LINEAR ALGEBRAIC GROUPS

Other than addition and multiplication the first group that most high school students get introduced to is the group whose multiplication is defined by

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 , the field of real numbers, there are real numbers θ and ψ such
that

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 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 )(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

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}, …, 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} ⋅ e_{j} = ∑
_{k}γ_{ij}^{k}e_{k}, or in other
words

Consider the equations

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^{×} and call the (algebraic) group of units of
A.

In particular, we can think of the group of invertible n × 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) × (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 × 2 invertible matrices as the locus of solutions of the equation (XW -Y Z)T = 1. To simplify matters further, let us take matrices of determinant 1 which are given by the equation XW - Y Z = 1. Multiplication is given in the usual way

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 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 × 2 matrices of determinant one given by the equation
XW - Y Z = 1. Over the field of complex numbers (or any field where the equation T^{2} + 1 = 0 has two
distinct roots ±ι), this equation can be transformed (reversibly) into the equation A^{2} + B^{2} + C^{2} + D^{2} = 1
by means of the substitutions

Now, in terms of these new co-ordinates, the multiplication is given by

While this multiplication may look somewhat mysterious, it is not really so. Regard (x,y,z,w) as
representing the quaternion x + yi + zj + wk, where i, j, k satisfy the rules i^{2} = j^{2} = k^{2} = ijk = -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},…, e_{n} with the relations e_{i}^{2} = -1 and e_{i}e_{j} = -e_{j}e_{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}^{×} of units in C_{n}. However, this is
also not enough for us as we also wish to generalise the action of C_{2}^{×} 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} ⋅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_{i}e_{j}. We let Γ_{n} be the subgroup of C_{n}^{×} that preserves H under the conjugation action of
this group on C_{n}. The group Γ_{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 Γ_{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

In terms of the description of C_{1} given above we note that

This suggests the parametrisation

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.