Math 251B: Linear Algebraic Groups.

Spring 2017

 Lectures: T-Th 12:30 PM--1:50 PM APM 7218 Office Hour: Send me an e-mail.
Send me an e-mail, we can meet and discuss math (possibly in a coffee shop).

Course description:

The following is a tentative list of topics to be covered:

• Understanding algebraic groups as representable group functors.
• Understanding the connection between algebraic groups and commutative Hopf algebras.
• Linearity of affine algebraic groups.
• Proving the existence and the uniquness of the quotient space G/H where H is a closed subgroup of an affine algebraic group G.
• Proving a version of the first isomorphism theorem for affine algebraic groups.
• Defining the Lie algebra of an algebraic group and the adjoint representation.
• Studying structure of a torus and a solvable group.
• Lie-Kolchin Theorem.
• Studying Borel subgroups.
Along the way, we have to cover some of the needed background from Algebraic Geometry:
• Algebraic sets and their connection with k-algebras of finite type.
• Nullstellensatz theorem.
• Variety: sheaf of regular functions; ringed space; separation axiom.
• Chevalley Theorem: constructible sets are mapped to constructible sets.
• Tangent spaces- three points of view for affine varieties: dual numbers; derivations; dual of mx/mx2.
• Tanget spaces of varieties.
• Module of differentials.
• Simple points: generically simple; separability criterion.
• A version of Zariski's main theorem.
• Complete varieties.

Prerequisite:

will be kept to a minimum.

Resources:

I will be using the following books and lecturenotes:

• Springer, Linear algebraic groups (2nd edition), Birkhauser.
• Humphreys, Linear algebraic groups, Springer-Verlag.
• Borel, Linear algebraic groups (2nd edition), Springer-Verlag.
• Waterhouse, Introduction to affine group schemes, Springer-Verlag.

Notes related to lectures:
• 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 DerR(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; Ωk[a]/k=0 if and only if a is algebraically separable over k; ΩF/R is isomorphic to ΩA/k tensor F if F is the field of fractions of the integral domain A; dim ΩF/k is at least trdegk(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 ΩE'/k tensor E embeds into ΩE/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 GLn(k), then Lie(G) is a subalgebra of gln(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

• the existence and the uniquness of the quotient of G by H. That means: there is a G-homogeneous space G/H:=X and a point x in X such that for any pair (Y,y) of a G-homogeneous Y and a point y in Y, if the stabilizer of y contains H, then there is a unique G-equivarient morphism f:X-->Y which sends x to y.
• if G is connected, then the quotient map G-->G/H is separable.
• a model of G/H is the set of left cosets of H in G, with the quotient topology, and its sheaf of regular functions is given by the H-invariant regular functions on G:

OG/H(U):=OG({g in G| gH in U}) H.

We pointed out that this implies G/H is affine if and only if k[G]H is of finite type and separates points of G/H.
• G/H is quasi projective.
• if N is a closed normal subgroup of an affine algebraic group, then G/N is an affine algebraic group; and there is an algebraic group representation ρ:G --> GL(V) such that (1) ker(ρ)=N (2) ker(dρ)=Lie(N); and these conditions are equivalent to saying N(k[ε])=ker(G(k[ε])-->GL(V)(k[ε]));
• a version of first isomorphism theorem.

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• Jordan decomposition:

• additive, finite-dimensional version; various properties of such a decomposition.
• 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 GLn(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, Gs:={g in G| g is semisimple}, and Gu:={g in G| g is unipotent}. Then Gs and Gu are closed subgroups of G and Gsx Gu --> G, f(s,u):=su is an isomorphism.
• G is called diagonalizable if it is isomorphic to a closed subgroup of GL1(k)n for some n; and it is called a torus if it is isomorphic to GL1(k)n for some n.
• We proved the following are equivalent
• G is commutative and consists of semisimple elements;
• G is diagonalizable;
• X*(G):=Hom(G,GL1(k)) is a finitely generated abelian group; and k[G] is the group algebra of X*(G);
• for any algebraic group representation ρ:G--> GL(V) there is a finite subset Φ of X*(G); such that

V is the direct sum of Vχ where χ varies in Φ, and Vχ={v in V| ρ(g)(v)=χ(g) v}.

• 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 GM. It is outlined why M=X*(GM), and deduced that GM is a diagonalizable group; One can see that for a diagonalizable group G we have G=GX*(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 NG(H)/CG(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 GLn(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 GLn(k); then for some x in GLn(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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