First we started by a naive approach towards
linear algebraic groups; then we viewed it as a representable
functor from k-algebras to groups; then we found out the
necessary and sufficient conditions for an algebra to
represent a group functor: commutative Hopf algebra.
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We defined the regular representation of a
representable group functor; proved that any representable
group functor is linear.
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Background on AG: algebraic sets are definied;
Nullstellensatz theorem is proved; Zariski topology on the n-dimensional
affine space is proved; irreducible topological spaces;
correspondence between spec(k[x_1,...,x_n]) and irreducible
algebraic sets; morphism of algebraic sets; product of two algebraic sets;
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Algebraic groups from algebraic sets point of view;
connected component of the identity; closed finite-index subgroups;
Zariski closure of an abstract subgroup;
kernel from two points of view: functorial and k-points;
Chevalley's theorem: image of a morphism contains a non-empty
open subset of its closure; image of an algebraic group is
an algebraic group;
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Background on AG: Sheaf of regular functions on an algebraic sets;
the stalk of the sheaf of regular functions at a point;
the isomorphism between the coordinate ring of an algerbaic set and
its ring of regular functions; defining prevariety;
separation axiom; quasi-affine varieties;
a criterion for being a variety;
projective and quasi-projective varities;
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Actions of affine algebraic groups; any orbit is quasi-affine;
existence of closed orbits; the induced action on the ring of regular functions;
locally finiteness of the ring of regular functions as a G-mod;
linearity of affine algebraic groups;
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The stabilizer of the defining ideal I(H) of a closed subgroup H
is H; getting a finite-dimensional representation V of G and a subspace H
such that the stabilizer of W is H; defining the exterior algebra of a
vector space; finding the standard basis of the exterior algebra;
computing the entries of the kth-wedge representation of GL_n;
Parts of Chevalley's theorem getting a finite-dimensional representation of G and a line l
such that the stabilizer of l is H; First step towards the quotient space:
Suppose G is an affine algebraic group and H is a closed subgroup of G. Then
there is a unique quasi-projective homogeneous G-space X together with a point x
such that (a) the stabilizer of x is H,
(b) the fibers of g --> gx are cosets of H.
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Background on AG: in this short note the dimension of a topological
space is defined; the connection between Krull dimension of the ring of
coordinates of an affine variety and its dimension is pointed out;
some of the relevant results from commutative algebra is mentioned.
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Background on AG: three definitions of the tangent space at a
point of an affine variety is given; the module of R-derivations
of an R-algebra A to an A-mod M Der_{R}(A,M) is defined;
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Background on AG: the existence and the uniquness of the
module of differentials of an R-algebra A is proved;
module of differentials of a finitely presented R-algebra
is found; Ω_{k[a]/k}=0 if and only if a is
algebraically separable over k; Ω_{F/R} is isomorphic
to Ω_{A/k} tensor F if F is the field of fractions of
the integral domain A; dim Ω_{F/k} is at least trdeg_{k}(F)
and equality holds if and only if F/k is separably generated;
a new definition of separability which makes sense for extensions with
transcendental elements; prove that F/k is separable if and only if
F/k is separabily generated; E/E' is separably generated if
and only if Ω_{E'/k} tensor E embeds into
Ω_{E/k} (under the natural homomophism).
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Background on AG: we proved that in an open dense set U the module of differential is a free module,
and so U is smooth; separability criteria is proved, too.
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Background on AG: we defined the differental of a morphism from three points of view and showed
their equivalence.
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In this short we observe what the separability criteria implies in the context
of G-equivarient morphisms between G-homogeneous varieties.
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Lie algebra Lie(G) of an affine algebraic group G is defined: (1) as a vector space
(2) as a G-module via the adjoint representation (3) defining the Lie algebra structure on Lie(G).
We proved that, if G is a closed subgroup of GL_{n}(k), then Lie(G) is a subalgebra of gl_{n}(k).
We defined the left and the right convolutions of an element of the Lie algebra and a regular function; Lie(G)
is characterized as the set of derivations of k[G] that commute with the right shift action of G on k[G].
