Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main...

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Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζ h and examples Conclusion 1

Transcript of Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main...

Page 1: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

1

Page 2: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Algebra and the geometryof nonlinear control systems

Matthias Kawski †

School of Mathematical and Statistical SciencesArizona State UniversityTempe, AZ 85287, USA

http://math.asu.edu/~kawski

Workshop in Celebration of the Life, Mathematics,and Memories of Christopher I. Byrnes

Lubbock, TX, September 2010

1This work was partially supported by the National Science Foundationthrough the grants DMS 09-08204 and DMS 09-60589,.

Page 3: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Trimester: Combinatorics & Control 2010, Madrid

Page 4: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Outline

• Background: From control to algebra and combinatorics

• Recall: Hall sets and coord’s of 2nd kind

• Hopf algebras and convolutions

• Coord’s of 1st kind w/ examples

• Conclusion and outlook

Page 5: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

In a nut-shell

• Recall: Chen Fliess series is an exponential Lie series

∑w∈Z∗

w ⊗ w = exp

(∑b∈B

ζb ⊗ b

)=←∏

b∈Bexp(ξb ⊗ b)

where B is ordered basis of free Lie algebra L(Z ) ⊂ A(Z )

• Known (coord’s of 2nd kind): ξHK = ξH ∗ ξK

• New/recent (coord’s of the 1st kind): ζ = π′1 ◦ ξ

• Use in control / geometric integration: explicit formula foriterated integral functionals Υζb and Υξb .

Page 6: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

In a nut-shell

• Recall: Chen Fliess series is an exponential Lie series

∑w∈Z∗

w ⊗ w = exp

(∑b∈B

ζb ⊗ b

)=←∏

b∈Bexp(ξb ⊗ b)

where B is ordered basis of free Lie algebra L(Z ) ⊂ A(Z )

• Known (coord’s of 2nd kind): ξHK = ξH ∗ ξK

• New/recent (coord’s of the 1st kind): ζ = π′1 ◦ ξ

• Use in control / geometric integration: explicit formula foriterated integral functionals Υζb and Υξb .

Page 7: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

In a nut-shell

• Recall: Chen Fliess series is an exponential Lie series

∑w∈Z∗

w ⊗ w = exp

(∑b∈B

ζb ⊗ b

)=←∏

b∈Bexp(ξb ⊗ b)

where B is ordered basis of free Lie algebra L(Z ) ⊂ A(Z )

• Known (coord’s of 2nd kind): ξHK = ξH ∗ ξK

• New/recent (coord’s of the 1st kind): ζ = π′1 ◦ ξ

• Use in control / geometric integration: explicit formula foriterated integral functionals Υζb and Υξb .

Page 8: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

In a nut-shell

• Recall: Chen Fliess series is an exponential Lie series

∑w∈Z∗

w ⊗ w = exp

(∑b∈B

ζb ⊗ b

)=←∏

b∈Bexp(ξb ⊗ b)

where B is ordered basis of free Lie algebra L(Z ) ⊂ A(Z )

• Known (coord’s of 2nd kind): ξHK = ξH ∗ ξK

• New/recent (coord’s of the 1st kind): ζ = π′1 ◦ ξ

• Use in control / geometric integration: explicit formula foriterated integral functionals Υζb and Υξb .

Page 9: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Basic set-up: splitting

Splitting into geometric state-dependent and time-varying parts

x = u1(t)f1(x) + . . .+ um(t)fm(x) = F (t , x)y = ϕ(x)

fi : M 7→ TM smooth vector fields on manifold Mn,

u : [0,T ] 7→ U ⊂⊂ Rm measurable controls/perturbations

,

and

φ ∈ Cω(M) measured output

.

not only control, but also geometric numerical integration,stochastic DE (rough paths [Lyons]) ldots.

Page 10: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Basic set-up: splitting

Splitting into geometric state-dependent and time-varying parts

x = u1(t)f1(x) + . . .+ um(t)fm(x) = F (t , x)y = ϕ(x)

fi : M 7→ TM smooth vector fields on manifold Mn,

u : [0,T ] 7→ U ⊂⊂ Rm measurable controls/perturbations

,

and

φ ∈ Cω(M) measured output

.

not only control, but also geometric numerical integration,stochastic DE (rough paths [Lyons]) ldots.

Page 11: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Basic set-up: splitting

Splitting into geometric state-dependent and time-varying parts

x = u1(t)f1(x) + . . .+ um(t)fm(x) = F (t , x)y = ϕ(x)

fi : M 7→ TM smooth vector fields on manifold Mn,

u : [0,T ] 7→ U ⊂⊂ Rm measurable controls/perturbations,

and

φ ∈ Cω(M) measured output

.

not only control, but also geometric numerical integration,stochastic DE (rough paths [Lyons]) ldots.

Page 12: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Basic set-up: splitting

Splitting into geometric state-dependent and time-varying parts

x = u1(t)f1(x) + . . .+ um(t)fm(x) = F (t , x)y = ϕ(x)

fi : M 7→ TM smooth vector fields on manifold Mn,

u : [0,T ] 7→ U ⊂⊂ Rm measurable controls/perturbations, and

φ ∈ Cω(M) measured output.

not only control, but also geometric numerical integration,stochastic DE (rough paths [Lyons]) ldots.

Page 13: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Long history

Combinatorics and algebra behind integration of ODEs

Just a few names

Campbell-Baker-Hausdorff, Magnus, Chen,

Schützenberger, Reutenauer, Fer, Ferfera,

Fliess, Lamnabhi-Lagarrigue, Crouch, Gray,

Grossmann, Larson, Rota and Baxter, Ebrahimi-Fard, . . .

