Dual Families in Enveloping Algebras

Matthieu Deneufchatel

Description

Dual Families in Enveloping Algebras Matthieu Deneufchatel, G. H. E. Duchamp et H. N. Minh Laboratoire d'Informatique de Paris Nord, Universite Paris 13 SLC 68, Ottrott

• enveloping algebras


• transport operators


• dual families


• universite de paris


• lipn - p13


• write schutzenberger's factorization

Sujets

• Université de Paris

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 16
• Langue: English

Extrait

Dual Families in Enveloping Algebras
Matthieu Deneufchaˆtel, G. H. E. Duchamp et H. N. Minh
Laboratoire d’Informatique de Paris Nord,
Universit´e Paris 13
SLC 68, OttrottMotivation
Plan
1 Motivation
2 Notations
3 General case
4 Some convergence remarks for physicists
5 Case of the free algebra
M. Deneufchaˆtel (LIPN - P13) Dual families 26/03/2012 2 / 29Motivation
Context :
Duality in Lie and enveloping algebras;
Numerical experimentations in the free algebra.
M. Deneufchˆatel (LIPN - P13) Dual families 26/03/2012 3 / 29Motivation
Context :
Duality in Lie and enveloping algebras;
Numerical experimentations in the free algebra.
Goals :
Discover the properties of the dual family of a basis from the
properties of the basis;
Write Schu¨tzenberger’s factorization.
M. Deneufchaˆtel (LIPN - P13) Dual families 26/03/2012 3 / 29Motivation
Context :
Duality in Lie and enveloping algebras;
Numerical experimentations in the free algebra.
Goals :
Discover the properties of the dual family of a basis from the
properties of the basis;
Write Schu¨tzenberger’s factorization.
Applications of Schut¨ zenberger’s factorization :
Polysystems and non linear differential equations (factorization of
transport operators);
Polyzetas and renormalization of divergent polyzetas.
M. Deneufchaˆtel (LIPN - P13) Dual families 26/03/2012 3 / 29Notations
Plan
1 Motivation
2 Notations
3 General case
4 Some convergence remarks for physicists
5 Case of the free algebra
M. Deneufchaˆtel (LIPN - P13) Dual families 26/03/2012 4 / 29Notations Multiindex notation
I a totally ordered set for <.
A an algebra with unit 1 .A
(I)If Y = (y ) is a totally ordered family inA and 2N ,i i2I
i i i 1 2 kY = y y y
i i i1 2 k
8J =fi ;i i g; i > i > > i such that supp() J.1 2 k 1 2 k
(I)If (e ) denotes the canonical basis ofN (e (j) = ), one has :i i2I i ij
eiY = y .i
Typically : I !Lyn(X);<!< .lex
M. Deneufchaˆtel (LIPN - P13) Dual families 26/03/2012 5 / 29Notations Poincar´e-Birkhoff-Witt theorem
Letg a k-Lie algebra and B = (b ) an ordered basis (for a total order <i i2I
on I) ofg.
Poincar´e-Birkhoff-Witt Basis
The elements
B ; = ( ;:::; ) with i > > i ;i i 1 p1 p
form a basis ofU (g) (called PBW basis).
M. Deneufchˆatel (LIPN - P13) Dual families 26/03/2012 6 / 29Notations Atomic elements
(B ) a basis, hji a scalar product.(I) 2N
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