In this previous post I recorded some (very standard) material on the structural theory of finite-dimensional complex Lie algebras (or Lie algebras for short), with a particular focus on those Lie algebras which were semisimple or simple. Among other things, these notes discussed the Weyl complete reducibility theorem (asserting that semisimple Lie algebras are the direct sum of simple Lie algebras) and the classification of simple Lie algebras (with all such Lie algebras being (up to isomorphism) of the form ,
,
,
,
,
,
,
, or
).
Among other things, the structural theory of Lie algebras can then be used to build analogous structures in nearby areas of mathematics, such as Lie groups and Lie algebras over more general fields than the complex field (leading in particular to the notion of a Chevalley group), as well as finite simple groups of Lie type, which form the bulk of the classification of finite simple groups (with the exception of the alternating groups and a finite number of sporadic groups).
In the case of complex Lie groups, it turns out that every simple Lie algebra is associated with a finite number of connected complex Lie groups, ranging from a “minimal” Lie group
(the adjoint form of the Lie group) to a “maximal” Lie group
(the simply connected form of the Lie group) that finitely covers
, and occasionally also a number of intermediate forms which finitely cover
, but are in turn finitely covered by
. For instance,
is associated with the projective special linear group
as its adjoint form and the special linear group
as its simply connected form, and intermediate groups can be created by quotienting out
by some subgroup of its centre (which is isomorphic to the
roots of unity). The minimal form
is simple in the group-theoretic sense of having no normal subgroups, but the other forms of the Lie group are merely quasisimple, although traditionally all of the forms of a Lie group associated to a simple Lie algebra are known as simple Lie groups.
Thanks to the work of Chevalley, a very similar story holds for algebraic groups over arbitrary fields ; given any Dynkin diagram, one can define a simple Lie algebra with that diagram over that field, and also one can find a finite number of connected algebraic groups over
(known as Chevalley groups) with that Lie algebra, ranging from an adjoint form
to a universal form
, with every form having an isogeny (the analogue of a finite cover for algebraic groups) to the adjoint form, and in turn receiving an isogeny from the universal form. Thus, for instance, one could construct the universal form
of the
algebraic group over a finite field
of finite order.
When one restricts the Chevalley group construction to adjoint forms over a finite field (e.g. ), one usually obtains a finite simple group (with a finite number of exceptions when the rank and the field are very small, and in some cases one also has to pass to a bounded index subgroup, such as the derived group, first). One could also use other forms than the adjoint form, but one then recovers the same finite simple group as before if one quotients out by the centre. This construction was then extended by Steinberg, Suzuki, and Ree by taking a Chevalley group over a finite field and then restricting to the fixed points of a certain automorphism of that group; after some additional minor modifications such as passing to a bounded index subgroup or quotienting out a bounded centre, this gives some additional finite simple groups of Lie type, including classical examples such as the projective special unitary groups
, as well as some more exotic examples such as the Suzuki groups or the Ree groups.
While I learned most of the classical structural theory of Lie algebras back when I was an undergraduate, and have interacted with Lie groups in many ways in the past (most recently in connection with Hilbert’s fifth problem, as discussed in this previous series of lectures), I have only recently had the need to understand more precisely the concepts of a Chevalley group and of a finite simple group of Lie type, as well as better understand the structural theory of simple complex Lie groups. As such, I am recording some notes here regarding these concepts, mainly for my own benefit, but perhaps they will also be of use to some other readers. The material here is standard, and was drawn from a number of sources, but primarily from Carter, Gorenstein-Lyons-Solomon, and Fulton-Harris, as well as the lecture notes on Chevalley groups by my colleague Robert Steinberg. The arrangement of material also reflects my own personal preferences; in particular, I tend to favour complex-variable or Riemannian geometry methods over algebraic ones, and this influenced a number of choices I had to make regarding how to prove certain key facts. The notes below are far from a comprehensive or fully detailed discussion of these topics, and I would refer interested readers to the references above for a properly thorough treatment.
— 1. Simple Lie groups over —
We begin with some discussion of Lie groups over the complex numbers
. We will restrict attention to the connected Lie groups, since more general Lie groups can be factored
into an extension of an (essentially arbitrary) discrete group by the connected component
(or, in the ATLAS notation of the previous post,
). One can interpret
as the minimal open subgroup of
, thus a Lie group is connected if and only if there are no proper open subgroups.
To each Lie group over
one can associate a complex Lie algebra
, which one can identify with the tangent space of
at the identity. This identification is however not injective; one can have non-isomorphic Lie groups with the same Lie algebra. For instance, the special linear group
and the projective special linear group
have the same Lie algebra
; intuitively, the Lie algebra captures all the “local” information of the Lie group but not the “global” or “topological” information. (This statement can be made more precise using the Baker-Campbell-Hausdorff formula, discussed in this previous post.) On the other hand, every connected Lie group
has a universal cover
with the same Lie algebra (up to isomorphism) as
, which is a simply connected Lie group which projects onto
by a short exact sequence
with being (an isomorphic copy of) the (topological) fundamental group of
. Furthermore, two Lie groups have the same Lie algebra (up to isomorphism) if and only if their universal covers agree (up to isomorphism); this is essentially Lie’s second theorem, discussed in this previous blog post (in the context of Lie groups and Lie algebras over the reals rather than the complex numbers, but the result holds over both fields). Conversely, every Lie algebra is the Lie algebra of some Lie group, and thus of some simply connected Lie group; this is essentially Lie’s third theorem, also discussed at the above post. Thus, the Lie groups associated to a given Lie algebra
can all be viewed as quotients of a universal cover
by a discrete normal subgroup
.
We can say a little more about the fundamental group . Observe that
acts by conjugation on
; however,
is discrete, and so the automorphism group of
is discrete also. Since
is connected, we conclude that the action of
on
is trivial; in other words,
is a central subgroup of
(and so
is a central extension of
). In particular, the fundamental group
of a connected Lie group
is always abelian. (Of course, fundamental groups can be non-abelian for more general topological spaces; the key property of Lie groups that are being used here is that they are H-spaces.)
Not every subgroup of a Lie group is again a Lie group; for instance, the rational numbers are a subgroup of the one-dimensional complex Lie group
but are clearly not a Lie group. However, a basic theorem of Cartan (proven in this previous post) says that any subgroup of a real Lie group which is topologically closed, is also a real Lie group. This theorem doesn’t directly apply in the complex case (for instance
is a subgroup of the complex Lie group
but is only a real Lie group rather than a complex one), but it does say that a closed subgroup of a complex Lie group is a real Lie group, and if in addition one knows that the real tangent space of the subgroup at the origin is closed under complex multiplication then it becomes a complex Lie group again.
We expect properties about the Lie algebra to translate to analogous properties about the Lie group
. In the case of simple Lie algebras, we have the following:
Lemma 1 Let
be a connected complex Lie group with Lie algebra
. Then the following are equivalent:
is a simple Lie algebra.
is non-abelian, and the only closed normal subgroups of
are discrete or all of
.
is non-abelian, and the only normal subgroups of
are discrete or all of
.
