Dual Families in Enveloping Algebras M. Deneufchˆatel, G. H. E. Duchamp and V. Hoang Ngoc Minh Laboratoire d’Informatique de Paris Nord, CNRS UMR 7030, Universit´e Paris 13, France matthieu.deneufchatel, ghed, minh @lipn.univ-paris13.fr

General Framework

Direct Problem: If Pα is a PBW basis and Sα its dual basis (hSα |Pβ i = δα,β ), a general theorem ([2]) states that Sh¨utzenberger’s factorisation holds. Converse Problem:

Notation: If Y = (yi)i∈I is a totally ordered family in an algebra A and α ∈ N(I), one defines Y α by

• Dualizing a Radford basis, what are the conditions for the factorisation to hold in its resolution of unity form? A necessary condition is that the basis obtained by dualization is of PBW type.

Introduction Given two bases (Pα)α∈N(I) and (Sα)α∈N(I) in duality for the scalar product h·|·i, Sch¨utzenberger’s factorisation → X Y Sα ⊗ Pα = (1) eSei ⊗Pei i∈I

α∈N(I)

is a well-known relation that holds under certain conditions in the case of every enveloping algebra. → Y X Sα ⊗ Pα = eSei ⊗Pei through the We are interested in its resolution of unity form: Id = α∈N(I)

i∈I

(

V ∗ ⊗ V → Endfinite(V ) mapping Φ : , form that adds some constraints on the bases. f ⊗ v 7→ Φ(f ⊗ v) : b 7→ f (b) · v We are also interested in Radford bases defined as transcendence bases of the shuffle algebra.

α(i ) α(i1) α(i2) yi2 · · · yik k

(2)

yi1

for every subset J = {i1, i2 · · · ik } , i1 > i2 > · · · > ik , of I which contains the support of α. Let g be a k-Lie algebra and B = (bi)i∈I be an ordered basis of it. Theorem 1 [Poincar´e-Birkhoff-Witt] The elements B α, α ∈ N(I), form a basis of U (g). A basis Pα has Poincar´e-Birkhoff-Witt type if, ∀ α ∈ N(I),

Y

• There exist Radford bases which are not dual to a PBW basis as shown by our counter-example (see below). Lemma 2 [DDM’12] Let (Sα)α∈N(I) be a basis of U (g) in duality with a basis (B α)α∈N(I) : hSα |B β i = δαβ . Then B β = B [β] (B is of Poincar´e-Birkhoff-Witt type) if and only if ∀i ∈ I, ∀β ∈ N(I), |β| ≥ 2, hSei |B β i = 0.

Peαi i = Pα.

i∈N

Counter-Example

′

Sw =

Classical Case: Shuffle Algebra ([1])

w  Pℓ1 , Pℓ2 Pw =   P α1 . . . P αk ℓ ℓ 1

k

if |w| = 1; if w = ℓ ∈ Lyn(X) and (ℓ1, ℓ2) = σ(ℓ); αk 1 . . . ℓ if w = ℓα ik with ℓ1 > · · · > ℓk . i1

(where σ(ℓ) denotes the standard factorization of ℓ). Dual basis:  w if |w| = 1;     xSu if w = xu and w ∈ Lyn(X); Sw = S α1 . . . S αk  ℓ i1 ℓ ik  αk  1  if w = ℓα . . . ℓ (decreasing factorization) . i i 1 k α1 ! . . . αk !

(4)

(5)

Thus X

w⊗w =

w∈X ∗

ց Y

Sℓ

α1 1

′

. . . Sℓ αn

α1 ! . . . αn !

Theorem 4 [DDM’12] Let P belong to khXi and ℓ ∈ Lyn(X). Then  X  P =ℓ+ hP |uiu;    ′ ℓ · · · > ℓ , ℓ ∈ Lyn(X) . 1 . . . ℓ if w = ℓα n i 1 n 1

n

Ba2b2 = a2b2 − 2abab + 2baba − b2a2.

Applications of Sch¨utzenberger’s factorisation

• Renormalization of divergent polyzetas.

′

Stuffle Algebra

if ℓ ∈ Lyn(X);

ℓ

′

hSα|Pβ i = δα,β .

Theorem 3

    

PBW Basis:   

(3)

(9)

Open Problem: Is there a pair of good dual bases? More precisely, is it possible to find the dual family of the basis defined by similar relations as we used for the family Sw (5) but with the stuffle product instead of the shuffle product?

Numerical Experimentations with SageMath (10) Two Sage worksheets present our code: • Shuffle algebra: http://sagenb.org/home/pub/4504/

is not of Poincar´e-Birkhoff-Witt type. Therefore Bw ∗ w∈X

• Stuffle algebra: http://sagenb.org/home/pub/4519/

Theorem 5 [DDM’12] Recursive Computation of the elements Bℓ′ , ℓ ∈ Lyn(X): Bℓ′ m = Pℓm ; Bℓ′ m−1 = Pℓm−1 − hPℓm−1 |ℓmiBℓ′ m ; .. Bℓ′ m−k = Pℓm−k −

k−1 X

hPℓm−j |ℓm−k iBℓ′ m−j .

j=0

(11)

References [1] Christophe Reutenauer, Free Lie Algebras. London Math. Soc. Monogr. (N.S.), 7, 1993. [2] Guy M´elan¸con, Christophe Reutenauer, Lyndon Words, Free Algebras and Shuffles. Can. J. Math., Vol. XLI, No. 4, 1989, pp. 577-591.