Jacques Helmstetter

Série de Hausdorff d'une algèbre de Lie et projections canoniques dans l'algèbre enveloppante

Soit A une algebre de Lie sur un corps K de caracteristique nulle. On etudie un calcul explicite de la serie de Hausdorff en passant par le calcul des projections canoniques

Meson Algebras of Order ≥3

Meson algebras of order 2 have already drawn much attention, and their study has brought plenty of interesting knowledge. This fact motivated the definition and the study of meson algebras of greater order. Unfortunately, these algebras finally proved to be disappointing; probably there is almost nothing to add to the information given in the present article. Almost all meson algebras of order ≥3 are trivial, and the only two cases that give nontrivial algebras, are completely described here.

The Group of Classes of Involutions of Graded Central Simple Algebras

Some properties of Clifford algebras are actually common properties of all graded central simple algebras A provided with an involution ρi; with ρ is associated a “complex divided trace” (a complex number r such that r 8 = 1), and thus all such involutions are classified by a cyclic group of order 8. Complex divided traces are also involved in the Brauer—Wall group of the field ℝ, and they bring efficiency and enlightenment in the study of bilinear forms on graded A-modules.

Relations Between Witt Rings and Brauer Groups

Let R be a commutative ring with unit element. To R we can associate the Witt ring W(R) which classifies the nondegenerate quadratic forms Q on finitely generated projective R-modules M and the Brauer group Br(R) which classifies the Azumaya algebras A over R, that is, A is a finitely generated projective R-module and, if A’ rev denotes the reversed algebra with multiplication (x, y) ↦ yx, the algebra A ⊗R A rev is canonically isomorphic to the algebra EndR(A) Let us recall that each nondegenerate quadratic form Q on a R-module M has an image in W(R), here denoted by w(M, Q) or w(Q), and that this image fullfils the following properties: w(Q) + w(Q’) = w(Q ⊗ Q’) (orthogonal sum), −w(Q) = w(−Q) and w(Q)w(Q’) = w(Q ⊗ Q’) (tensor product), for all nondegenerate quadratic forms Q and Q’ Besides, each Azumaya algebra A has an image b(A) in the Brauer group Br(R) and b(A)b(A’) = b(A ⊗R A’), b(A)-1 = b(Arev), for all R-Azumaya algebras A and A’. Moreover, the unit element in the Brauer group Br(R) is the image of the ring R.

Application of Clifford Algebras to *-Products

Some theorems involving symmetric bilinear forms and Clifford algebras may be “translated” to theorems involving symplectic forms and Moyal-products of distributions. The translation of some properties of Clifford groups is carried out.