Page 14: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

A closer look at manipulating series expansions

φ(x(t ,u)) = 1 · φ(x0)

+∫ t

0ua(s)ds (faφ)(x0)

+∫ t

0ub(s)ds (fbφ)(x0)

+ 12

∫ t0

∫ s10

∫ s10 ua(s1)ua(s2)ds2ds1 (fafaφ)(x0)

+ 12

∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafbφ)(x0)

+ 12

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 (fbfaφ)(x0)

+ 12

∫ t0

∫ s10 ub(s1)ub(s2)ds2ds1 (fbfbφ)(x0)

+ 16

∫ t0

∫ s10

∫ s20 ua(s1)ua(s2)ua(s3)ds3ds2ds1 (fafafaφ)(x0)

+ . . .

Objective: Collect first order differential operators, and minimizenumber of higher order differential operators involved

Page 15: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

A closer look at manipulating series expansions

φ(x(t ,u)) = 1 · φ(x0)

+∫ t

0ua(s)ds (faφ)(x0)

+∫ t

0ub(s)ds (fbφ)(x0)

+ 12

∫ t0

∫ s10

∫ s10 ua(s1)ua(s2)ds2ds1 (fafaφ)(x0)

+ 12

∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafbφ)(x0)

+ 12

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 (fbfaφ)(x0)

+ 12

∫ t0

∫ s10 ub(s1)ub(s2)ds2ds1 (fbfbφ)(x0)

+ 16

∫ t0

∫ s10

∫ s20 ua(s1)ua(s2)ua(s3)ds3ds2ds1 (fafafaφ)(x0)

+ . . .Objective: Collect first order differential operators, and minimizenumber of higher order differential operators involved

Page 16: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Integrate by parts: The wrong way to do it∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb+

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa =

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb −

∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa

+∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa +

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+∫ t

0

(ua(s1)

∫ s10 ub(s2) + ub(s1)

∫ s10 ua(s2)ds2

)ds1 fbfa

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+

(∫ t0ua(s) ds

)·(∫ t

0ua(s) ds)

fbfa

Lie brackets together w/ iterated integrals in right orderhigher order deriv’s (wrong order) w/ pointwise prod’s of int’s

Page 17: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Integrate by parts: The wrong way to do it∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb+

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa =

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb −

∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa

+∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa +

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+∫ t

0

(ua(s1)

∫ s10 ub(s2) + ub(s1)

∫ s10 ua(s2)ds2

)ds1 fbfa

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+

(∫ t0ua(s) ds

)·(∫ t

0ua(s) ds)

fbfa

Lie brackets together w/ iterated integrals in right orderhigher order deriv’s (wrong order) w/ pointwise prod’s of int’s

Page 18: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Integrate by parts: The wrong way to do it∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb+

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa =

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb −

∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa

+∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa +

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+∫ t

0

(ua(s1)

∫ s10 ub(s2) + ub(s1)

∫ s10 ua(s2)ds2

)ds1 fbfa

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+

(∫ t0ua(s) ds

)·(∫ t

0ua(s) ds)

fbfa

Lie brackets together w/ iterated integrals in right orderhigher order deriv’s (wrong order) w/ pointwise prod’s of int’s

Page 19: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Integrate by parts: The wrong way to do it∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb+

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa =

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb −

∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa

+∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa +

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+∫ t

0

(ua(s1)

∫ s10 ub(s2) + ub(s1)

∫ s10 ua(s2)ds2

)ds1 fbfa

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+

(∫ t0ua(s) ds

)·(∫ t

0ua(s) ds)

fbfa

Lie brackets together w/ iterated integrals in right orderhigher order deriv’s (wrong order) w/ pointwise prod’s of int’s

Page 20: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Integrate by parts: The wrong way to do it∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb+

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa =

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb −

∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa

+∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa +

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+∫ t

0

(ua(s1)

∫ s10 ub(s2) + ub(s1)

∫ s10 ua(s2)ds2

)ds1 fbfa

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+

(∫ t0ua(s) ds

)·(∫ t

0ua(s) ds)

fbfa

Lie brackets together w/ iterated integrals in right orderhigher order deriv’s (wrong order) w/ pointwise prod’s of int’s

Page 21: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Integrate by parts: The wrong way to do it∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb+

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa =

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 fafb −

∫ t0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa

+∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 fbfa +

∫ t0

∫ s10 ub(s1)ua(s2)ds2ds1 fbfa

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+∫ t

0

(ua(s1)

∫ s10 ub(s2) + ub(s1)

∫ s10 ua(s2)ds2

)ds1 fbfa

=∫ t

0

∫ s10 ua(s1)ub(s2)ds2ds1 (fafb − fbfa)

+

(∫ t0ua(s) ds

)·(∫ t

0ua(s) ds)

fbfa

Lie brackets together w/ iterated integrals in right orderhigher order deriv’s (wrong order) w/ pointwise prod’s of int’s

Page 22: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Integrate by parts, smart way. Hopf algebra preview

Do not manipulate iterated integrals and iterated Lie brackets ofvector fields by hand – work on level of “words” (their indices)∑

I∈{a,b}∗I ⊗ I = 1⊗ 1

+ a ⊗ a

+ b ⊗ b

+ 12 aa ⊗ aa

+ 12 ab ⊗ ab

+ 12 ba ⊗ ba

+ 12 bb ⊗ bb

+ 16 aaa ⊗ aaa

+ . . .

= 1⊗ 1

+ a ⊗ a

+ b ⊗ b

+ 12 aa ⊗ aa

+ 12 ab ⊗ (ab − ba)

+ 12 (ab + ba) ⊗ ba

+ 12 bb ⊗ bb

+ 16 aaa ⊗ aaa

+ . . .

