Déformation des groupes de Lie-Poisson riemanniens ab 39 EURO Déformations non commutatives
In this book we'll be using results and technics from deformation quantization of Poisson manifold theory in the sense Kontsevich and Cattaneo-Felder. The goal is to make suitable adaptations in order to use them in the Lie algebra case. This way we confront old problems of Lie theory and non commutative harmonic analysis. The first chapter is a detailed introduction to the part of the theory on (nilpotent) Lie groups and Lie algebras that we need. The second one is also a detailed introduction on deformation (bi)quantization and tools that we'll use in the sequence. Towards the end of chapter 2 we explain how these results will be used to prove theorems in the Lie case and introduce some central objects of study. Chapter 3 contains a detailed proof of a non-canonical isomorphism between a well known algebra of invariant differential operators and the corresponding to these data reduction algebra from deformation quantization. In chapter 4 the question of equivalence between characters from deformation quantization and harmonic analysis on Lie groups is answered positively. Finally in chapter 5 a central worked out example provides an overview of the above put in action.
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Operad theory is a field of abstract algebra concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. Operads generalize the various associativity properties already observed in algebras and coalgebras such as Lie algebras or Poisson algebras by modeling computational trees within the algebra. Algebras are to operads as group representations are to groups. Originating from work in algebraic topology by Boardman and Vogt, and J. Peter May (to whom their name is due), it has more recently found many applications, drawing for example on work by Maxim Kontsevich on graph homology.
High Quality Content by WIKIPEDIA articles! In differential geometry, the Schouten Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two different versions, both rather confusingly called by the same name. The most common version is defined on alternating multivector fields and makes them into a Gerstenhaber algebra, but there is also another version defined on symmetric multivector fields, which is more or less the same as the Poisson bracket on the cotangent bundle. It was discovered by Jan Arnoldus Schouten (1940, 1953) and its properties were investigated by his student Albert Nijenhuis (1955). It is related to but not the same as the Nijenhuis Richardson bracket and the Frölicher Nijenhuis bracket.
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In a field of mathematics known as differential geometry, the Courant bracket is a generalization of the Lie bracket from an operation on the tangent bundle to an operation on the direct sum of the tangent bundle and the vector bundle of p-forms. The case p = 1 was introduced by Theodore James Courant in his 1990 doctoral dissertation as a structure that bridges Poisson geometry and presymplectic geometry, based on work with his advisor Alan Weinstein. The twisted version of the Courant bracket was introduced in 2001 by Pavol Severa, and studied in collaboration with Weinstein.
High Quality Content by WIKIPEDIA articles! In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of G on the space L2(G/ ) of square-integrable functions, where G is a Lie group and a cofinite discrete group. The character is given by the trace of certain functions on G.The simplest case is when is cocompact, when the representation breaks up into discrete summands. Here the trace formula is an extension of the Frobenius formula for the character of an induced representation of finite groups. When is the cocompact subgroup Z of the real numbers G=R, the Selberg trace formula is essentially the Poisson summation formula.
The book consists of three chapters. In chapter I, we introduce some notions and definitions for basic concepts of the theory of integrable bi-Hamiltonian systems. Several open problems related to our main results are also mentioned in this part. In chapter II. We applied the so-called Jordan-Kronecker decomposition theorem to study algebraic properties of the pencil generated by two constant compatible Poisson structures on a vector space. In particular, we study the linear automorphism group that preserves the pencil. In classical symplectic geometry, many fundamental results are based on the symplectic group, which preserves the symplectic structure. Therefore in the theory of bi-Hamiltonian structures, we hope the linear automorphism group also plays a fundamental role. In chapter III, We describe the Lie group of linear automorphisms of the pencil, and obtain an explicit formula for the dimension of the Lie group and discuss some other algebraic properties such as solvability and Levi-Malcev decomposition.