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 Seminaire Combinatoire & Algorithmes, LITIS

• transport operators


• dual families


• universite de paris


• lipn - p13


• write schutzenberger's factorization


• lie algebras

Sujets

• Université de Paris
• Lie algebra

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 19

Extrait

Dual Families in Enveloping Algebras
Matthieu Deneufchˆatel, G. H. E. Duchamp et H. N. Minh
Laboratoire d’Informatique de Paris Nord,
Universit´e Paris 13
S´eminaire Combinatoire & Algorithmes, LITISMotivation
Plan
1 Motivation
2 Notations
3 General case
4 Topology
5 Case of the free algebra
M. Deneufchˆatel (LIPN - P13) Dual families 22/03/2012 2 / 29Motivation
Context :
Duality in Lie algebras;
Numerical experimentations in Free algebras.
M. Deneufchaˆtel (LIPN - P13) Dual families 22/03/2012 3 / 29Motivation
Context :
Duality in Lie algebras;
Numerical experimentations in Free algebras.
Goals :
Discover the properties of the dual family of a basis from the
properties of the basis;
Write Schu¨tzenberger’s factorization.
M. Deneufchˆatel (LIPN - P13) Dual families 22/03/2012 3 / 29Motivation
Context :
Duality in Lie algebras;
Numerical experimentations in Free algebras.
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 22/03/2012 3 / 29Notations
Plan
1 Motivation
2 Notations
3 General case
4 Topology
5 Case of the free algebra
M. Deneufchˆatel (LIPN - P13) Dual families 22/03/2012 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 22/03/2012 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 22/03/2012 6 / 29Notations Atomic elements
(B ) a basis, hji a scalar product.(I) 2N
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