Page 23: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Drop everything except the indices

• The iterated integral

Υi1i2...in =

∫ t

0

∫ t1

0· · ·∫ tn−1

0ui1(t1)ui2(t2) · · · uin (tn) dtn dtn−1 · · · dt1

is uniquely identified by the multi-index (“word”) i1i2 . . . in

• The n-th order partial differential operator fin ◦ fin−1 ◦ . . . fi1is uniquely identified by the multi-index (“word”) i1i2 . . . in

• The Chen series is identified with the identity map on freeassociative algebra A(Z ) over set of indices Z = {1, . . .m}

CF ∼ IdA(Z ) =∑n≥0

∑w∈Z n

w ⊗ w ∈ A(Z )⊗ A(Z )

with shuffle product on left and concatenation on right

Page 24: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Drop everything except the indices

• The iterated integral

Υi1i2...in =

∫ t

0

∫ t1

0· · ·∫ tn−1

0ui1(t1)ui2(t2) · · · uin (tn) dtn dtn−1 · · · dt1

is uniquely identified by the multi-index (“word”) i1i2 . . . in

• The n-th order partial differential operator fin ◦ fin−1 ◦ . . . fi1is uniquely identified by the multi-index (“word”) i1i2 . . . in

• The Chen series is identified with the identity map on freeassociative algebra A(Z ) over set of indices Z = {1, . . .m}

CF ∼ IdA(Z ) =∑n≥0

∑w∈Z n

w ⊗ w ∈ A(Z )⊗ A(Z )

with shuffle product on left and concatenation on right

Page 25: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Drop everything except the indices

• The iterated integral

Υi1i2...in =

∫ t

0

∫ t1

0· · ·∫ tn−1

0ui1(t1)ui2(t2) · · · uin (tn) dtn dtn−1 · · · dt1

is uniquely identified by the multi-index (“word”) i1i2 . . . in

• The n-th order partial differential operator fin ◦ fin−1 ◦ . . . fi1is uniquely identified by the multi-index (“word”) i1i2 . . . in

• The Chen series is identified with the identity map on freeassociative algebra A(Z ) over set of indices Z = {1, . . .m}

CF ∼ IdA(Z ) =∑n≥0

∑w∈Z n

w ⊗ w ∈ A(Z )⊗ A(Z )

with shuffle product on left and concatenation on right

Page 26: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Homomorphisms I

• For fixed smooth vector fields fi

F : A(Z ) 7→ partial diff operators on C∞(M)

F : (i1i2 . . . in) 7→ fi1 ◦ fi2 ◦ . . . finassociative algebras: concatenation 7→ composition

• For fixed control u ∈ UZ

Υ(u) : A(Z ) 7→ AC([0,T ],R)

Υ(u) : (i1i2 . . . in) 7→∫ T

0

∫ t1

0· · ·∫ tp−1

0uip (tp) . . . ui1(t1) dt1 . . . dtp

associative algebras (Ree’s theorem):shuffle of words 7→ pointwise multiplication of functions

Page 27: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Homomorphisms I

• For fixed smooth vector fields fi

F : A(Z ) 7→ partial diff operators on C∞(M)

F : (i1i2 . . . in) 7→ fi1 ◦ fi2 ◦ . . . finassociative algebras: concatenation 7→ composition

• For fixed control u ∈ UZ

Υ(u) : A(Z ) 7→ AC([0,T ],R)

Υ(u) : (i1i2 . . . in) 7→∫ T

0

∫ t1

0· · ·∫ tp−1

0uip (tp) . . . ui1(t1) dt1 . . . dtp

associative algebras (Ree’s theorem):shuffle of words 7→ pointwise multiplication of functions

Page 28: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Homomorphisms II

• Restriction is Lie algebra homomorphism

F : L(Z ) ⊆ A(Z ) 7→ Γ∞(M) (vector fields)

• Do not fix controls: iterated integral functionals

Υ: ∈ A(Z ) 7→ IIF(UZ )

Υ: (i1i2 . . . in) 7→

(u 7→

∫ T

0

∫ t1

0· · ·∫ tp−1

0uip (tp) . . . ui1(t1) dt1 . . . dtp

)associative algebras: shuffle of words 7→ pointwisemultiplication of iterated integral functionals

• Much better:

Theorem: If U = L1([0,T ], [−1,1]) thenΥ: (A(Z ), ∗) 7→ IIF(UZ ) is a Zinbiel algebra isomorphism.

Page 29: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Homomorphisms II

• Restriction is Lie algebra homomorphism

F : L(Z ) ⊆ A(Z ) 7→ Γ∞(M) (vector fields)

• Do not fix controls: iterated integral functionals

Υ: ∈ A(Z ) 7→ IIF(UZ )

Υ: (i1i2 . . . in) 7→

(u 7→

∫ T

0

∫ t1

0· · ·∫ tp−1

0uip (tp) . . . ui1(t1) dt1 . . . dtp

)associative algebras: shuffle of words 7→ pointwisemultiplication of iterated integral functionals

• Much better:

Theorem: If U = L1([0,T ], [−1,1]) thenΥ: (A(Z ), ∗) 7→ IIF(UZ ) is a Zinbiel algebra isomorphism.

