algorithms for finding the lie superalgebra...

Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Algorithms for Finding the Lie SuperalgebraStructure of Regular Super Differential Equations

Xuan Liu

Department of Applied MathematicsWestern University

May 15, 2014


Contents

Outline

Background

Defining SystemMONO Expansion

Algorithms and Examples

Super KdV

Discussion and Future Work

Acknowledgement and References


Presentation Outline

1. What is supersymmetry?

2. Apply supersymmetry to super differential equations.

3. Main work: algorithms for finding the Lie superalgerba struc-ture.

4. Two examples.


Applications of Super Differential Equations

1. Advanced math physics models (e.g. string theory).

2. Even non-super (traditional models) can be embeded in a su-per model that makes determination of analytic solutions eas-ier (e.g. SUSY quantum mechanic; and it has relations toBluman’s potential symmetries).

3. Even the LHC has eliminated some popular supersymmetrymodels, supersymmetries are still useful.


Super Objects

Even and Odd

even · even = even

even · odd = odd

odd · odd = even

Lots of SUPER objects are involved!


Grassman Algebra

Consider a set of N generators θ1, θ2, . . . , θN , which are as-sumed to have products θiθj such that

(i) for all i , j , k = 1, ...,N,

(θiθj)θk = θi (θjθk); (1)

(ii) for all i , j = 1, ...,N,

θiθj = −θjθi (implies θ2i = 0); (2)

(iii) each non-zero product

θj1θj2 · · · θjr

linearly independent of products involving less than r generators.


Super Differential Equations

Consider the general case of a nonlinear system of Grassmann-valued differential equations or superequations of s equations of or-der k = (k1; k2) denoted by

∆ν(X ,Θ,A(k1),Q(k2)) = 0, ν = 1, . . . , s, (3)

with m independent even variables X = {x1, . . . , xm}, n independentodd variables Θ = {θ1, . . . , θn}, q dependent even variables A ={A1, . . . ,Aq} and p dependent odd variables Q = {Q1, . . . ,Qp}.

NOTE: θ2 = 0 and Q2 = 0.


Determining equations for supersymmetries

1. Reduce to one-parameter Lie super transforamtion about theidentity.

2. Apply superprolongation formula to the super DEs.

3. Replace the highest derivatives.

4. Pick the coeffiecients of monomials of the dependent variblesand their derivatives.

5. Get the determining equations for supersymmetries.

[See Edgardo Cheb-Terrab’s implementations in Maple.]


Parametric and Principle Derivatives

For a (super) differential system, the set of (super) derivativescan be divided into two disjoint subsets:

• Parametric derivatives

those which can not be obtained as the derivatives of any leadingderivative,

• Principle derivatives

those are a derivative of some leading derivative.


Regular Super DEs

REGULAR super differential equations have even coef-ficients of their highest derivatives and even parameters andparametric functions.

Written in solved form with respect to their highest deriva-tives −→ non-trivial.

[For example, Q ∗ HD = b, it can’t solved for HD when Q 6= 0.]


Decompose by odd variablesMAPLE procedure: MONO

• Decompose a super function by its odd variable monomials.

Example

• Even function f (x , θ1, θ2),

fθ1θ1 = 0, fθ2θ2 = 0,

it can be expanded as

PE1(x) + PO1(x)θ1 + PO2(x)θ2 + PE2(x)θ1θ2.

• For an odd function f (x , θ1, θ2), it can be expanded as

PO1(x) + PE1(x)θ1 + PE2(x)θ2 + PO2(x)θ1θ2.


Advantage of MONO Expansion

• A super function can be written more percisely.

• Eliminate the odd independences.

• Apply MAPLE commmutative commands.


Reduce Defining System Algorithm

Input: Defining system S.

1. Decompose each of the infinitesmals by MONO expansion.

2. Substitute them into the input system S.

3. Equating all the coefficients of odd variable monomials to bezero forms the new defining system Sred.

4. Send Sred to the commutative Maple commands rifsimp andinitialdata

Output: Return the size of the symmetry group.


A Simple ExampleConsider a simple example Qxx = 0.

Input: Defining system

S =

{Λ(x ,Q)xx = 0,−Ξ(x ,Q)xx + 2Λ(x ,Q)xQ = 0.

(4)

1. MONO expansion of infinitesimals{Ξ(x ,Q) = PE11(x) + PO11(x) ∗ Q,Λ(x ,Q) = PO21(x) + PE21(x) ∗ Q. (5)

2. Subsitution{PO21(x)xx + PE21(x)xx ∗ Q = 0,−PE11(x)xx + Q ∗ PO11(x)xx + 2PE21(x)x = 0.

(6)

3. PO21(x)xx = 0,PE21(x)xx = 0,−PE11(x)xx + 2PE21(x)x = 0,PO11(x)xx = 0.

(7)

4. Send (7) to the commutative Maple commands rifsimp and initialdata


Maple Output

Output:


Finding Structure Constant Algorithm

Input: S or Sred in standard form, number of even infinitesi-mals m1, number of odd infinitesimals m2.

