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Dual Families in Enveloping Algebras

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39 pages
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


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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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