Page 30: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Homomorphisms II

• Restriction is Lie algebra homomorphism

F : L(Z ) ⊆ A(Z ) 7→ Γ∞(M) (vector fields)

• Do not fix controls: iterated integral functionals

Υ: ∈ A(Z ) 7→ IIF(UZ )

Υ: (i1i2 . . . in) 7→

(u 7→

∫ T

0

∫ t1

0· · ·∫ tp−1

0uip (tp) . . . ui1(t1) dt1 . . . dtp

)associative algebras: shuffle of words 7→ pointwisemultiplication of iterated integral functionals

• Much better:

Theorem: If U = L1([0,T ], [−1,1]) thenΥ: (A(Z ), ∗) 7→ IIF(UZ ) is a Zinbiel algebra isomorphism.

Page 31: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Homomorphisms II

• Restriction is Lie algebra homomorphism

F : L(Z ) ⊆ A(Z ) 7→ Γ∞(M) (vector fields)

• Do not fix controls: iterated integral functionals

Υ: ∈ A(Z ) 7→ IIF(UZ )

Υ: (i1i2 . . . in) 7→

(u 7→

∫ T

0

∫ t1

0· · ·∫ tp−1

0uip (tp) . . . ui1(t1) dt1 . . . dtp

)associative algebras: shuffle of words 7→ pointwisemultiplication of iterated integral functionals

• Much better: Theorem: If U = L1([0,T ], [−1,1]) thenΥ: (A(Z ), ∗) 7→ IIF(UZ ) is a Zinbiel algebra isomorphism.

Page 32: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Zinbiel product, Hall sets, and formula for ξ

The product of control theory (U ∗ V )(t) =∫ t

0 U(s) V ′(s) dssatisfies the right Zinbiel identity

U ∗ (V ∗W ) = (U ∗ V ) ∗W + (V ∗ U) ∗W

Theorem (Schützenberger, Sussmann, Grossman,Reutenauer-Melançon)For a Hall set H over Z and letters a ∈ Z and H,K ,HK ∈ H

ξa = aξHK = µHK ξH ∗ ξK

Page 33: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Zinbiel product, Hall sets, and formula for ξ

The product of control theory (U ∗ V )(t) =∫ t

0 U(s) V ′(s) dssatisfies the right Zinbiel identity

U ∗ (V ∗W ) = (U ∗ V ) ∗W + (V ∗ U) ∗W

Theorem (Schützenberger, Sussmann, Grossman,Reutenauer-Melançon)For a Hall set H over Z and letters a ∈ Z and H,K ,HK ∈ H

ξa = aξHK = µHK ξH ∗ ξK

Page 34: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Illustration: Normal form for nilpotent systemNormal from for a free nilpotent system (of rank r = 5)using a typical Hall set on the alphabet Z = {a,b}

ξa = ua

ξb = ubξab = ξa · ξb = ξa ubξaab = ξa · ξab = ξ2

a ub using ψ−b(aab) = (a(ab))

ξbab = ξb · ξab = ξbξa ub using ψ−b(bab) = (b(ab))

ξaaab = ξa · ξaab = ξ3a ub using ψ−b(aaab) = (a(a(ab)))

ξbaab = ξb · ξaab = ξbξ2a ub using ψ−b(baab) = (b(a(ab)))

ξbbab = ξb · ξbab = ξ2bξa ub using ψ−b(bbab) = (b(b(ab)))

ξaaaab = ξa · ξaaab = ξ4a ub using ψ−b(aaaab) = (a(a(a(ab))))

ξbaaab = ξb · ξaaab = ξbξ3a ub using ψ−b(baaab) = (b(a(a(ab))))

ξbbaab = ξb · ξbaab = ξ2bξ

2a ub using ψ−b(bbaab) = (b(b(a(ab))))

ξabaab = ξab · ξaab = ξabξ3a ub using ψ−b(abaab) = ((ab)(a(ab)))

ξabbab = ξab · ξbab = ξabξ2bξa ub using ψ−b(abbab) = ((ab)(b(ab)))

Cost: accept label coord’s not by integers, but by Hall wordsBenefit: One-line formula for all iterated integrals! ξHK = ξH ∗ ξK

Page 35: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Illustration: Normal form for nilpotent systemNormal from for a free nilpotent system (of rank r = 5)using a typical Hall set on the alphabet Z = {a,b}

ξa = ua

ξb = ubξab = ξa · ξb = ξa ubξaab = ξa · ξab = ξ2

a ub using ψ−b(aab) = (a(ab))

ξbab = ξb · ξab = ξbξa ub using ψ−b(bab) = (b(ab))

ξaaab = ξa · ξaab = ξ3a ub using ψ−b(aaab) = (a(a(ab)))

ξbaab = ξb · ξaab = ξbξ2a ub using ψ−b(baab) = (b(a(ab)))

ξbbab = ξb · ξbab = ξ2bξa ub using ψ−b(bbab) = (b(b(ab)))

ξaaaab = ξa · ξaaab = ξ4a ub using ψ−b(aaaab) = (a(a(a(ab))))

ξbaaab = ξb · ξaaab = ξbξ3a ub using ψ−b(baaab) = (b(a(a(ab))))

ξbbaab = ξb · ξbaab = ξ2bξ

2a ub using ψ−b(bbaab) = (b(b(a(ab))))

ξabaab = ξab · ξaab = ξabξ3a ub using ψ−b(abaab) = ((ab)(a(ab)))

ξabbab = ξab · ξbab = ξabξ2bξa ub using ψ−b(abbab) = ((ab)(b(ab)))

Cost: accept label coord’s not by integers, but by Hall words

Benefit: One-line formula for all iterated integrals! ξHK = ξH ∗ ξK

Page 36: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Illustration: Normal form for nilpotent systemNormal from for a free nilpotent system (of rank r = 5)using a typical Hall set on the alphabet Z = {a,b}

ξa = ua

ξb = ubξab = ξa · ξb = ξa ubξaab = ξa · ξab = ξ2

a ub using ψ−b(aab) = (a(ab))