Proof: Suppose first that is simple (which implies that
, and hence
, is non-abelian), but
has a closed normal subgroup
which is not discrete or all of
, then by Cartan’s theorem it is a real Lie group with positive dimension. Then the Lie algebra
of
is a non-trivial real Lie algebra which is preserved by the adjoint action of
. If
then
contains a neighbourhood of the identity in
and is thus all of
as
is connected, so
is a proper subalgebra of
. Note that
is a complex Lie algebra ideal of
, so by simplicity this ideal is trivial, thus
lies in the centre of
, which is again trivial by simplicity, a contradiction.
If is normal but not closed, one can adapt the above argument as follows. If
is central then it is discrete (because
is centreless) so assume that
is not central, then it contains a non-trivial conjugacy class; after translation this means that
contains a curve through the identity whose derivative at the identity is a non-zero vector
in
. As
is simple,
is the minimal ideal generated by
, which implies that the orbit of
under the adjoint action of
spans
as a linear space, thus there are a finite number of
-conjugates of
that form a basis for
. Lifting back up to
and using the inverse function theorem, we conclude that
contains an open neighbourhood of the identity and is thus all of
.
Now suppose that is not simple. If it has a non-trivial abelian ideal, then one can exponentiate this ideal and take closures to obtain a closed normal abelian subgroup of
, which is not all of
as
is non-abelian, and which is complex because the ideal is a complex vector space. So we may assume that no such ideal exists, which means (see Theorem 1 from the previous set of notes) that
is semisimple and thus the direct sum
of simple algebras for some
. If we then take
to be the subgroup of
whose adjoint action on
is the identity on
, then
is a closed subgroup of
, thus a real Lie group, and also a complex Lie group as the tangent space is
, giving a closed normal subgroup of intermediate dimension.
In view of this lemma, we call a connected complex Lie group simple if it is non-abelian and the only closed normal subgroups of are discrete or all of
. This differs slightly from the group-theoretic notion of simplicity, which asserts instead that the only normal subgroups of
(including the non-closed normal subgroups) are trivial or all of
. However, these two notions are actually not that far apart from each other. Firstly, given a simple Lie algebra
, one can form the adjoint form
of the associated Lie group, defined as the closed subgroup of the general linear group
on
generated by the transformations
for
. This is group is clearly connected. Because all such transformations are derivations on
, and derivations on a simple Lie algebra are inner (see Lemma 8 from previous notes), we see that the tangent space of this group is
, which is isomorphic to
as
is simple (and thus centerless). In particular,
is a complex Lie group whose Lie algebra is
. Furthermore, any other connected complex Lie group
with Lie algebra
will map by a continuous homomorphism to
by the conjugation action of
on
; this map is open near the origin, and so this homomorphism is surjective. Thus,
is a discrete cover of
, much as
is a discrete cover of
, and so all the Lie groups
with Lie algebra
are sandwiched between the universal cover
and the adjoint form
. The same argument shows that
itself has no non-trivial discrete normal subgroups, as one could then have non-trivial quotients of
which still somehow cover
by an inverse of the quotient map, which is absurd. Thus the adjoint form
of the Lie group is simple in the group-theoretic sense, but none of the other forms are (since they can be quotiented down to
). In particular,
is centerless, so given any of the other covers
of
, the kernel of the projection of
to
is precisely
, thus
for any of the Lie group forms
.
Note that for any form of the Lie group associated to the simple Lie algebra
, the commutator group
contains a neighbourhood of the origin (as
is perfect) and so is all of
. Thus we see that while any given form
of the Lie group is not necessarily simple in the group-theoretic sense, it is quasisimple, that is to say it is a perfect central extension of a simple group.
It is now of interest to understand the fundamental group of the adjoint form
, as this measures the gap between
and
and will classify all the intermediate forms
of the Lie group associated to
(as these all arise from quotienting
by some subgroup of
). For this we have the following very useful tool:
Lemma 2 (Existence of compact form) Let
be a simple complex Lie algebra, and let
be its adjoint form. Then there exists a compact subgroup
of
with Lie algebra
, where
is a real Lie algebra that complexifies to
, thus
. Furthermore, every element
in
has a unique polar decomposition
, where
and
.
Proof: Before we begin the proof, we give a (morally correct) example of the lemma: take , and replace
by
(this is not the adjoint form of
, but never mind this). Then the obvious choice of compact form is the special unitary group
, which has as Lie algebra the real algebra
of skew-adjoint transformations of trace zero. This suggests that we need a notion of “adjoint”
for more general Lie algebras
in order to extract the skew-adjoint ones.
We now perform this construction. As discussed in the previous set of notes, has a Cartan-Weyl basis consisting of vectors
for roots
as well as co-roots
for simple roots
(with the
for other roots
then expressed as linear combinations of the simple co-roots
, and where we have fixed some direction
in which to define the notions of positive and simple roots), obeying the relations
and
and
as well as the relation
when and some integers
, with the convention that
vanishes when
is not a root. We can also arrange matters so that
; see Lemma 31 of the previous notes. If we then define the adjoint map
to be the antilinear map that preserves all the co-roots
, but maps
to
for all
, one easily verifies that
is an anti-homomorphism, so that
for all
. Furthermore, one can now make
into a complex Hilbert space with the Hermitian form
(with
being the Killing form), which one can verify using the Cartan-Weyl basis to be positive definite (indeed the Cartan-Weyl basis becomes an orthogonal basis with this Hermitian form). For any
, one can also verify that the maps
and
are adjoints with respect to this Hermitian form.
If we now set to be the self-adjoint elements of
, and
to be those elements of
that are unitary with respect to the Hermitian form, we see that
complexifies to
and
is a compact group with real Lie algebra
. Also, since
is the adjoint of
, we see that
is closed under the operation of taking adjoints.
Now we obtain the polar decomposition. If , then
is a self-adjoint positive definite map on the Hilbert space
, which also lies in
and thus respects the Lie bracket:
. By diagonalising
and working with the structure constants of the Lie bracket in the eigenbasis of
we conclude that all powers
for
also respect the Lie bracket; sending
we conclude that
is a derivation of
, and thus inner, which implies that
for all
. In particular the square root
lies in
. Setting
we obtain the required polar decomposition; the uniqueness can be obtained by observing that
implies
.
