the subspace of the three-dimensional sphere that
consists of ``purely imaginary'' quaternions, i.~e.\ quaternions of
the form $(0,b,c,d)$. This can be identified with the two-dimensional
sphere. Moreover, this sub-variety of the quaternions is stable under
the conjugation action of the quaternions on itself. The collection
of all quaternions $(x,y,z,w)$ that commute with $(0,b,c,d)$ can be
identified as $(u,vb,vc,vd)$ where $(u,v)$ is a point of the circle.
It follows that we obtain the universal geometric identity
$S^3/S^1=S^2$.
How does one generalise the quaternions? One natural way (invented by
Clifford) is to consider the $k$-algebra $C_n$ (with unit)
generated by $e_1$,\dots, $e_n$ with the relations $e_i^2=-1$ and
$e_ie_j=-e_je_i$. When $n=2$ we get the algebra of quaternions.
We note that not every non-zero element of this algebra is
invertible when $n>2$; so we should further restrict our attention to
the group $C_n^{\times}$ of units in $C_n$. However, this is also not
enough for us as we also wish to generalise the action of $C_2^{\times}$
on the two-dimensional sphere. In terms of this description when $n=2$,
the subspace of imaginary quaternions is spanned by $e_1$, $e_2$ and
$e_1\cdot e_2$. A generalisation that suggests itself is to consider
the subspace $H$ of $C_n$ which is spanned by the elements $e_i$ and
$e_ie_j$. We let $\Gamma_n$ be the subgroup of $C_n^{\times}$ that
preserves $H$ under the conjugation action of this group on $C_n$. The
group $\Gamma_n$ is called the (non-normalised) spin group. There is a
natural ``norm'' on $C_n$ and the group $\Spin(n)$ is the subgroup of
$\Gamma_n$ that consists of elements of norm 1.
Let us return to the cases $n=1$ and $n=2$. There is a natural
parametrisation of the circle $x^2+y^2=1$ given by
\[ t \mapsto \left(\frac{1-t^2}{1+t^2},\frac{2t}{1+t^2}\right) \]
In terms of the description of $C_1$ given above we note that
\[
\frac{1-te_1}{1+te_1} = \frac{(1-te_1)^2}{1+t^2} =
\frac{1-t^2}{1+t^2} + \frac{2t}{1+t^2} e_1
\]
This suggests the parametrisation
\[
t \mapsto \frac{1-pe_1-qe_2-re_1e_2}{1+pe_1+qe_2+re_1e_2}
\]
for the unit quaternions in $C_2$. These parametrisations are due to
Cayley and were generalised for $C_n$ by R.~Lipschitz who showed that
the corresponding expression in that case indeed gives a
parametrisation of $\Spin(n)$.
The study of the spin group and its representations is important for
a lot of modern physics, geometry, analysis and number theory. Learn
more about it as soon as possible! R. Sridharan, R. Parimala and
their many students have undertaken an extensive study of quadratic
forms over different kinds of fields using (among other things) the
study of the associated Clifford algebra and algebraic groups.
Many mathematicians such as M. S. Raghunathan have studied algebraic
groups and associated number theoretic problems extensively.
\end{document}