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We proved
Chevalley's theorem for any closed subgroup H of an affine algebraic group G there is an
algebraic group representation ρ:G --> GL(V) and a line l in V such that
- H={g in G| ρ(g)l=l} and
- Lie(H)={x in Lie(G)| dρ(x)l ⊂ l}.
Quotient: Let G be an affine algebraic group and H a closed subgroup; then
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Jordan decomposition:
- additive, finite-dimensional version; various properties of such a decomposition.
- additive, locally-finite endomorphism;
- multiplicative, locally-finite automorphism;
- affine algebraic groups;
- compactibility of the Jordan decomposition in an algebraic subgroup of GL(V) and
the Jordan decomposition as an linear transformation.
Proved that any unipotent subgroup of GL_{n}(k) is upper-triangularizable; and deduced that
when a unipotent algebraic group acts on an affine variety, then all the orbits are closed.
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Commutative affine algebraic groups; diagonalizable groups; torus:
- Let G be a commutative affine algebraic group, G_{s}:={g in G| g is semisimple}, and
G_{u}:={g in G| g is unipotent}. Then G_{s} and G_{u} are closed subgroups of G
and G_{s}x G_{u} --> G, f(s,u):=su is an isomorphism.
- G is called diagonalizable if it is isomorphic to a closed subgroup of GL_{1}(k)^{n} for some n;
and it is called a torus if it is isomorphic to GL_{1}(k)^{n} for some n.
- We proved the following are equivalent
- We proved that, if H is a closed subgroup of a diagonalizable group G, then the restriction map X^{*}(G)-->X^{*}(H)
is onto.
- We pointed out that X^{*}(G) does not have a p-element where p=char(k)
- For a finitely generated abelian group M with no p-elements, we defined a Hopf algebra structure on the group algebra kM where
the co-product is induced by M-->MxM, y-->(y,y) and the co-inverse map is induced by M-->M, y--> -y. We proved that kM is a reduced algebra
and so we get an affine algebraic group G_{M}. It is outlined why M=X^{*}(G_{M}), and deduced that G_{M} is a
diagonalizable group; One can see that for a diagonalizable group G we have G=G_{X*(G)}.
- We indecated that the above contrcution implies the functor G-->X^{*}(G) from diagonalizable groups to finitely generated groups is
fully faithful.
- We proved that any diagonalizable group G is isomoprhic to T x F where T is a torus and F is a finite diagonalizable group;
- We proved that a diagonalizable group G is a torus if and only if it is connected if and only if X^{*}(G) is torsion free.
- Rigidity of diagonalizable groups: a homomophism between two diagonalizable groups cannot be algebraically deformed; deduced that
if G is an affine algebraic group and H is a diagonalizable subgroup of G, then N_{G}(H)/C_{G}(H) is finite.
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Parabolic and Borel subgroups:
- Complete varieties are defined and their basic properties are proved;
- Mentioned that any projective variety is complete;
- A subgroup P of an affine algebraic group is called parabolic if G/P is complete;
- We prove that
- P parabolic in G implies G/P is projective;
- Q parabolic in P and P parabolic in G if and only if Q parabolic in G;
- P parabolic in G and P ⊂ H ⊂ G.
- Upper-triangular matrixes form a parabolic subgroup of GL_{n}(k): flag variety is complete;
- A connected affine algebraic group has a proper parabolic subgroup if and only if it is not solvable;
- Borel fixed point theorem: suppose G is a connected solvable affine group and it acts on a complete variety X.
Then it has a fixed point.
- Lie-Kolchin theorem: suppose G is a connected solvable closed subgroup of GL_{n}(k); then for some x in
GL_{n}(k) we have xGx^{-1} ⊂ the set of upper-triangular matrixes.
- A Borel subgroup B of G is a connected solvable subgroup of maximum possible dimension.
- We proved that
- P is parabolic in G if and only if it contains a Borel subgroup of G;
- Any two Borel subgroups of G are conjugate;
- Any connected solvable subgroup of G is contained in a Borel subgroup.
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