From the polar decomposition we see that can be contracted onto
(by deforming
as
as
goes from
to
). In particular,
is connected and has the same fundamental group as
. On the other hand, the Hermitian form
restricts to a real positive definite form on the tangent space of
that is invariant with respect to the conjugation action of
, and thus defines a Riemannian metric on
. The definiteness of the Killing form then impolies (after some computation) that this metric has strictly positive sectional curvature (and hence also strictly positive Ricci curvature), and so any cover of
also has a metric with Ricci and sectional curvatures uniformly bounded from below. Applying Myers’ theorem (discussed in this previous blog post), we conclude that any cover of
is necessarily compact also; this implies that the fundamental group of
, and hence of
, is finite. Thus there are only finitely many different forms of
between
and
, with the latter being a finite cover of the former. For instance, in the case of
(i.e. the type
case), one can show that the adjoint form
is isomorphic to
and the universal cover
is isomorphic to
, so that
(since the central elements of come from the
roots of unity), and all the intermediate forms of
then come from quotienting out
by some subgroup of the
roots of unity. Actually, as it turns out, for all Lie algebras other than the
family, the fundamental group
is very small, having order at most
; see below. For instance, in the orthogonal algebras
(coming from the
and
families) the adjoint form is
and the universal cover is the spin group
, which is a double cover of
; in particular, there are no other models of the Lie groups associated to the
and
diagrams. This is in marked contrast with the case of abelian Lie groups, in which there is an infinity of Lie groups associated to a given abelian Lie algebra. For instance, with the one-dimensional Lie algebra
, every lattice
in
gives a different Lie group
with the specified Lie algebra.
The compact form of the adjoint form of course lifts to compact forms for all other Lie groups with the given Lie algebra. Among other things, it demonstrates (by the Weyl unitary trick) the representation version of Weyl’s complete reducibility theorem: every finite-dimensional representation
of
splits as the direct sum of a finite number of irreducible representations. Indeed, one can lift this representation to a representation
of the universal cover
, which then restricts to a representation of the compact form
of
. But then by averaging some Hermitian form on
with respect to the Haar measure on
one can then construct a Hermitian form with respect to which
acts in a unitary fashion, at which point it is easy to take orthogonal complements and decompose
into
-irreducible components, which on returning to the infinitesimal action establishes a decomposition into complex vector spaces that are irreducible with respect to the action of
and hence (on complexifying)
. A similar theorem applies for actions of simple (or semisimple) Lie groups, showing that such groups are reductive.
Another application of the unitary trick reveals that every simple complex Lie group is linear, that is to say it is isomorphic to a Lie subgroup of
for some
(this is in contrast to real Lie groups, which can be non-linear even when simple; the canonical example here is the metaplectic group
that forms the double cover of the symplectic group
for any
). Indeed, letting
be the compact form of
, the Peter-Weyl theorem (as discussed in this previous blog post) we see that
can be identified with a unitary Lie group (i.e. a real Lie subgroup of
for some
); in particular, its real Lie algebra can be identified with a Lie algebra
of skew-Hermitian matrices. Note that
can be identified with the complexification
. The set
can then be seen to be a connected smooth manifold which locally is a Lie group with Lie algebra
, and by a continuity argument contains the group generated by a sufficiently small neighbourhood of the identity, and is therefore a Lie group with the same compact form as
, and thus descends from quotienting the universal cover
by the same central subgroup, and so is isomorphic to
. This argument also shows that the compact form of a connected simple complex Lie group is always connected, and that every complex form of a Lie group is associated to some linear representation of the underlying Lie algebra
. (For instance, the universal form is associated to the sum of the representations having the fundamental weights (the dual basis to the simple coroots) as highest weights, although we will not show this here.)
If one intersects a Cartan subalgebra with
and then exponentiates and takes closures, one obtains a compact abelian connected subgroup of
whose Lie algebra is again
(from the self-normalising property of Cartan algebras); these groups are known as (real) maximal tori. As all Cartan subalgebras are conjugate to each other, all maximal tori are conjugate to each other also. On a compact Lie group, the exponential map is surjective (as discussed in this previous blog post); as every element in
lies in a Cartan algebra, we obtain the useful fact that every element of
lies in a maximal torus. The same statement lifts to other models
of the Lie group, and among other things implies that the centre
of such a model is equal to the intersection of all the maximal tori in that model.
We can push the above analysis a bit further to give a more explicit description of the fundamental group of in terms of the root structure. We will be a bit sketchy in our presentation; details may be found for instance in the text of Sepanski.
We first need a basic lemma. Let be the compact form of a simple Lie group, and let
be a maximal torus in
. Let
be the normaliser of
in
; as Cartan algebras are self-normalising, we see that
has the same Lie algebra as
, and so
is a finite group, which acts on the Lie algebra
of
by conjugation, and similarly acts on the dual
. It is easy to see that this action preserves the roots of
. Note that the Weyl group
of the root system, defined in the previous set of notes, also acts (faithfully) on
. It turns out that the two groups coincide:
Lemma 3 (Equivalence of Weyl groups) We have
, with the actions on
(or equivalently,
) being compatible.
Proof: It will suffice to show that
- (a) the action of
on
is faithful;
- (b) to every element of
one can find an element of
that acts the same way on
; and
- (c) for every element of
there is an element of
that acts the same way on
.
To prove (a), we establish the stronger statement that any element of
that preserves a given Weyl chamber
of
(for some regular
) is necessarily in
. If
preserves the
-Weyl chamber
, then it permutes the
-simple roots, and thus fixes the sum
of these
-simple roots. Thus, the one-parameter group
lies in the connected component
of the centraliser
of
. Of course,
also lies in
, as does any maximal torus of
that contains
. In particular, any maximal torus of
containing
is also a maximal torus in
; since all maximal tori in
are conjugate, we conclude that all maximal tori in
are also maximal tori in
; they also all contain
since
is central in
. In particular,
lies in a maximal torus
of
(and hence in
) that contains
. In particular, the adjoint action of
fixes the Lie algebra
of
. But
is regular in
, so its centraliser in
is
. Thus
; since
, we have
as required.
The proof of (c) is similar. Here, need not preserve
, but one can select an element
of
to maximise
; arguing as in the proof of Lemma 28 of these previous notes, we see that
maps the
-Weyl chamber to itself, and the claim follows from the previous discussion.
To prove (b), it suffices to show that every reflection comes from an element of
. But in the rank one case (when
is isomorphic
) this can be done by direct computation, and the general rank case can then be obtained by looking at the embedded copy of the rank one Lie group associated to the pair of roots
.
Call an element of regular if it is conjugate (under the adjoint action of
) to a regular element of
(and hence, by the Weyl group action, to an element in the interior
of the (adjoint of the) Weyl chamber); this conjugation element can be viewed as an element of
, which is unique by the discussion in the previous section. This gives a bijection
to the regular elements of
, which can be seen to be a homeomorphism. The non-regular elements can be computed to have codimension at least three in
(because the centraliser of non-regular elements have at least two more dimensions than in the regular case), so
is simply connected; as this space retracts onto
, we conclude that
is simply connected.
From this we may now compute the fundamental group of (or equivalently, of
). By inspecting the adjoint action of
on
, we see that for
,
is trivial in
if and only if
lies in the coweight lattice
, so the torus
may be identified with the quotient
. Inside
we have the coroot lattice
generated by the coroots
; these are both full rank in
and so the quotient
is finite.