ξbab = ξb · ξab = ξbξa ub using ψ−b(bab) = (b(ab))

ξaaab = ξa · ξaab = ξ3a ub using ψ−b(aaab) = (a(a(ab)))

ξbaab = ξb · ξaab = ξbξ2a ub using ψ−b(baab) = (b(a(ab)))

ξbbab = ξb · ξbab = ξ2bξa ub using ψ−b(bbab) = (b(b(ab)))

ξaaaab = ξa · ξaaab = ξ4a ub using ψ−b(aaaab) = (a(a(a(ab))))

ξbaaab = ξb · ξaaab = ξbξ3a ub using ψ−b(baaab) = (b(a(a(ab))))

ξbbaab = ξb · ξbaab = ξ2bξ

2a ub using ψ−b(bbaab) = (b(b(a(ab))))

ξabaab = ξab · ξaab = ξabξ3a ub using ψ−b(abaab) = ((ab)(a(ab)))

ξabbab = ξab · ξbab = ξabξ2bξa ub using ψ−b(abbab) = ((ab)(b(ab)))

Cost: accept label coord’s not by integers, but by Hall wordsBenefit: One-line formula for all iterated integrals! ξHK = ξH ∗ ξK

Page 37: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Hopf algebra and dynamical systems

Long history of using combinatorial Hopf algebras for analysisof dynamical systems.

E.g. Grossmann and Larson, Hopf algebra structure of labeledtrees, and of differential operators

Main reference for this work: Reutenauer, Free Lie Algebras

Recent related work, numerical “geometric” integration:Murua, Ebrahimi-Fard, Munthe-Kaas, . . .

Page 38: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Easy intro to Hopf algebra: shuffle

Combinatorially: for words w , z ∈ Z ∗ and letters a,b ∈ Z

( w a ) X ( z b ) = (( w a ) X z ) b + ( w X ( z b )) a

Example: (ab) X (cd) = a b c d + a c b d + c a b d +a c d b + c a d b + c d a b

Algebraically: transpose of the coproduct ∆

< v X w , z >=< v ⊗ w , ∆(z) >

where

∆: A(Z ) 7→ A(Z )⊗ A(Z ) by ∆(a) = 1⊗a+a⊗1 for a ∈ Z

Page 39: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Easy intro to Hopf algebra: shuffle

Combinatorially: for words w , z ∈ Z ∗ and letters a,b ∈ Z

( w a ) X ( z b ) = (( w a ) X z ) b + ( w X ( z b )) a

Example: (ab) X (cd) = a b c d + a c b d + c a b d +a c d b + c a d b + c d a b

Algebraically: transpose of the coproduct ∆

< v X w , z >=< v ⊗ w , ∆(z) >

where

∆: A(Z ) 7→ A(Z )⊗ A(Z ) by ∆(a) = 1⊗a+a⊗1 for a ∈ Z

Page 40: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Two Hopf algebra structures on A(Z )

• Products: Concatenation and shuffle

• Coproducts:∆: A(Z ) 7→ A(Z )⊗ A(Z ) concatenation homo∆′ : A(Z ) 7→ A(Z )⊗ A(Z ) shuffle homomorphismFor a ∈ Z , ∆(a) = ∆′(a) = 1⊗ a + a⊗ 1, and for w ∈ Z ∗

∆(w) =∑

u,v∈Z∗〈w ,u x v〉 u ⊗ v .∆′(w) =

∑u,v∈Z∗〈w ,uv〉 u ⊗ v .

• Antipode (same for both): linear map α : A(Z ) 7→ A(Z )if w = a1a2 . . . ak ∈ Z k then α(w) = (−)k ak . . . a2a1.

• Example: If a,b ∈ Z then

∆′(ab) = 1⊗ ab + a⊗ b + ab⊗ 1and

∆(ab) = ∆(a)∆(b) = 1⊗ ab + a⊗ b + b ⊗ a + ab ⊗ 1.

Page 41: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Two Hopf algebra structures on A(Z )

• Products: Concatenation and shuffle

• Coproducts:∆: A(Z ) 7→ A(Z )⊗ A(Z ) concatenation homo∆′ : A(Z ) 7→ A(Z )⊗ A(Z ) shuffle homomorphismFor a ∈ Z , ∆(a) = ∆′(a) = 1⊗ a + a⊗ 1, and for w ∈ Z ∗

∆(w) =∑

u,v∈Z∗〈w ,u x v〉 u ⊗ v .∆′(w) =

∑u,v∈Z∗〈w ,uv〉 u ⊗ v .

• Antipode (same for both): linear map α : A(Z ) 7→ A(Z )if w = a1a2 . . . ak ∈ Z k then α(w) = (−)k ak . . . a2a1.

• Example: If a,b ∈ Z then

∆′(ab) = 1⊗ ab + a⊗ b + ab⊗ 1and

∆(ab) = ∆(a)∆(b) = 1⊗ ab + a⊗ b + b ⊗ a + ab ⊗ 1.

Page 42: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Two Hopf algebra structures on A(Z )

• Products: Concatenation and shuffle

• Coproducts:∆: A(Z ) 7→ A(Z )⊗ A(Z ) concatenation homo∆′ : A(Z ) 7→ A(Z )⊗ A(Z ) shuffle homomorphismFor a ∈ Z , ∆(a) = ∆′(a) = 1⊗ a + a⊗ 1, and for w ∈ Z ∗

∆(w) =∑

u,v∈Z∗〈w ,u x v〉 u ⊗ v .∆′(w) =

∑u,v∈Z∗〈w ,uv〉 u ⊗ v .

