Manipulating exponential products Instead of working with complicated concatenations of flows like z (t)= e t 9 t 8 (f 0 +f 1 +f 2 ) dt ◦ ...e t 2 t 1 (f 0 +f 1 −f 2 ) dt ◦ e t 1 0 (f 0 +f 1 +f 2 ) dt (p) it is desirable to rewrite the solution curve using a minimal number of vector fields f π k that span the tangent space (typically using iterated Lie brackets of the system fields f 0 ,f 1 ,...f m ) Coordinates of the first kind z (t)= e b 1 (t,u)f π 1 +b 2 (t,u)f π 1 +b 3 (t,u)f π 3 +...+b n (t,u)f π n (p) Coordinates of the second kind z (t)= e c 1 (t,u)f π 1 ◦ e c 2 (t,u)f π 1 ◦ e c 3 (t,u)f π 3 ◦ ... ◦ e c n (t,u)f π n (p) Using the Campbell-Baker-Hausdorff formula, this is possible, but a book-keeping nightmare. Moreover, the CBH formula does not use a basis, but uses linear combinations of all possible iterated Lie brackets. Yet, by the Jacobi identity (and anticommutativity), in ever Lie algebra e.g. [X, [Y, [X, Y ]]] + [Y, [[X, Y ],X ] ]+[[X, Y ], [X, Y ]]] = 0 and hence [X, [Y, [X, Y ]]] = [Y, [X, [X, Y ]]] Plan: • Work with bases for (free) Lie algebras. • Find useful formulae for the coefficients b k (t, u) or c k (t, u). 12
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Manipulating exponential products
Instead of working with complicated concatenations of flows like
z(t) = e∫ t9t8
(f0+f1+f2) dt ◦ . . . e∫ t2t1
(f0+f1−f2) dt ◦ e∫ t10 (f0+f1+f2) dt(p)
it is desirable to rewrite the solution curve using a minimal number of
vector fields fπkthat span the tangent space (typically using iterated
Lie brackets of the system fields f0, f1, . . . fm)
Morale: When working with repeated integrations by parts, omit“integrals and similar notational ballast”. Instead work purely
combinatorially in shuffle algebra.
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Product expansion of CF-series
Using Ree’s theorem the existence of exponential product expansions
of the CF-series is assured. for suitable bases B of L(X1, . . . Xm).
SCF (T, u) =←∏B∈B
exp (βB(T, u) B)
Recall remaining issues/questions:
• Need explicit basis for the free Lie algebra
• Want explicit formulae for the iterated integral coefficients
αB(T, u) and/or βB(T, u).
Using different bases for the free Lie algebra explicit formulae for the
dual bases (iterated integral functionals βB(T, u) have been redis-
covered several times in different contexts:
• Schutzenberger (1958), Seminaire Dubreil
• Sussmann (1986), Nonlinear control
• Melancon and Reutenauer (1989), Combinatorics
• Grayson and Grossman (1991),
Realizations of free nilpotent Lie algebras
=⇒ See historical slide
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Lazard elimination
Theorem [Lazard elimination]:
Suppose k a field of scalars, X is a set and c ∈ X .
Then the free Lie algebra Lk(X ) over k gener-
ated by X is the direct sum of the one-dimensional
subspace {λc:λ ∈ k} and of a Lie-subalgebra
of Lk(X ) that is freely generated by the set
{(adjc, b): b ∈ X \ {c}, j ≥ 0}.
This principle is at the heart of constructions involving Hall basesfor free Lie algebras, for Sussmann’s derivation of the exponentialproduct expansion by solving DEs by iteration, and thereby it is
closely connected to Zinbiel structures.
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Hall and Lyndon bases
Ph. Hall, 1930s, calculus of commutator groups
M. Hall, 1950s, first bases for free Lie algebras
Lyndon, 1950s, different (?) bases
Sirsov, 1950s, different (?) bases
Viennot, 1970s, only one kind of practical basis
A Hall set over a set X is any strictly ordered subset H of the free
magma M(X ) (i.e. the set of all parenthesized words, or labelled
binary trees) that satisfies
• X ⊆ H• Suppose a ∈ X . Then (w, a) ∈ H iff w ∈ H, w < a and
a < (w, a).