Example 1 In the
example,
is the space
of vectors
with
; the coweight lattice
is then generated by
for
, and the root lattice
is spanned by
for
and has index
in
.
Call an element of
non-integral if one has
for all
; this is a stronger condition than being regular, which corresponds to
being non-zero for all
. The set of non-integral elements of
is a collection of open polytopes, and is acted upon by the group
of affine transformations generated by the Weyl group and translations by elements of the coroot lattice
. A fundamental domain of this space is the Weyl alcove
, in which
for positive roots and
for the maximal root
; this is a simplex in the Weyl chamber consisting entirely of non-integral elements, such that the reflection along any of the faces of the alcove lies in
, which shows that it is indeed a fundamental domain. (In the
case, the alcove consists of tuples
with
.)
Call an element of regular if it is conjugate to
for some non-integral
; as before, the regular elements have codimension at least three in
, and so the fundamental group of
is the same as the fundamental group of the non-integral elements of
. (In the case of
,
is the projective special unitary group
, and the equivalence class of an unitary matrix is regular if its eigenvalues are all distinct.) Observe that
and
are conjugate whenever
; in fact the same is true for all
in
, the group of affine transformations on
generated by the Weyl group and translations by elements of the coweight lattice
. Because of this, we see that every element of
can be expressed in the form
where
lies in the Weyl alcove
,
lies in
, and
is the conjugate of
by (any representative of)
. By lifting, we can then write any loop
in
in the form
for some continuous and
. If we fix the base point
of
, then we can fix the initial point
of
, and normalise
to be the identity; we then have
which places in
(since
and
, being non-integral, do not lie in any maximal torus other than
, as can be seen by inspecting its adjoint action on
). Thus there is an element
of
and
such that
and
; this assigns an element
of
to
with the property that
; one can check that this assignment is preserved under homotopy of
. From the simply connected nature of both
and
one can check that this assignment is injective; and by the connected nature of
and
the assignment is surjective. On the other hand, as
is a fundamental domain for
, we see that each (right) coset of
in
has exactly one representative
for which
, so we have obtained a bijective correspondence between
and
. In fact it is not difficult to show that this bijection is a group isomorphism, thus
With this formula one can now compute the fundamental group or centre (1) associated to any Dynkin diagram group quite easily, and it usually ends up being very small:
- For
,
, or
, the group (1) is trivial.
- For
,
, or
, the group (1) has order two.
- For
, the group (1) has order three.
- For
, the group (1) has order four, and is cyclic for odd
and the Klein group for even
.
- As mentioned previously, for
, the group (1) is cyclic of order
.
Remark 1 The above theory for simple Lie algebras extends without difficulty to the semisimple case, with a connected Lie group defined to be semisimple if its Lie algebra is semisimple. If one restricts to the simply connected models
, then every simply connected semisimple Lie group is expressible as the direct sum of simply connected simple Lie groups. A general semisimple Lie groups might not be a direct product of simple Lie groups, but will always be a central product (a direct product quotiented out by some subgroup of the center).
Remark 2 The compact form
(and its lifts) are usually not the only real Lie groups associated to
, as there may be other real forms of
than
. These can be classified by a somewhat messier version of the arguments given previously, but we will not pursue this matter here; see e.g. Knapp’s book.
— 2. Chevalley groups —
The theory of connected Lie groups works well over the reals or complexes
, as these fields are themselves connected in the topological sense, but becomes more problematic when one works with disconnected fields, such as finite fields or the
-adics. However, there is a good substitute for the notion of a Lie group in these settings (particularly when working with algebraically complete fields
), namely the notion of an algebraic group. Actually, in analogy to how complex Lie groups are automatically linear groups (up to isomorphism), we will be able to restrict attention to (classical) linear algebraic groups, that is to say Zariski-closed subgroups of a general linear group
over an algebraically closed field
. (Remarkably, it turns out that all affine algebraic groups are isomorphic to a linear algebraic group, though we will not prove this fact here.)
The following result allows one to easily generate linear algebraic groups:
Theorem 4 Let
be algebraically closed. All topological notions are with respect to the Zariski topology, and notions of constructibility and irreducibility are in the algebraic geometry sense. If
is a connected constructible subset of
containing the identity, then the group
generated by
is closed (and is thus a linear algebraic group) and also irreducible.
In particular, this theorem implies that linear algebraic groups are connected if and only if they are irreducible.
Proof: By combining with its reflection
we may assume that
is symmetric:
. The product sets
are all constructible and increasing, so at some point the dimension must stabilise, thus we can find
such that
and
both have dimension
. Let
be the
-dimensional irreducible components of
, and
be the
-dimensional irreducible components of
, thus every element of
lies in one of the sets
for some
. As these sets are closed and disjoint and
is connected, only one of the
, say
, is non-empty; as
contains the identity, we conclude that
and
, thus
is an open dense subset of
, which is symmetric, contains the identity, is Zariski closed, and closed under multiplication and is thus an algebraic group. This implies that
is all of
(because
and
intersect for all
as they are both open dense subsets of
) and the claim follows.
This already gives a basic link between the category of complex Lie groups and the category of algebraic groups:
Corollary 5 Every complex simple Lie group
in adjoint form is an linear algebraic group over
.
The same statement is in fact true (up to isomorphism) for the other forms of a complex simple Lie group (by essentially the same argument, and using the fact that the Jordan decomposition for a simple Lie algebra is universal across all representations), though we will focus here on the adjoint form for simplicity. Note though that not every real simple Lie group is algebraic; for instance, the universal cover of has an infinite discrete centre (the fundamental group of
is isomorphic to
) and is therefore non-algebraic. To emphasise the algebraicity of the complex simple Lie group
(and in order to distinguish it from the more general Chevalley groups
which we will introduce shortly) we will now write it as
.
Proof: Recall (see this previous post) that the complex Lie algebra has a Cartan-Weyl basis – a complex-linear basis
indexed by the roots
and the simple roots
respectively, obeying the Cartan-Weyl relations
where we extend to all roots
by making
linear in the coroot of
,
are integers, and
are structure constants. Among other things, this shows that
is generated by the one-parameter unipotent subgroups
and toral subgroups
for various
. The unipotent groups
are algebraic because
is nilpotent. The toral groups
are not quite algebraic (they aren’t closed), but they are constructible, because the Cartan-Weyl relations show that
is given by a diagonal matrix whose entries are monomials in
, so by reparameterising in terms of
we obtain the desired constructibility. The claim then follows from Theorem 4.