• Antipode (same for both): linear map α : A(Z ) 7→ A(Z )if w = a1a2 . . . ak ∈ Z k then α(w) = (−)k ak . . . a2a1.

• Example: If a,b ∈ Z then

∆′(ab) = 1⊗ ab + a⊗ b + ab⊗ 1and

∆(ab) = ∆(a)∆(b) = 1⊗ ab + a⊗ b + b ⊗ a + ab ⊗ 1.

Page 43: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Two Hopf algebra structures on A(Z )

• Products: Concatenation and shuffle

• Coproducts:∆: A(Z ) 7→ A(Z )⊗ A(Z ) concatenation homo∆′ : A(Z ) 7→ A(Z )⊗ A(Z ) shuffle homomorphismFor a ∈ Z , ∆(a) = ∆′(a) = 1⊗ a + a⊗ 1, and for w ∈ Z ∗

∆(w) =∑

u,v∈Z∗〈w ,u x v〉 u ⊗ v .∆′(w) =

∑u,v∈Z∗〈w ,uv〉 u ⊗ v .

• Antipode (same for both): linear map α : A(Z ) 7→ A(Z )if w = a1a2 . . . ak ∈ Z k then α(w) = (−)k ak . . . a2a1.

• Example: If a,b ∈ Z then ∆′(ab) = 1⊗ ab + a⊗ b + ab⊗ 1and ∆(ab) = ∆(a)∆(b) = 1⊗ ab + a⊗ b + b ⊗ a + ab ⊗ 1.

Page 44: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Two Hopf algebra structures on A(Z )

• Products: Concatenation and shuffle

• Coproducts:∆: A(Z ) 7→ A(Z )⊗ A(Z ) concatenation homo∆′ : A(Z ) 7→ A(Z )⊗ A(Z ) shuffle homomorphismFor a ∈ Z , ∆(a) = ∆′(a) = 1⊗ a + a⊗ 1, and for w ∈ Z ∗

∆(w) =∑

u,v∈Z∗〈w ,u x v〉 u ⊗ v .∆′(w) =

∑u,v∈Z∗〈w ,uv〉 u ⊗ v .

• Antipode (same for both): linear map α : A(Z ) 7→ A(Z )if w = a1a2 . . . ak ∈ Z k then α(w) = (−)k ak . . . a2a1.

• Example: If a,b ∈ Z then ∆′(ab) = 1⊗ ab + a⊗ b + ab⊗ 1and ∆(ab) = ∆(a)∆(b) = 1⊗ ab + a⊗ b + b ⊗ a + ab ⊗ 1.

Page 45: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Two convolution productsEach Hopf algebra structure yields an associative convolutionproduct of linear endomorphisms f ,g : A(Z ) 7→ A(Z )

f ? g = conc ◦ (f ⊗ g) ◦∆, and

f ?′ g = shu ◦ (f ⊗ g) ◦∆′,

For w ∈ A(Z )

(f ? g)(w) =∑

u,v∈Z+

〈w ,u x v〉 f (u)g(v) and

(f ?′ g)(w) =∑

u,v∈Z+

〈w ,uv〉 f (u) x g(v)

Note: Apparent confusion of ? and ?′ led to incorrect formulasfor ζh in the literature.

Page 46: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Fundamental subspaces and projections• Direct sum decomposition A(Z ) =

⊕∞k=1 Uk where for

k ∈ Z+, Uk ⊆ A(Z ) is linear subspace spanned by `k

(concatenation) for ` ∈ L(Z ). Alternatively, Uk is spannedby symmetric products of Lie polynomials `1, . . . `k ∈ L(Z )

Sym(`1, . . . `k ) = 1k!

∑σ∈Sk

`σ(1)`σ(2) . . . `σ(k)

• associated projections (NOT orthogonal) πk : A(Z ) 7→ Uk

π1 =∑k≥1

(−)k−1 1k I?k

=∑k≥1

(−)k−1 1k conck ◦ I⊗k ◦∆k , and

πk = 1k!π

?k1 .

where I : A(Z ) 7→ A(Z ), I(1) = 0, I(w) = w for w ∈ Z ∗.

Page 47: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Fundamental subspaces and projections• Direct sum decomposition A(Z ) =

⊕∞k=1 Uk where for

k ∈ Z+, Uk ⊆ A(Z ) is linear subspace spanned by `k

(concatenation) for ` ∈ L(Z ). Alternatively, Uk is spannedby symmetric products of Lie polynomials `1, . . . `k ∈ L(Z )

Sym(`1, . . . `k ) = 1k!

∑σ∈Sk

`σ(1)`σ(2) . . . `σ(k)

• associated projections (NOT orthogonal) πk : A(Z ) 7→ Uk

π1 =∑k≥1

(−)k−1 1k I?k

=∑k≥1

(−)k−1 1k conck ◦ I⊗k ◦∆k , and

πk = 1k!π

?k1 .

where I : A(Z ) 7→ A(Z ), I(1) = 0, I(w) = w for w ∈ Z ∗.

Page 48: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Fundamental subspaces and projections• Direct sum decomposition A(Z ) =

⊕∞k=1 Uk where for

k ∈ Z+, Uk ⊆ A(Z ) is linear subspace spanned by `k

(concatenation) for ` ∈ L(Z ). Alternatively, Uk is spannedby symmetric products of Lie polynomials `1, . . . `k ∈ L(Z )

Sym(`1, . . . `k ) = 1k!