• Suppose u, v, w, (u, v) ∈ H.
Then (u, (v, w)) ∈ H iff v ≤ u ≤ (v, w) and u < (u, (v, w)).
Original Hall bases as in Bourbaki require that ordering be compat-
ible with the length. Viennot showed that is not necessary.
The image of a Hall set under the canonical map ϕ:M(X ) �→ Lk(X )
from the free magma into the free Lie algebra is a basis for Lk(X ).
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Lie brackets and formal brackets
Need to distinguish formal brackets and elements of a Lie algebra.
E.g., consider {x, y, (x, y), (y, (x, (x, y)))} ⊆ H ⊆ M({x, y}). Then
Compare to the mix of products from different algebras in Reutenauer’s
and Melancon’s formula for the dual-PBW bases.
In the shuffle algebra (A(X ), X ) the transposes of both the left
and right translation by a letter λa:w �→ aw, and �a:w �→ wa are
derivations. However:
On the Zinbiel algebra, only λ†a is a derivation, �†a is not.
λ†a(w ∗ z) = (λ†aw) ∗ z + w ∗ (λ†az)
�†a(w ∗ z) = w ∗ (�†az)
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Realization of free Zinbiel algebra
Compare the standard realization of polynomial algebras:
k[X1, . . . Xn] polynomials w/ coeff’s in k
↓k[x1, . . . xn] polynomial functions kn �→ k
= the subalgebra of Map(kn,k) = kkn
generated by the projections
xk = πk: (p1, . . . , pn) �→ pk
Similarly realize the free Zinbiel algebra as a Zinbiel algebra of time-
varying scalars. E.g for an index set X of letters
U = ACloc([0,∞),R)
πa:UX �→ U , πa({ub: b ∈ X}) = ua
IIF(X ) ⊆ Map(UX ,U)
subalgebra generated by projections πa, a ∈ XTheorem[Kawski and Sussmann]: The map ΥΥ(:C(X ) �→ IIF(X )
defined by ΥΥ(: a �→ πa is a Zinbiel algebra isomorphism.
Zinbiel algebra surjective homomorphism is rather clear by now. The
one-to-one-ness requires a sufficiently rich coefficient ring and some
analysis (Nagano’s theorem . . . ).
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CF series as identity map
The Chen Fliess series of iterated integral functionals cor-responds to the identity map under the Zinbiel algebraisomorphism ΥΥ, i.e. it is natural object.
idA ∼= ∑w w ⊗ w
idA⊗ΥΥ←−−→ ∑
w w ⊗ ΥΥ(w)
‖ ‖Schutzenberger, Sussmann
Melancon & R.
Combinatorics Diff Equns proof
‖ ‖←∏H exp ([H ]⊗ SH)
idA⊗ΥΥ←−−→ ←∏
H exp ([H ]⊗ βH)
Hom(A,A) ∼= A⊗A idA⊗ΥΥ←−−→ A⊗AIIF
free Zinbiel Zinbiel.algebra iterated integral
algebra isomorphism functionals
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Koszul duality and Leibniz operad
The Zinbiel “operad” is “dual” to the Leibniz “operad”.
(“pre-Lie algebra structure”, or “noncommutative Lie algebra”)
(Left) Leibniz identity
v � (w � z) = (v � w) � z + v � (w � z) for all v, w, z (3)
Compare / recall: (right) Zinbiel identity
v ∗ (w ∗ z) = (v ∗ w) ∗ z + (w ∗ v) ∗ z for all v, w, z (4)
and the ( left) chronological identity of Gamkrelidze and Agrachev)
x · (y · z)− (x · y) · z = y · (x · z)− (y · x) · z⇔ x · (y · z)− y · (x · z) = (x · y) · z − (y · x) · z
sometimes suggestively written as L[x,y] = [Lx, Ly]
Puzzle: The anti-commutativity of the Lie-brackets appears so nat-
ural in control – yet algebraically it appears to be only a coincidence.
What in control corresponds geometrically to the Zinbiel algebra – it
must be connections; but they are not much used in controllability
and optimality – should they? What do they add?
For details on Koszul duality of operads see Ginzburg and Kapranov, Duke. J.
Math, 1997. For details on Leibniz algebras, and their role in cyclic homology,