Somewhat miraculously, the same construction works for any other algebraically closed fields (and even to non-algebraically closed fields, as discussed below), to construct an algebraic group
that is the analogue over
of the adjoint form of the complex Lie group
. Whereas
consisted of linear transformations from the complex vector space
to itself,
consists of linear transformations on the
-vector space
, which has the same Cartan-Weyl basis
but now viewed as a basis over
rather than
. The analogue
of the toral subgroups
are then the group of linear transformations
on
that map
to
for all roots
and annihilate all the
, for some
; this is a connected constructible subgroup of
. As for the
-analogue
of the unipotent subgroups
, we use crucially the fact (established in this previous post) that one can ensure that
, where
is the largest integer such that
is a root. This implies in the complex setting that
where the series terminates once stops being a root. The point here is that the coefficients
, etc. are all integers, and so one can take this as a definition for
for
and any
regardless of what characteristic
is, and one still obtains a connected unipotent group in this way. If we then let
be the group generated by these one-parameter subgroups
,
, we see that this is a connected linear algebraic group defined over
, known as the (adjoint form) Chevalley group over
associated to the given root system (or Dynkin diagram).
The same construction works over fields that are not algebraically closed, giving groups
that are also denoted
where
is the Dynkin diagram associated to
; for instance
is the projective special linear group
. The resulting groups
are then not algebraic groups, since we only define the notion of a (classical) algebraic variety over algebraically closed fields. Nevertheless, these groups still retain a great deal of the other structure of the complex Lie group
, and in particular inherit the Bruhat decomposition which we now pause to recall. We first identify some key subgroups of
. We first locate the maximal torus
, defined as the group generated by the one-parameter toral subgroups
for
; this is an abelian subgroup of
. Next, we locate the Borel subgroup
, defined as the group generated by
and the unipotent groups
for positive roots
; this can be seen to be a solvable subgroup of
. Then, for each reflection
in the Weyl group
associated to a simple root
, we define the elements
for , one can check using the Cartan-Weyl relations that
determines an element in a coset of
in its normaliser
which is independent of the choice of
. Letting
be the group generated by the
and
, we thus see that
normalises
, and with some further application of the Cartan-Weyl relations one sees that
is isomorphic to
(with each
projecting down to
); cf. Lemma 3. Indeed, if
is a representative of
, one sees that the operation of conjugation
maps
to
for any root
.
For notational reasons we now fix an assignment of a representative in
to each element
, although all of the objects we will actually study will not be dependent on this choice of assignment.
The following axioms can then be verified from further use of the Cartan-Weyl relations:
-
is generated by
and
.
-
is the intersection of
and
, and is normalised by
.
-
is generated by the reflections
, which are of order two.
- No reflection
(or more precisely, no representative in
of that reflection) normalises
.
For each element of the Weyl group, we can form the double coset
; this is easily seen to be independent of the choice of representative
. Thus for instance
. It is also clear that any two double cosets
are either equal or disjoint, and one has the inclusion
for any , as well as the symmetry
. We also have the important further inclusion relation:
Proof: First suppose that is a positive root. Then we observe the factorisation
where is the group generated by all the
for positive
. From the positivity of
one has
and from the simplicity of one has
and thus
multiplying on the left by and on the right by
we conclude that
as the left-hand side is a non-empty union of double cosets, we in fact have equality
Now suppose instead that is a negative root. Applying the previous equality with
replaed by
we conclude that
and thus
On the other hand, direct calculation with the Cartan-Weyl relations reveals that
and the claim then follows from (3).
Lemma 6 and the preceding four axioms form the axiom system, introduced by Tits, for a -pair. This axiom system is convenient for abstractly achieving a number of useful facts, such as the Bruhat decomposition, and the simplicity of
(in most cases). We begin with the Bruhat decomposition:
Proposition 7 (Bruhat decomposition)
is the disjoint union of
as
ranges over
. (Thus there is a canonical bijection between
and
, which by slight abuse of notation can be written as
.)
Proof: We first show that the cover
. As the
cover both
and
(which together generate
) and their union is symmetric, it suffices to show that
is closed under multiplication, thus
. But this is easily achieved by iterating Lemma 6 (inducting on the length of
, that is to say the minimal number of reflections
needed to generate
, noting that the case
is trivial).
Now we show that the are disjoint. Since double cosets are either equal or disjoint, it suffices to show that
implies
for all
. We induct on the length of
. The case when
is trivial, so suppose that
and that the claim has already been proven for all shorter
. We write
for some shorter
. Then
and hence is either equal to
or
. By induction we then either have
or
. The former is absurd, thus
and thus
as required.
By further exploitation of the -pair axioms and some other properties of
, we can show that this group is simple in the group-theoretic sense in almost all cases (there are a few exceptions in very low characteristic). This generalises the discussion of complex Lie groups in the previous section, except now we do not need to pass through the simplicity of the associated Lie algebra (and instead work with the irreducibility of the root system).
We use an argument of Iwasawa and Tits. We first need some structural results about parabolic subgroups of – subgroups that contain the Borel subgroup
(or a conjugate thereof).
Lemma 8 Let
be an element of the Weyl group, with a minimal-length representation
in terms of representations. Then
lie in the group generated by
and
.
Proof: We may assume inductively that and that the claim has been proven for smaller values of
. From minimality we know that
is a negative root, and so
and , hence
, being in the group generated by
and
, is contained in the group generated by
and
. Writing
, this implies that this group contains the group generated by
and
, and the claim then follows from induction.
Corollary 9 (Classification of parabolic groups) Every parabolic group
containing
takes the form
for some
, where
is the subgroup of
generated by the
for
, and conversely each of the
is a parabolic subgroup of
. Furthermore all of these
parabolic groups are distinct.
Proof: The fact that is a group follows from Lemma 6. To show distinctness, it suffices by the Bruhat decomposition to show that the
are all distinct, but this follows from the linear independence of the simple roots. Finally, if
is a parabolic subgroup containing
, we can set
, then clearly
contains
. On the other hand, as
,
is the union of double cosets
, and from Lemma 8 if
contains
, then
is generated by reflections from
. The claim follows.
This, together with the previously noted solvability of and the irreducibility of the root system, gives a useful criterion for simplicity:
Lemma 10 (Criterion for simplicity) Suppose that
is a perfect group and that
does not contain any non-trivial normal subgroup of
(i.e.
). Then
is simple.
Proof: Let be a non-trivial normal subgroup of
. Then by hypothesis
is not contained in
, so the group
is a parabolic subgroup of
that is strictly larger than
, thus
for some non-empty
. If
and
, then
intersects
, and thus (by the normality of
) also intersects
. By Lemma 6 (and (3)), we have
and so at least one of and
lies in
. But as
and
, we conclude that
. From this and Lemma 8 we see that any minimal representation of
has generators both in
and
, which forces
(note that
cannot vanish). Thus we see that
commutes with
, contradicting the irreducibility of the root system unless
. We thus have
. As
is perfect, this implies that
is also perfect; but this is a quotient of the solvable group
and is thus solvable also. As only the trivial group is both perfect and solvable, we conclude that
, and the claim follows.
