Lectures In Geometry: Lie Groups and Lie Algebras (semester V)

Argumento de Lectures In Geometry: Lie Groups and Lie Algebras (semester V)

The theory of Lie groups relies on Cartan's theorem on the equivalence of the category of simply connected Lie groups to that of Lie algebras. This book presents the proof of the Cartan theorem and the main results. The branches of the theory of Lie groups which rest on the Cartan theorem remain outside the limits of our exposition. The theory of Lie algebras has been developed to an extent necessary for the Cartan theorem to be proved.

This book like the previous ones of this series* is a nearly faithful record of the Lectures delivered by the author at Moscow University to students (and postgraduates) of the Faculty of Mathematical Mechanics. However, while books I and II were based on Lectures of a compulsory course, this book is a record of an elective course, which makes it essentially different in a number of respects.0Preface.

Lecture 1 Smooth and topological groups. Relaxing the conditions denning Lie groups. Examples of Lie groups. Cayley transformation. Further examples of Lie groups. Connected and arcwise connected spaces and groups. Reduction of any smooth groups to connected groups. Examples of connected Lie groups.

Lecture 2 Left-invariant vector fields. Parallelizability of Lie groups. Integral curves of left-invariant vector fields and one-parameter subgroups. Lie functor. An example: a group of in-vertible elements of an associative algebra. Functions with values in an associative algebra. One-parameter subgroups of the group G (At).

Lecture 3 Matrix Lie groups admitting the Cayley construction. A generalization of the Cayley construction. Groups possessing In-images. Lie algebras. Examples of Lie algebras. Lie algebras of vector fields. The Lie algebra of a Lie group. An example: the Lie algebra of a group of invertible elements of an associative algebra. Locally isomorphic Lie groups. Local Lie groups. Lie functor on the category of local Lie groups.

Lecture 4 The exponent of a linear differential operator. A formula for the values of smooth functions in the normal neighbourhood of the identity of a Lie group. A formula for the values of smooth functions on the product of two elements. The convergence of a Campbell-Hausdorff series. The reconstruction of a local Lie group from its Lie algebra. Operations in the Lie algebra of a Lie group and one-parameter subgroups. Differentials of internal automorphisms. The differential of an exponential mapping. Canonical coordinates. The uniqueness of the structure of a Lie group. Groups without small subgroups and Hilbert's fifth problem.

Lecture 5 Free associative algebras. Free Lie algebras. The basic lemma. The universal enveloping algebra. The embedding of a Lie algebra into its universal enveloping algebra. Proof of the fact that the algebra I is free. The Poin-care-Birkhoff-Witt theorem. Tensor products of vector spaces and of algebras. Hopf algebras.

Lecture 6 The Friedrichs theorem. The proof of Statement B of Lecture 4. The Dynkin theorem. The linear part of a Camp-bell-Hausdorff series. The convergence of a Campbell-Hausdorff series. Lie group algebras. The equivalence of the categories of local Lie groups and of Lie group algebras. Isomorphism of the categories of Lie group algebras and of Lie algebras. Lie's third theorem.

Lecture 7 Local subgroups and subalgebras. Invariant local subgroups and ideals. Local factor groups and quotient algebras. Reducing smooth local groups to analytic ones. Pfafnan systems. Subfiberings of tangent bundles. Integrable subfiberings. Graphs of a Pfaffian system. Involutory subfiberings. The complete univalence of a Lie functor. The involuted-ness of integrable subfiberings. Completely integrable subfiberings.

Lecture 8 Coverings. Sections of coverings. Pointed coverings. Coamal-gams. Simply connected spaces. Morphisms of coverings. The relation of quasi-order in the category of pointed coverings. The existence of simply connected coverings. Questions of substantiation. The functorial property of a universal covering.

Lecture 9 Smooth coverings. Isomorphism of the categories of smooth and topological coverings. The existence of universal smooth coverings. The coverings of smooth and topological groups. Universal coverings of Lie groups. Lemmas on topological groups. Local isomorphisms and coverings. The description of locally isoraorphic Lie groups.

Lecture 10 Local isomorphisms and isomorphisms of localizations. The Cartan theorem. A final diagram of categories and functors. Reduction of the Cartan theorem. The globaliza-bility of embeddable local groups. Reducing the Cartan theorem to the Ado theorem.