∑σ∈Sk

`σ(1)`σ(2) . . . `σ(k)

• associated projections (NOT orthogonal) πk : A(Z ) 7→ Uk

π1 =∑k≥1

(−)k−1 1k I?k

=∑k≥1

(−)k−1 1k conck ◦ I⊗k ◦∆k , and

πk = 1k!π

?k1 .

where I : A(Z ) 7→ A(Z ), I(1) = 0, I(w) = w for w ∈ Z ∗.

Page 49: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Fundamental subspaces and projections• Direct sum decomposition A(Z ) =

⊕∞k=1 Uk where for

k ∈ Z+, Uk ⊆ A(Z ) is linear subspace spanned by `k

(concatenation) for ` ∈ L(Z ). Alternatively, Uk is spannedby symmetric products of Lie polynomials `1, . . . `k ∈ L(Z )

Sym(`1, . . . `k ) = 1k!

∑σ∈Sk

`σ(1)`σ(2) . . . `σ(k)

• associated projections (NOT orthogonal) πk : A(Z ) 7→ Uk

π1 =∑k≥1

(−)k−1 1k I?k

=∑k≥1

(−)k−1 1k conck ◦ I⊗k ◦∆k , and

πk = 1k!π

?k1 .

where I : A(Z ) 7→ A(Z ), I(1) = 0, I(w) = w for w ∈ Z ∗.

Page 50: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Fundamental subspaces and projections• Direct sum decomposition A(Z ) =

⊕∞k=1 Uk where for

k ∈ Z+, Uk ⊆ A(Z ) is linear subspace spanned by `k

(concatenation) for ` ∈ L(Z ). Alternatively, Uk is spannedby symmetric products of Lie polynomials `1, . . . `k ∈ L(Z )

Sym(`1, . . . `k ) = 1k!

∑σ∈Sk

`σ(1)`σ(2) . . . `σ(k)

• associated projections (NOT orthogonal) πk : A(Z ) 7→ Uk

π1 =∑k≥1

(−)k−1 1k I?k

=∑k≥1

(−)k−1 1k conck ◦ I⊗k ◦∆k

, and

πk = 1k!π

?k1 .

where I : A(Z ) 7→ A(Z ), I(1) = 0, I(w) = w for w ∈ Z ∗.

Page 51: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Fundamental subspaces and projections• Direct sum decomposition A(Z ) =

⊕∞k=1 Uk where for

k ∈ Z+, Uk ⊆ A(Z ) is linear subspace spanned by `k

(concatenation) for ` ∈ L(Z ). Alternatively, Uk is spannedby symmetric products of Lie polynomials `1, . . . `k ∈ L(Z )

Sym(`1, . . . `k ) = 1k!

∑σ∈Sk

`σ(1)`σ(2) . . . `σ(k)

• associated projections (NOT orthogonal) πk : A(Z ) 7→ Uk

π1 =∑k≥1

(−)k−1 1k I?k

=∑k≥1

(−)k−1 1k conck ◦ I⊗k ◦∆k , and

πk = 1k!π

?k1 .

where I : A(Z ) 7→ A(Z ), I(1) = 0, I(w) = w for w ∈ Z ∗.

Page 52: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

The adjoint π′1

Define the adjoint map π′1 : A(Z ) 7→ A(Z ) by

〈w , π1(z)〉 = 〈π′1(w), z〉 for all w , z ∈ Z.

Since I is selfadjoint, conc and ∆′ are adjoints of each other, asare x and ∆. Hence, using the second convolution product ?′

π′1 =∑k≥1

(−)k−1 1k I?

′k . =∑k≥1

(−)k−1 1k shuk ◦ I⊗k ◦∆′k

On words w ∈ Z , calculate (sum over uj 6= 1)

π′1(w) =∑k≥1

1k (−)k−1

∑u1...uk=w

u1 x . . . x uk

Example: π′1(ξab) = π′1(ab) = ab − 12 a x b = 1

2 (ab − ba).

Page 53: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Formulas for the ζh

Theorem[E. Gehrig].On every Hall set ζ = π′1 ◦ ξ.

Moreover: (recall PBW basis and Viennot) every w ∈ Z ∗ factorsuniquely w = h1h2 · · · hk with Hall words h1 ≥ h2 ≥ . . . ≥ hk

Naturally extending ξ and ζ to A(Z ),

ξw = 1r1!···rk !ξ

x r1h1

x ξx r2h2

x . . . x ξx rkhk

(known)

ζw = π′|r |(ξw ) = 1r1!···rk !ζ

x r1h1

x . . . x ζx rkhk

(new)

In general the maps π′k do not commute with shuffles.But surprisingly they do on products of this special form.

Page 54: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Formulas for the ζh

Theorem[E. Gehrig].On every Hall set ζ = π′1 ◦ ξ.

Moreover: (recall PBW basis and Viennot) every w ∈ Z ∗ factorsuniquely w = h1h2 · · · hk with Hall words h1 ≥ h2 ≥ . . . ≥ hk

Naturally extending ξ and ζ to A(Z ),

ξw = 1r1!···rk !ξ

x r1h1

x ξx r2h2

x . . . x ξx rkhk

(known)

ζw = π′|r |(ξw ) = 1r1!···rk !ζ

x r1h1

x . . . x ζx rkhk

(new)

In general the maps π′k do not commute with shuffles.But surprisingly they do on products of this special form.

Page 55: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Sample calculation π′1(aaabb) and ζabaab

π′1(aaabb) = aaabb − 12 (aaab x b + aaa x bb + aa x aab

+a x aabb) + 13(aaa x b x b + aa x ab x b

+2aa x a x bb + a x aab x b + a x a x abb)

−14(aa x a x b x b + a x aa x b x b + a x a x ab x b

+a x a x a x bb) + 15a x a x a x b x b.