In the specific case of the adjoint form, the second hypothesis in Lemma 10 can be verified:
Lemma 11
does not contain any non-trivial normal subgroup of
.
As in the complex case, it turns out that non-adjoint forms of a Chevalley group have non-trivial centre that lies in every maximal torus and hence in every Borel group, so this lemma is specific to the adjoint form.
Proof: Let be a normal subgroup of
that lies in
. Conjugating by the long word in
(that maps all positive roots to negative roots) we see that
actually lies in the torus
. In particular, for any root
,
lies in both
and
and is thus trivial; this shows that
is central. But by the Cartan-Weyl relations we see that there are no elements of
that commute with all the
, and the claim follows.
We remark that the above arguments can also be adapted to show that always has trivial centre
(because the above lemma and the proof of Lemma 10 then shows that
, making
normal in
, which can be shown to lead to a contradiction).
From the above discussion we see that will be simple whenever it is perfect. Establishing perfection is relatively easy in most cases, as it only requires enough explicit examples of commutators to encompass a generating subset of
. It is only when the field
and the Dynkin diagram are extremely small that one has too few commutators to make a generating subset, and
fails to be perfect (and thus also fails to be simple); the specific failures turn out to be
,
,
, and
. See the text of Carter for details.
We have focused primarily on the adjoint form of the Chevalley groups, but much as in the complex Lie group case, to each Dynkin diagram and field
one can associate a finite number of forms of the Chevalley group, ranging from the minimal example of the adjoint form
to the maximal example
of the universal form. When
is algebraically closed, these are all linear algebraic groups, and every form of the Chevalley group has an isogeny (the algebraic group analogue of a finite cover) to the adjoint form (arising from quotienting out by the centre) and receives an isogeny from the universal form, much as in the complex case. We still have the basic identity (1), but the lattices
now lie over
rather than
or
(which can make the order of
smaller than in the complex case if
has small positive characteristic
by quotienting out the elements of order a prime power of
, thus collapsing the number of distinct forms of the Chevalley group in some characteristics), and the fundamental group
has to be interpreted as an étale fundamental group rather than a topological fundamental group. See for instance the notes of Steinberg or the text of Gorenstein-Lyons-Solomon for details. As an example of the collapse phenomenon mentioned earlier,
(the universal form for
) and
(the adjoint form for
) are distinct for most fields
, but coincide when
has characteristic two.
We also caution that a Chevalley group over a non-algebraically closed field is not necessarily the same as the set of
-points of the Chevalley group
of the algebraic closure
, as the latter may be strictly larger. For instance, the real elements of
are the elements of
, which a larger group than
(it also contains the projectivisation of matrices with negative determinant). Thus Chevalley groups and algebraic groups are slightly different concepts when specialised to non-algebraically closed fields.
The Chevalley construction gives some specific families of algebraic groups over algebraically closed fields that are either simple (in the adjoint form) or almost simple (which means that the only normal groups are zero-dimensional); in the latter case they are also quasisimple as in the complex case. It is natural to ask whether there are any other (non-abelian) simple algebraic groups over an algebraically closed field. It turns out (quite remarkably) that one can perform the entirety of the classification of complex Lie algebras in the category of algebraic groups over a given algebraically closed field (regardless of its characteristic!), to arrive at the conclusion that the Chevalley groups are (up to isomorphism) the only non-abelian simple or almost simple connected linear algebraic groups. This is despite the lack of any reasonable analogue of the compact form over arbitrary fields, and also despite the additional subtleties present in the structural theory of Lie algebras when the characteristic is positive and small. Instead, one has to avoid use of Lie algebras or compact forms, and try to build the basic ingredients of the
-pair structure mentioned above (e.g. maximal tori, Borel subgroups, roots, etc.) directly. This result however requires a serious amount of algebraic geometry machinery and will not be discussed here; see e.g. this text of Humphreys for details.
Remark 3 The Bruhat decomposition gives a parameterisation of
as
where
is the group generated by the
for all positive roots
, and
is the subgroup generated by the
for those positive roots
for which
is negative; every element
of
then has a unique representation of the form
for some
,
,
, and
. Among other things, this allows for a computation of the order of the Chevalley group
over a finite field of
elements:
where
is the number of positive roots,
is the rank (the dimension of the maximal torus), and
is the number of positive roots
with
negative. If suggestively writes
, this becomes
suggesting that in the limit
, the Chevalley group
over the “field with one element” should degenerate to something like
, an extension of the Weyl group by some sort of torus over the field with one element. Now, this calculation does not make actual rigorous sense – the currently accepted definition of a field does not allow the possibility of fields of order equal to one (or arbitrarily close to one) – but there are tantalising hints in various areas of mathematics that these sorts of formal computations can sometimes to tied to interesting rigorous mathematical statements. However, it appears that we are still some ways off from a completely satisfactory understanding of the extent to which the “field with one element” actually exists, and what its nature is.
— 3. Finite simple groups of Lie type —
As discussed above, the (adjoint form of the) Chevalley group construction , when applied to a finite field, usually gives a finite simple group. However, this construction does not give all of the finite simple groups that are associated to Lie groups. A basic example is the projective special unitary group
over a finite field whose order
is a perfect square:
. This field supports a Frobenius automorphism
which behaves much like complex conjugation
does on the complex field (for instance,
fixes the index two subfield
, much as complex conjugation fixes the index two subfield
). We can then define
as the quotient of the matrix group
by its centre, where is the matrix formed by applying the Frobenius automorphism
to each entry of the transpose
of
. This resembles Chevalley groups such as
, but the group
requires the additional input of the Frobenius automorphism, which is available for some fields
but not for others, and destroys the algebraic nature of the group. For instance,
is not a complex algebraic group, because complex conjugation
is not a complex algebraic operation; it is similarly not a complex Lie group because complex conjugation is not a complex analytic operation. One can view this groups as algebraic (or analytic) over an index two subgroup – for instance,
is a real Lie group, and can also be (carefully) viewed as a real algebraic group, as long as one bears in mind that the reals are not algebraically closed. While this can certainly be a profitable way to view group of this type (known as Steinberg groups), there is another perspective on such groups which extends to the most general class of finite simple groups of Lie types, which contains not only the Chevalley groups and the Steinberg groups but an additional third class, namely the Suzuki-Ree groups. To motivate this different viewpoint, observe that the definition (4) of the special unitary group
can be rewritten as
where ,
is the Frobenius map defined earlier (acting componentwise on each matrix entry), and
is the transpose inverse map
Observe that and
are commuting automorphisms on
of order two, and so
is also an automorphism of order two (i.e. it is an involution). Thus we see that the special unitary group is the subgroup of the Chevalley group
which is fixed by the involution
.
This suggests that we can locate other finite simple (or at least finite quasisimple) groups of Lie type by looking at the fixed points
of automorphisms in a Chevalley group
. One should look for automorphisms with a fairly small order (such as two or three), as otherwise the fixed point set might be so small as to generate a trivial group.