1. Write two supesymmetry vector fields Li , Lj , where

Li =m1∑l1=1

EvenInfil1∂EvenVarl1+

m2∑l2=1

OddInfil2∂OddVarl2

and Lj has the same form.

2. Take their Lie superbracket [Li , Lj ] and it can be written as

[Li , Lj ] =m1∑l1=1

Al1∂EvenVarl1+

m2∑l2=1

B l2∂OddVarl2. (8)


3. Compute

[Li , Lj ] =

m1+m2∑k=1

ckij Lk (9)

=

m1∑l1=1

(

m1+m2∑k=1

ckij EvenInfkl1)∂EvenVarl1

+

m2∑l2=1

(

m1+m2∑k=1

ckij OddInfkl2)∂OddVarl2

.

4. The equations in (8) and (9) form a linear system with m1 +m2equations.

m1 equations

∑m1+m2

k=1ckij EvenInfk1 = A1,

...,∑m1+m2k=1

ckij EvenInfkm1= Am1 ,

(10)

m2 equations

∑m1+m2

k=1ckij OddInfk1 = B1,

...,∑m1+m2k=1

ckij OddInfkm2= Bm2 ,

(11)

5. Select an ordering � of parametric derivatives6. Keep differentiating (10) and (11) w.r.t. parametric derivatives.7. Reduce w.r.t rifsimp form and set the system as the given

order.8. Provide two copies of intial data a1, ..am1+m2 and b1, ..bm1+m2

for the parametric derivatives and read off the nonzero structureconstants.

Output: Nonzero structure constants ckij .


ExampleStart with the defining system of the same example Qxx = 0.

Input: S in standard form

S = {Λxx = 0,ΛxQ =1

2Ξxx ,ΞQQ = 0,ΛQQ = 0,Ξxxx = 0,ΞxxQ = 0},

number of even infinitesimal m1 = 1, number of odd infinites-imal m2 = 1.

1. Write two supersymmetry vector fields Li , Lj ,

Li = Ξi∂x + Λi∂Q and Lj = Ξj∂x + Λj∂Q .

2. Take their Lie superbracket

[Li , Lj ] = (

A︷︸︸︷ΞiΞj

x − ΞjΞix + ΛiΞj

Q − ΛjΞiQ)∂x

+(

B︷︸︸︷ΞiΛj

x − ΞjΛix + ΛiΛj

Q − ΛjΛiQ)∂Q .


3. Compute

[Li , Lj ] =8∑

k=1

ckij Lk

= (8∑

k=1

ckij Ξk)∂x + (

8∑k=1

ckij Λk)∂Q .

4. The results in step 2 and 4 form the linear system with 1 + 1equations.

8∑k=1

ckij Ξk = ΞiΞj

x − ΞjΞix + ΛiΞj

Q − ΛjΞiQ , (12)

8∑k=1

ckij Λk = ΞiΛj

x − ΞjΛix + ΛiΛj

Q − ΛjΛiQ . (13)

5. Set parametric derivatives in a certain ordering �,

δ = Ξ � Ξx � ΛQ � Ξxx � Λ � ΞQ � Λx � ΞxQ .


6.7. Keep differentiating the two equation in step 5 w.r.t. parametricderivatives until we have 8 equations, and modulo them w.r.tparametric derivatives and set them as ordering �,

8∑k=1

ckij Ξk = ΞiΞjx − ΞjΞi

x + ΛiΞjQ− ΛjΞi

Q ,

8∑k=1

ckij Ξkx = ΞiΞj

xx − ΞjΞjxx + Ξi

QΛjx − Ξ

jQ

Λix + ΛiΞ

jxQ− ΛjΞi

xQ ,

8∑k=1

ckij ΛkQ = Ξi

QΛjx − Ξ

jQ

Λix +

1

2(ΞiΞj

xx − ΞjΞixx ) + Λi

QΛjQ− Λ

jQ

ΛiQ ,

8∑k=1

ckij Ξkxx = Ξi

xΞjxx − Ξj

xΞixx + 2(Λi

xΞjxQ− Λj

xΞixQ ),

8∑k=1

ckij Λk = ΞiΛjx − ΞjΛi

x + ΛiΛjQ− ΛjΛi

Q ,

8∑k=1

ckij ΞkQ = Ξi

QΞjx − Ξ

jQ

Ξix + ΞiΞ

jxQ− ΞjΞi

xQ + ΛiQΞ

jQ− Λ

jQ

ΞiQ ,

8∑k=1

ckij Λkx = Ξi

xΛjx − Ξj

xΛix + Λi

xΛjQ− Λj

xΛiQ +

1

2(ΛiΞj

xx − ΛjΞixx ),

8∑k=1

ckij ΞkxQ = −

1

2(Ξi

xxΞjQ− Ξj

xxΞiQ ) + Λi

QΞjxQ− Λ

jQ

ΞixQ .


