## Lectures in Geometry – Semester 5 Lie Groups and Lie Algebras – Postnikov

In this post, we will see the book Lectures in Geometry – Semester 5 Lie Groups and Lie Algebras by M. Postnikov. This book is the first one of a five part Lectures in Geometry series. So far we have volumes 1, 2 and 5, volumes 3 and 4 are missing. If anyone has access to volumes 3 and 4 please consider posting them. This textbook like the rest of the series is compiled from lectures delivered by the author to students at Moscow State University. The audience was made up of final year students and postgraduates, and as a consequence, a certain level of knowledge (for example, the properties of manifolds) is assumed. The lectures fall into five parts. The first covers the basic concepts of Lie groups, Lie algebras, and the Lie algebra of a given Lie group. The second part deals with “ locality theory”, the third generalizes these ideas. The first three sections could be used as a foundation course in Lie algebras for beginners. The fourth part considers Lie subgroups and Lie factor groups. The final part has a purely algebraic character, and is practically independent of the preceding four parts; it considers a proof of Ado’s theorem, which is interesting in itself. This book will be useful for mathematics students taking courses in Lie algebras, because it contains all the major results and proofs in compact and accessible form.

The book was translated from Russian by by Vladimir Shokurov and was published in 1986 by Mir.

You can get the book here.

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# Contents

Preface 10

## Lecture 1 15

Smooth and topological groups. Relaxing the conditions defining 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 33

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 invertible elements of an associative algebra. Functions with values in an associative algebra. One-parameter subgroups of the group G (𝓐)

## Lecture 3 51

Matrix Lie groups admitting the Cayley construction. A generalization of the Cayley construction. Groups possessing ln-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 71

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 Campbell-Hausdorff series and Dynkin polynomials. 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 98

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 <X> is free. The Poincaré-Birkhoff-Witt theorem. Tensor products of vector spaces and of algebras. Hopf algebras

## Lecture 6 116

The Friedrichs theorem. The proof of Statement B of Lecture 4. The Dynkin theorem. The linear part of a Campbell-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 7 134

Local subgroups and sub-algebras. Invariant local subgroups and ideals. Local factor groups and quotient algebras. Reducing smooth local groups to analytic ones. Pfaffian systems. Sub-fiberings of tangent bundles. Integrable sub-fiberings. Graphs of a Pfaffian system. Involutory sub-fiberings. The complete univalence of a Lie functor. The involutedness of integrable sub-fiberings. Completely integrable sub-fiberings

## Lecture 8 159

Coverings. Sections of coverings. Pointed coverings. Coamalgams. 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 9 183

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 isomorphic Lie groups

## Lecture 10 197

Local isomorphisms and isomorphisms of localizations. The Cartan theorem. A final diagram of categories and functors. Reduction of the Cartan theorem. The globalizability of embeddable localigroups. Reducing the Cartan theorem to the Ado theorem

## Lecture 11 208

Sub-manifolds of smooth manifolds. Subgroups of Lie groups.
Integral manifolds of integrable sub-fiberings. Maximal integral manifolds. The idea of the proof of Theorem 14. The local structure of sub-manifolds. The uniqueness of the structure of a locally rectifiable sub-manifold with a countable base. Sub-manifolds 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 12 228

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 U(n)

## Lecture 13 247

The Clifford algebra of a quadratic functional. ℤ_2-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 SO(n). Groups Spin(n) with n < 4. Homomorphism 𝜒. 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 14 286

Doubling of algebras. Metric algebras. Normed algebras. Automorphisms and differentiations of metric algebras. Differentiations of a doubled algebra. Differentiations and automorphisms of the algebra ℍ. The algebra of octaves. The Lie algebra g_2ℂ. Structural constants of the Lie algebra g_2ℂ. Representation of the Lie algebra g_2ℂ, by generators and relations

## Lecture 15 306

Identities in the octave algebra ℂ_a. Sub-algebras of the octave algebra ℂa. The Lie group G_2. The SH teat principle for the group Spin(8). The analogue of the triplicity principle for the group Spin(9). The Albert algebra 𝔸l. The octave projection plane

## Lecture 16 327

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 F_4. The Lie algebra f_4. The structure of the Lie algebra fℂ_4

## Lecture 17 346

Solvable Lie algebras. The radical of a Lie algebra. Abelian Lie algebras. The centre of a Lie algebra. Nilpotent Lie algebras. The nil-radical of a Lie algebra. Linear Lie nil-algebras. The Engel theorem. Criteria of nil-potency. Linear irreducible Lie algebras. Reductive Lie algebras. Linear solvable Lie algebras. The nilpotent radical of a Lie algebra

## Lecture 18 360

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 semisimplicity. Casimir operators

## Lecture 19 377

Cohomologies of Lie algebras. The Whitehead theorem. The Fitting decomposition. The generalized Whitehead theorem. The Whitehead lemmas. The Weyl complete reducibility theorem. Extensions of Abelian Lie algebras

## Lecture 20 391

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 cats 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 21 404

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 co-dimension 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) 418

Bibliography 431
Subject Index 433 