As the example of the special unitary group suggests, one can obtain such automorphisms by composing two types
of automorphisms. On the one hand, we have the field automorphisms
, where
is some power of the characteristic
of the field
, applied to each matrix entry of Chevalley group elements. On the other hand, we have graph automorphisms
, arising from automorphisms of the Dynkin diagram (which, as noted in Theorem 29 of this previous post, induces an automorphism of Lie algebras, and can also be used to induce an automorphism of Chevalley groups), which commute with field automorphisms. The transpose inverse map
defined in (5) is, strictly speaking, not of this form: it is associated to the Lie algebra involution
, which maps each root
to its negation
, so in particular does not map simple roots to simple roots. However, if one composes
with the conjugation action of the long word in the Weyl group (an example of an inner automorphism), which in the case of
is represented by an antidiagonal matrix, the associated Lie algebra involution now maps each root
to its reflection
, and corresponds to the Dynkin diagram automorphism of
formed by reflection. With this conjugation by the long word, the fixed points of the resulting automorphism
is still a special unitary group
, but the sesquilinear form that defines unitarity is not the familiar form
but rather an antidiagonal version
It turns out that up to group isomorphism, we still obtain the same projective special unitary group regardless of choice of sesquilinear form, so this reversal in the definition of the form is ultimately not a difficulty.
If the graph automorphism has order
, and one takes the field automorphism
to also have order
by requiring that
, take the fixed points
of the resulting order
automorphism
, we (essentially) obtain the standard form of a Steinberg group
, where
is the Dynkin diagram. By “essentially”, we mean that we may first have to pass to a bounded index subgroup, and then quotient out by the centre, before one gets a finite simple group; this is a technical issue which be will briefly discuss later. Thus for instance
is denoted
. (In some texts such a group would be denoted
instead.) In a similar vein, the Dynkin diagrams
and
also obviously support order two automorphisms, leading to additional Steinberg groups
when
is a perfect square. The
class can be interpreted as a class of projective special orthogonal groups, but the
family does not have a classical interpretation. A noteworthy special case is
, which is the unique Dynkin diagram that also supports an automorphism of order three, leading to the final class of Steinberg groups, the triality groups
when
is a perfect cube.
In large characteristic (five and higher), the Chevalley and Steinberg groups are (up to isomorphism) turn out to be the only way to generate finite simple groups of Lie type; one can experiment with other cocmbinations of automorphisms on Chevalley groups but they end up either giving the same groups up to isomorphism as the preceding constructions, or groups that are not simple (they do not obey the axioms that one can use to easily test for simplicity). But in small characteristic, where the distinction between short and long roots can become blurred, there are additional Dynkin diagram automorphisms. Specifically, for the Dynkin diagrams
and
in perfect fields of characteristic two, there is a projective Dynkin diagram automorphism of order two that swaps the long and short roots, which induces a automorphism
of the Chevalley group which is order two modulo a Frobenius map (in that
is given by the Frobenius map
); see the text of Carter for the construction. If one combines this automorphism with a field automorphism
with
equal to
, we obtain an order two automorphism
that generates the families of Suzuki groups
and Ree groups
. Similarly, the Dynkin diagram
in perfect fields of characteristic three has an automorphism that swaps the short and long root, and if
leads to the final class of Ree groups,
. In contrast to the Steinberg groups, the Suzuki-Ree groups cannot be easily viewed as algebraic groups over a suitable subfield; morally, one “wants” to view
and
as being algebraic over the field of
elements (and similarly view
as algebraic over the field of
elements), but such fields of course do not exist. (Despite superficial similarity, this issue appears unrelated to the “field with one element” discussed in Remark 3, although both phenomena do suggest that there is perhaps a useful generalisation of the concept of a field that is currently missing from modern mathematics.) One can also view the Steinberg and Suzuki-Ree groups
(collectively referred to as twisted groups of Lie type) as being “fractal” subgroups (modulo quotienting by the centre) of the associated Chevalley group
, of relative “fractal dimension” about
, with the former group lying in “general position” with respect to the latter in some algebraic geometry sense; for instance one could view
as a subgroup of
of approximately “half the dimension”, and in general position in the sense that it does not lie in any (bounded complexity) algberaic subgroup of
. This type of viewpoint was formalised quite profitably in this paper of Larsen and Pink (and is also used in a forthcoming paper of Breuillard, Green, Guralnick, and myself).
Remark 4 We have oversimplified slightly the definition of a twisted finite simple group of Lie type: in some cases the group
is not quite a simple group. As in the previous section, this can happen for very small groups (the Chevalley group examples
,
,
mentioned earlier, but also
,
,
, and
). Another issue (which already arises in the Chevalley group case if one does not use the adjoint form) is that the fixed points
contain a non-trivial centre and are only a quasisimple group rather than a simple group. Usually one can quotient out by the centre (which will always be quite small) to recover the finite simple group, or work exclusively with adjoint forms which are automatically centreless. But there is one additional technicality that arises even in the adjoint form, which is that sometimes there are some extraneous fixed points of
of
that one does not actually want (for instance, they do not lie in the group generated by the natural analogues of the
and
groups in this setting, thus violating the
-axioms). So one sometimes has to restrict attention to a bounded index subgroup of
, such as the group
generated by those “unipotent” elements whose order is a power of the characteristic
; an alternative (and equivalent, except in very small cases) approach is to work with the derived group
of
, which turns out to kill off the extraneous elements (which are associated to another type of automorphism we did not previously discussed, namely the diagonal automorphisms). See the text of Gorenstein-Lyons-Solomon for a detailed treatment of these issues.
15 comments
Comments feed for this article
6 September, 2013 at 2:39 am
amarashiki
Tao, this was a really amazing post!!!!!
6 September, 2013 at 6:03 am
John Mangual
At this point all discussions of Lie Algebras / Groups and Root Systems give me deja-vu.
Don’t get me wrong, I love groups and wish there were more of them!!
I didn’t know the “lie groups” over Finite Fields were simple.
6 September, 2013 at 9:53 am
Allen Knutson
I was hoping to find out why _you_ now want to know this stuff well…
6 September, 2013 at 10:05 pm
Alireza
I’d rather view PGL_n as the adjoint form of SL_n. In some sense PSL_n is not a representable functor. Of course, over an algebraically closed field K we have PGL_n(K)=PSL_n(K), but not over other fields (e.g. finite fields).
7 September, 2013 at 1:30 pm
Terence Tao
Hmm, there seems to be a tradeoff here between “functoriality” across different fields, and “connectedness” for a single choice of field. PGL_n may be a representable functor, but PGL_n(k) need not be simple or even quasisimple, and does not play the role of the “minimal” model of a group of Lie type the same way that PSL_n does. I could see it going both ways: PSL_n is the image of the adjoint representation of SL_n, but then again PGL_n is the image of the adjoint representation of GL_n. I believe my use of the term “adjoint form” is consistent with that in Gorenstein, Lyons, and Solomon, but I’d have to double-check that (my copy of GLS is at work).