Lecture 11 Submanifolds of smooth manifolds. Subgroups of Lie groups. Integral manifolds of integrable subfiberings. Maximal integral manifolds. The idea of the proof of Theorem 1. The local structure of Submanifolds. The uniqueness of the structure of a locally rectifiable submanifold with a countable base. Submanifolds of manifolds with a countable base. Connected Lie groups have a countable base. The local rectifiability of maximal integral manifolds. The proof of Theorem 1.

Lecture 12 Alternative definitions of a subgroup of a Lie group. Topological subgroups of Lie groups. Closed subgroups of Lie groups. Algebraic groups. Groups of automorphisms of algebras. Groups of automorphisms of Lie groups. Ideals and invariant subgroups. Quotient manifolds of Lie groups. Quotient groups of Lie groups. The calculation of fundamental groups. The simple-connectedness of groups SU (n) and Sp (n). The fundamental group of a group.

Lecture 13 The Clifford algebra of a quadratic functional, I2-graduation of a Clifford algebra. More about tensor multiplication of vector spaces and algebras. Decomposition of Clifford algebras into a skew tensor product. The basis of a Clifford algebra. Conjugation in a Clifford algebra. The centre of a Clifford algebra. A Lie group Spin(n). The fundamental group of a group S0(n). Groups Spin(n) with n << 4. Ho-momorphism x- The group Spin(6). The group Spin(5). Matrix representations of Clifford algebras. Matrix representations of groups Spin(n). Matrix groups in which groups Spin(n) are represented. Reduced representations of groups Spin(n). Additional facts from linear algebra.

Lecture 14 Doubling of algebras. Metric algebras. Normed algebras. Automorphisms and differentiations of metric algebras. Differentiations of a doubled algebra. Differentiations and automorphisms of the algebra t. The algebra of octaves. The Lie algebra 02 . Structural constants of the Lie algebra p2C. Representation of the Lie algebra p2C by generators and relations.

Lecture 15 Identities in the octave algebra Ca. Subalgebras of the octave algebra Ca. The Lie group G2 . The triplicity principle for the group Spin(8). The analogue of the triplicity principle for the group Spin(9). The Albert algebra A. The octave projection plane Contents.

Lecture 16 Scalar products in the algebra Al. Automorphisms and differentiations of the algebra Al. Adjoint differentiations of the algebra Al. The Freudenthal theorem. Consequences of the Freudenthal theorem. The Lie group F4. The Lie algebra f4. The structure of the Lie algebra f4C.

Lecture 17 Solvable Lie algebras. The radical of a Lie algebra. Abe-lian Lie algebras. The centre of a Lie algebra. Nilpotent Lie algebras. The nilradical of a Lie algebra. Linear Lie nilal-gebras. The Engel theorem. Criteria of nilpotency. Linear irreducible Lie algebras. Reductive Lie algebras. Linear solvable Lie algebras. The nilpotent radical of a Lie algebra.

Lecture 18 Trace functional. Killing's functional. The trace functional of a representation. The Jordan decomposition of a linear operator. The Jordan decomposition of the adjoint operator. The Cartan theorem on linear Lie algebras. Proving Cartan's criterion for the solvability of a Lie algebra. Linear Lie algebras with a nonsingular trace functional. Semisimple Lie algebras. Cartan's criterion for semisim-plicity. Casimir operators.

Lecture 19 Cohomologies of Lie algebras. The Whitehead theorem. The Fitting decomposition. The generalized Whitehead theorem. The Whitehead lemmas. The Weyl complete reduc-ibility theorem. Extensions of Abelian Lie algebras.

Lecture 20 The Levi theorem. Simple Lie algebras and simple Lie groups. Cain and unimodular groups. Schur's lemma. The centre of a simple matrix Lie group. An example of a non-matrix Lie group. De Rham cohomologies. Cohomologies of the Lie algebras of vector fields. Comparison between the cohomologies of a Lie group and its Lie algebra.

Lecture 21 Killing's functional of an ideal. Some properties of differentiations. The radical and nilradical of an ideal. Extension of differentiations to a universal enveloping algebra. Ideals of finite codimension of an enveloping algebra. The radical of an associative algebra. Justification of the inductive step of the construction. The proof of the Ado theorem. Conclusion.

Supplement to the English Translation. Proof.of the Cartan theorem by V. V. Gorbatsevich)
Bibliography Subject Index.