Introduction to lie algebras and representation theory

In this chapter we describe the six families of so-called ‘classical’ simple groups. These are the linear, unitary and symplectic groups, and the three families of orthogonal groups. All may be obtained from suitable matrix groups G by taking G′/Z(G′). Our main aims again are to define these groups, prove they are simple, and describe their automorphisms, subgroups and covering groups.

The classical groups

I have read with much interest and pleasure “What is tensor analysis?” by Doctor Banesh Hoffmann and his article is the best elementary analysis of the subject which I have seen.

What is tensor analysis

In Chap. 9, we studied in detail the operad Ass encoding the associative algebras. It is a paradigm for nonsymmetric operads, symmetric operads, cyclic operads. In this chapter we present several other examples of operads. First, we present the two other “graces”, the operads Com and Lie encoding respectively the commutative (meaning commutative and associative) algebras, and the Lie algebras. Second, we introduce more examples of binary quadratic operads: Poisson, Gerstenhaber, pre-Lie, Leibniz, Zinbiel, dendriform, magmatic, several variations like Jordan algebra, divided power algebra, Batalin–Vilkovisky algebra. Then we present various examples of operads involving higher ary-operations: homotopy algebras, infinite-magmatic, brace, multibrace, Jordan triples, Lie triples. The choice is dictated by their relevance in various parts of mathematics: differential geometry, noncommutative geometry, harmonic analysis, algebraic combinatorics, theoretical physics, computer science. Of course, this list does not exhaust the examples appearing in the existing literature. The reader may have a look at the cornucopia of types of algebras 2012 to find more examples.

Examples of Algebraic Operads

https://orbilu.uni.lu/bitstream/10993/14127/1/LieInfinityAlgebroids.pdf

On the category of Lie n-algebroids

https://orbilu.uni.lu/bitstream/10993/9369/1/Higher_Trace_and_Berezinian%28v3%29.pdf

Higher trace and Berezinian of matrices over a Clifford algebra

In physics and in mathematics Z2n-gradings, n ≥ 2, appear in various fields. The corresponding sign rule is determined by the “scalar product” of the involved Z2n-degrees. The Z2n-supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (respectively, odd) coordinates do not necessarily commute (respectively, anticommute) pairwise. In this article we develop the foundations of the theory: we define Z2n-supermanifolds and provide examples in the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the class of Z2•-supermanifolds is closed with respect to the tangent and cotangent functors. We explain that any n-fold vector bundle has a canonical “superization” to a Z2n-supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the Z2n-context.

The category of Z2n-supermanifolds

Geometry and Mathematical Physics

We prove Pursell-Shanks type results for the Lie algebra D(M ) of all linear differential operators of a smooth manifold M , for its Lie subalgebra D 1 (M ) of all linear first-order differential operators of M and for the Poisson algebra S(M )=P ol(T ∗ M ) of all poly- nomial functions on T ∗ M, the symbols of the operators in D(M ). Chiefly, however, we provide explicit formulas completely describing the automorphisms of the Lie algebras D 1 (M ) ,S (M )a ndD(M ). The initial objective of the present paper was to prove that the Lie algebra D(M ) of all linear differential operators D : C ∞ (M ) → C ∞ (M )o f as mooth manifoldM , determines the smooth structure of M. Beyond this conclusion, we present a description of all automorphisms of the Lie algebra D(M ) and even of the Lie subalgebra D 1 (M ) of all linear first-order differential operators of M and of the Poisson algebra S(M )=P ol(T ∗ M ) of polynomial functions on the cotangent bundle T ∗ M (the symbols of the operators in D(M )), the automorphisms of the two last algebras of course being canonically related with those of D(M ). In each situation we obtain an explicit formula, for instance - in the case of D(M ) - in terms of the automorphism of D(M ) implemented by a diffeomorphism of M , the conjugation-automorphism of D(M ) and the automorphism of D(M ) generated by the derivation of D(M ) associated to a closed 1-form of M. In the first part of our work, the approach is purely algebraic. In § 2 ,w e heaveD(M )a nd S(M ) on a general algebraic level and define the notions 'quantum Poisson algebra' D and 'classical Poisson algebra' S, the classical limit of D.I n§ 3 we show that if two (quantum or classical) Poisson algebras are isomorphic as Lie algebras, their 'basic algebras of functions' are isomorphic as associative algebras - an algebraic Shanks-Pursell type result, which naturally implies our previously described initial goal. The leading idea of the proof is the algebraic characterization, under a minimal

/pdf/automorphisms-of-quantum-and-classical-poisson-algebras-37lg98um8f.pdf

Norbert Poncin

Papers

On the category of Lie n-algebroids

Higher trace and Berezinian of matrices over a Clifford algebra

The category of Z2n-supermanifolds

Geometry and Mathematical Physics

Automorphisms of quantum and classical Poisson algebras