Technical Books

The aim of this note is to develop the basic general theory of Lie algebras to give a first insight into the basics of the structure theory and representation theory of semi simple Lie algebras. Topics covered includes, Group actions and group representations, General theory of Lie algebras, Structure theory of complex semisimple Lie algebras, Cartan subalgebras, Representation theory of complex semisimple Lie algebras, Tools for dealing with finite dimensional representations.

This note focus on the socalled matrix Lie groups since this allows us to cover the most common examples of Lie groups in the most direct manner and with the minimum amount of background knowledge. Topics covered includes, Matrix Lie groups, Topology of Lie groups, Maximal tori and centres, Lie algebras and the exponential map, Covering groups.

This note explains the following topics, Representations of sl2, Structure and classification of simple lie algebras, Structure theory of semisimple lie algebras and root systems.

This note contains below topics, Classical and quantum OlshanetskyPerelomov systems for finite Coxeter groups, The rational Cherednik algebra, The MacdonaldMehta integral, Parabolic induction and restriction functors for rational Cherednik algebras, The KnizhnikZamolodchikov functor, Rational Cherednik algebras and Hecke algebras for varieties with group actions, Symplectic reflection algebras, CalogeroMoser spaces, Quantization of ClaogeroMoser spaces.

This note contains below topics, Lie Groups in tangent categories, The Lie functor of a group object, The representability of the Lie functor, The differential Lie algebra of a Lie group, The Lie groupLie algebra correspondence.

This note contains below topics, A summary of the KillingCartan classification via roots, A first linear model of split G2, Understanding G2 better, Generic 3 forms in F7, G2 and the spheres in dimension 6, G2 and spinors.

This note contains below topics, Review of algebraic schemes over a field, Algebraic groups over a field geometric properties, Examples of algebraic groups, Group theory, Representations, Some basic constructions, Tannaka duality and applications, The Lie algebra of an algebraic group, Groups of multiplicative type, Algebraic groups acting on schemes, Homogeneous spaces, Tori acting on schemes, Unipotent groups, Solvable algebraic groups, Borel and Cartan subgroups, Isogenies and universal covers, Semisimple and reductive groups, Split semisimple groups and their root systems.

This book is an introduction to Lie algebras, algebraic groups, and Lie groups in characteristic zero, emphasizing the relationships between these objects visible in their categories of representations. This book contains below topics, Lie Algebras, Algebraic Groups, III Lie groups, A Arithmetic Subgroups.

This note introduces the definitions of the Ktheory groups such as The group K0, The group K1, The group K2 and Higher Ktheory groups.

The primary purpose of this note is to examine many of these Ktheoretic invariants, not from a historical point of view, but rather a posteriori, now that Ktheory is a mature subject.

This book explains the following topics, Topological Ktheory, Ktheory of C algebras, Geometric and Topological Invarients, THE FUNCTORS K1 K2, K1, SK1 of Orders and Grouprings, Higher Algebraic Ktheory, Higher Dimensional Class Groups of Orders and Group rings, Higher Ktheory of Schemes, Modm Higher Ktheory of exact Categories, Schemes and Orders, Profinite Higher Ktheory of Exact Categories, Schemes and Orders, Equivariant Higher Ktheory Together with Relative Generalizations, Interpretation in Terms of Grouprings.

This note covers the following topics, Some homotopy theory, Exact categories, Qconstruction, Fundamental groupoid, Waldhausens constructions, Additivity, The Ktheory spectrum, Products, Group completion, Q theorem, The defining acyclic map, Homotopy fibres, Resolution theorem, D�vissage, Abelian category localization, Coherent sheaves and open subschemes, Product formulas, Ktheory with finite coefficients, Homology, Ktheory of graded rings, Homotopy property, Rigidity, Ktheory of finite fields.

This lecture note covers the following topics,Projections and Unitaries, The K0Group for Unital C Algebras, K1Functor and the Index Map, Bott Periodicity and the Exact Sequence of KTheory, Tools for the computation of Kgroups.

This note covers the following topics, Recollections and preliminaries, Symmetric monoidal and stable categories, The group completion theorem and the K theory of finite fields, The K theory of stable categories.

These are lecture notes for an introductory course on Whitehead, Reidemeister, and Ray Singer torsions. This note contains below topics, Whitehead group, Torsion of a chain complex, Whitehead torsion for CW complexes, Reidemeister torsion, Lens spaces, Some applications and properties of torsions, Analytic torsion.