Similarly, starting fromξabaab = 1

2 (ab ∗ (a ∗ (a ∗ b))) = 3aaabb + 2aabab + abaab

ζabaab = π′1(ξabaab) = π′1(3aaabb + 2aabab + abaab)

= 110aaabb + 1

15aabab + 115aabba + 1

15abaab − 110ababa

+ 115abbaa− 1

10baaab + 115baaba + 1

15babaa− 110bbaaa.

Page 56: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

The first 14 terms as nilpotent cascade system I

Iterated integrals zH(t ,U ′) = µH Υ(ζH)(U)(t)

za = ua

zb = ub

zab = −16zbua + 1

6zaub

zaab =(−1

2zab − 112zazb

)ua + 1

12z2aub

zbab = − 112z2

bua +(−1

2zab + 112zazb

)ub

zaaab =(−1

2zaab − 112zabza

)ua

zbaab =(−1

2zbab − 112zabzb

)ua −

(12zaab + 1

12zabza)

ub

zbbab =(−1

2zbab − 112zabzb

)ub

Page 57: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

The first 14 terms as nilpotent cascade system II

zaaaab =(−1

2zaaab − 112zazaab − 1

720z3azb)

ua − 1720z4

aub

zbbbab = 1720z4

bua −(1

2zbbab + 112zbzbab + 1

720zaz3b

)ub

zbaaab =( 1

240z2ay2

b −12zbaab − 1

12zbzaab − 112zazbab

)ua

−(1

2zaaab + 112zazaab + 1

240z3azb)

ub

zabaab =(−1

2zbaab + 112zbzaab − 1

12zazbab − 112y2

ab

+ 1360z2

ay2b)

ua +(−1

6zazaab − 1360z3

azb)

ub

zabbab =(−1

2zbbab1

12zbzbab + 1720zaz3

b

)ua

−(1

6zazbab + 112y2

ab + 1720z2

ay2b)

ub

zbbaab =(−1

2zbbab − 112zbzbab + 1

240zaz3b

)ua

−(1

2zbaab + 112zbzaab + 1

12zazbab + 1240z2

ay2b)

ub

Page 58: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Summary and outlook

• Formulas to obtain coord’s of the first kind ζhfrom known formulas for ξh in terms of Zinbiel product

• The main contribution is not the existence of the map,but expressing it via elementary objects in Hopf algebra.

• Formulas for ζh are explicit and easily computable,and amenable for implementation for path planning,or for geometric integration procedures.

• Combinatorial Hopf algebra: Connections betweennonlinear control, geometric numerical integration,stochastic DEs (rough paths), more to come.Next: geometric, no letters, but use full tensor algebra

For more information, visit http://math.asu.edu/~kawski

Thank you.

This presentation was prepared with LaTeX using the beamer package.

Page 59: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Summary and outlook

• Formulas to obtain coord’s of the first kind ζhfrom known formulas for ξh in terms of Zinbiel product

• The main contribution is not the existence of the map,but expressing it via elementary objects in Hopf algebra.

• Formulas for ζh are explicit and easily computable,and amenable for implementation for path planning,or for geometric integration procedures.

• Combinatorial Hopf algebra: Connections betweennonlinear control, geometric numerical integration,stochastic DEs (rough paths), more to come.Next: geometric, no letters, but use full tensor algebra

For more information, visit http://math.asu.edu/~kawski

Thank you.

This presentation was prepared with LaTeX using the beamer package.

Page 60: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Summary and outlook

• Formulas to obtain coord’s of the first kind ζhfrom known formulas for ξh in terms of Zinbiel product

• The main contribution is not the existence of the map,but expressing it via elementary objects in Hopf algebra.

• Formulas for ζh are explicit and easily computable,and amenable for implementation for path planning,or for geometric integration procedures.

• Combinatorial Hopf algebra: Connections betweennonlinear control, geometric numerical integration,stochastic DEs (rough paths), more to come.Next: geometric, no letters, but use full tensor algebra

For more information, visit http://math.asu.edu/~kawski

Thank you.

This presentation was prepared with LaTeX using the beamer package.

Page 61: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Summary and outlook

• Formulas to obtain coord’s of the first kind ζhfrom known formulas for ξh in terms of Zinbiel product

• The main contribution is not the existence of the map,but expressing it via elementary objects in Hopf algebra.

• Formulas for ζh are explicit and easily computable,and amenable for implementation for path planning,or for geometric integration procedures.

• Combinatorial Hopf algebra: Connections betweennonlinear control, geometric numerical integration,stochastic DEs (rough paths), more to come.Next: geometric, no letters, but use full tensor algebra

For more information, visit http://math.asu.edu/~kawski

Thank you.

This presentation was prepared with LaTeX using the beamer package.

Page 62: Intro and main Background, Zinbiel, and Hopf …gilliam/ttu/cib_mem_webpage/Chris...Intro and main Background, Zinbiel, and ˘ Hopf algebra, convolution h and examples Conclusion In

Intro and main Background, Zinbiel, and ξ Hopf algebra, convolution ζh and examples Conclusion

Summary and outlook

• Formulas to obtain coord’s of the first kind ζhfrom known formulas for ξh in terms of Zinbiel product

• The main contribution is not the existence of the map,but expressing it via elementary objects in Hopf algebra.

• Formulas for ζh are explicit and easily computable,and amenable for implementation for path planning,or for geometric integration procedures.

• Combinatorial Hopf algebra: Connections betweennonlinear control, geometric numerical integration,stochastic DEs (rough paths), more to come.Next: geometric, no letters, but use full tensor algebra

For more information, visit http://math.asu.edu/~kawski

Thank you. This presentation was prepared with LaTeX using the beamer package.