7 September, 2013 at 9:02 pm
Alireza
In some sense, PSL_n is NOT an algebraic group. As you said, for any (commutative) ring R, PSL_n(R)=Ad(SL_n(R)), but Ad(SL_n) as an algebraic group is PGL_n. So the adjoint form of SL_n is PGL_n.
7 September, 2013 at 9:13 pm
Alireza
Just adding to the previous comment, in general I prefer to work with simply connected forms. Then I can view the adjoint form as a subgroup of the group of automorphisms of G(F) (for instance, one does the same for the construction and the study of Bruhat-Tits buildings).
8 September, 2013 at 11:54 am
Terence Tao
I think what is going on here is that the “correct” notion of an “adjoint form” (or “universal form”, etc.) depends on the precise category of groups (or group-like objects) one is working in. If one is working as an algebraic group theorist, then presumably one wants to work in the category of algebraic groups (with the underlying field
or ring
viewed as a parameter rather than being fixed, or equivalently one can take a scheme-theoretic point of view), in which case I agree that
is not an algebraic group, and that
is then the more natural choice for the adjoint form of
. But the category of algebraic groups is not perfectly well-behaved with respect to quotients which leads to some complications (e.g. the adjoint form
is not a quotient of the universal form
in this category).
But there are other categories one could work in instead. For instance, if one wants to work in the category of real Lie groups, then
is a real Lie group, and is the connected Lie group associated to the adjoint action of
(or of the Lie algebra
). The group
is also a real Lie group, with
as its connected component, but (when
is even) it has an additional component which one cannot “reach” if one works purely over the reals, but one can see if one allows the underlying field to change temporarily to be complex (or if one works in the category of algebraic groups).
If one is thinking as a finite group theorist, and then works in the category of finite groups (which may, but do not necessarily have to, be the
-points of an algebraic group for a finite field
, or more generally the fixed points of some automorphism on such a group), then again
is the more natural candidate for the adjoint form for
than
; for instance, on page 68 of Gorenstein-Lyons-Solomon’s “The classification of finite simple groups, Number 3”
is identified as the adjoint form of
(with
being the associated inner diagonal group).
[This seems to be related to a technical issue when constructing finite simple groups of Lie type, in that one does not just take the fixed points
of some Steinberg endomorphism
on an algebraic group, but rather just the subgroup
of those fixed points generated by the
-elements.]
In the category of Chevalley groups over a given field
, it again seems that
is the better candidate for the adjoint form than
(see e.g. the table on page 45 of these notes of Steinberg). For instance,
is generated by the one-parameter unipotent subgroups coming from the Chevalley basis, whereas
is not.
[Again, this seems to relate to a divergence between a Chevalley group
and an algebraic group
, when the underlying field
is viewed as a parameter; a Chevalley group and an algebraic group may agree for algebraically closed fields
but be slightly different for non-algebraically closed fields
, and in particular they have different functorial properties. For the Dynkin diagram
, it appears that
is the adjoint form algebraic group, and
is the adjoint form Chevalley group.]
I’m not 100% sure on this, but I believe that if one enlarges the category of algebraic groups to the category of group schemes (which is better behaved with respect to quotients than the category of algebraic groups), then
becomes a valid object in the category, and I think one could make the case that this then becomes the natural adjoint form for
in this category.
8 September, 2013 at 1:00 pm
Rtistry & Rbominations :: linklist, sunday 2013-09-08
[…] Notes on simple groups of Lie type […]
8 September, 2013 at 8:18 pm
Alireza
I do not know what you mean that “PGL_n is not the quotient of SL_n in the category of k-algebraic groups”. PGL_n is the quotient of SL_n as k-algebraic groups for any field k. We have the following short exact sequence
1–> \mu_n –> SL_n –> PGL_n –>1
where \mu_n is a k-group of n-th roots of unity. When it comes to looking at the l-rational points of these groups for a field extension l (or an k-algebra), one has to use either Galois or flat cohomology to get the needed information about the the image of the adjoint map. For instance in the case of a field extension using flat cohomology we have the following long exact sequence
1–> \mu_n(l) –> SL_n(l) –> PGL_n(l) –> H^1_f(l,\mu_n) –> H^1_f(l,SL_n)=1,
and so
1–> \mu_n(l) –> SL_n(l) –> PGL_n(l) –> l^*/(l^*)^n — > 1.
This exact sequence shows what you described in the real case. One uses the same line of argument for any other group as well. In the other cases, one needs to use extra information about the cohomology of simply connected cases. For instance when l is finite, Lang’s theorem give us the needed information. Or when l is a local field Kneser-Bruhat-Tits theorem is needed and when l is global, local-global theorems give us the needed information and so on.
It is true that PSL_n(k) is generated by its (good) unipotent elements, but it is due to the fact that SL_n(k) has this property. As I said it in my previous comment, it is better to work on the simply connected case and using that one can get the desired result on the image of the k-points. It is worth pointing out the Kneser-Tits conjecture in connection to this point:
The conjecture essentially says if this fact is true for any other semisimple simply connected K-isotropic group or not; namely is G(K) generated by its (good) unipotent elements? [the quotient group is called the Whitehead group of (G,K)] And the answer is NO in general. However it is true for local and global fields. And of course it is true for the K-split groups a.k.a. Chevalley groups. Moral of the story, working with simply connected groups is much better ;)
8 September, 2013 at 9:55 pm
Terence Tao
Ah, sorry about that; I had forgotten about the role of cohomology and was thinking (somewhat carelessly) that if H was a quotient of G as an algebraic group, then H(k) was a quotient of G(k) for any k; the fact I had in mind was that PGL_n(k) was not always a quotient of SL_n(k), but as you say this does not prohibit PGL_n from being a quotient of SL_n. (I’m much more used to working over a fixed field than over all fields simultaneously.)
In any event, I agree that simply connected forms are often easier to work with than adjoint forms; I certainly prefer working with SL_n than either PSL_n or PGL_n, for instance :)
10 September, 2013 at 8:15 am
Expansion in finite simple groups of Lie type | What's new
[…] this paper is to establish the analogous result for finite simple groups of Lie type (as defined in the previous blog post) and bounded rank, namely that almost all pairs of elements of such a group generate a Cayley […]
13 September, 2013 at 12:02 am
Jesper Grodal
Thanks for the post! Apart from the references you give (Carter, GLS,..), I’ve also found the recent book Linear Algebraic Groups and Finite Groups of Lie Type by Malle and Testerman a nice reference.
2 October, 2013 at 4:10 pm
Vít Tuček
Nitpick: The metaplectic group is not a double cover of the special linear group but of the symplectic group.
[Corrected, thanks – T.]
19 October, 2013 at 5:19 am
hassan
What is the para-complexification of $U(n)$