8. Provide two copies of intial data a1, ..a8 and b1, ..b8 to the sys-tem in step 7,

8∑k=1

ckij Ξk = ΞiΞjx − ΞjΞi

x︸︷︷︸a1b2−b1a2

+ ΛiΞjQ− ΛjΞi

Q︸︷︷︸a5b6−b5a6

−→ c112 = 1, c1

56 = 1,

8∑k=1

ckij Ξkx = ΞiΞj

xx − ΞjΞjxx︸︷︷︸

a1b4−b1a4

+ ΞiQΛj

x − ΞjQ

Λix︸︷︷︸

a6b7−b6a7

+ ΛiΞjxQ− ΛjΞi

xQ︸︷︷︸a5b8−b5a8

−→ c214 = 1, c2

67 = 1, c258 = 1,

8∑k=1

ckij ΛkQ = Ξi

QΛjx − Ξ

jQ

Λix︸︷︷︸

a6b7−b6a7

+1

2(ΞiΞj

xx − ΞjΞixx︸︷︷︸

a1b4−b1a4

)

−→ c367 = 1, c3

14 = 1/2,

8∑k=1

ckij Ξkxx = ... −→ c4

24 = 1, c478 = 2,

8∑k=1

ckij Λk = ... −→ c517 = 1, c5

53 = 1,

8∑k=1

ckij ΞkQ = ... −→ c6

62 = 1, c618 = 1, c6

36 = 1,

8∑k=1

ckij Λkx = ... −→ c7

27 = 1, c773 = 1, c7

54 = 1/2,

8∑k=1

ckij ΞkxQ = ... −→ c8

46 = −1/2, c838 = 1.


Output: Read off the nonzero structure constans ckij and get

L1 L2 L3 L4 L5 L6 L7 L8L1 0 L1 0 L2 + 1/2L3 0 0 L5 L6L2 −L1 0 0 L4 0 −L6 L7 0L3 0 0 0 0 −L5 L6 −L7 L8L4 −L2 − 1/2L3 −L4 0 0 −1/2L7 −1/2L8 0 0L5 0 0 L5 1/2L7 0 L1 0 L2L6 0 L6 −L6 1/2L8 L1 0 L2 + L3 0L7 −L5 −L7 L7 0 0 L2 + L3 0 2L4L8 −L6 0 −L8 0 L2 0 2L4 0

.


Finding the Structure for Super KdV

Super KdV is Qt = Qxxx − aθQQxx + aQQθx + (6− 3a)QθQxQ = 0.Its defining system is

Λt − Λxxx − aQΛxθ + aθQΛxx = 0,

Γt + aQΛxQ − (3a − 6)Λx = 0,

3Ξ1x − Ξ2

t = 0,

2aQΞ1x − aQΓθ + aΛ = 0,

aθQΞ1x − aQΓ + aθΛ + 3Ξ1

xx − 3ΛxQ + aQΞθ = 0,

(3a − 6)Λθ + 2aθQΛxQ − aθQΞ1xx

−Ξ1t − 3ΛxxQ + Ξ1

xxx + aQΛθQ + aQΞ1xθ = 0,

(3a − 6)ΛQ − (3a − 6)Γθ + (6a − 12)Ξ1x = 0,

Ξ1xx = 0, Ξ1

xθ = 0,

Ξ1t = 0,

QΞ1θ = QΓ− θΛ− θQΞ1

x , irregular

Ξ1Q = 0,

Ξ2x = 0,

Ξ2t = 3Ξ1

x ,

Ξ2θ = 0, Ξ2

Q = 0,

Γx = 0, Γt = 0,

Γθ = Λ + 2QΞ1x ,

ΓQ = 0,

Λx = 0, Λt = 0,

Λθ = 0,

ΛQ = −2Ξ1x + Γθ.

where Ξ1 = Ξ1(x , t, θ),Ξ2 = Ξ2(t), Γ = Γ(t, θ), and Λ = Λ(x , t, θ,Q).


Necessary MONO Expansion

• The MONO expansion for all four infinitesimals is

Ξ1

Ξ2

ΓΛ

=

PO11 PO12 PE12PO21 PO22 PE22PE31 PE32 PO32PE41 PE42 PO42

(x,t)

θQθQ

+

PE11PE21PO31PO41

(x,t)

.

• Subs the expansion back to the defining system, we have Sred =

(PO11)x = (PO11)t = PO12 = PE12 = 0, (PE11)x = 2PE31, (PE11)t = 0;

PO21 = PO22 = PE22 = 0, (PE21)x = 0, (PE21)t = 6PE31;

(PE31)x = (PE31)t = PE32 = PO32 = 0, PO31 = −PO11;

PE41 = PO42 = 0, PE42 = −3PE31, PO41 = 0,

being REGULAR!


• Hence we have

Ξ1 = PO11θ + PE11, Ξ2 = PE21, Γ = PE31θ − PO11, Λ = −3PE31Q.

• Work out the Lie superbracket

[Li , Lj ] = [Ξ1i∂x + Ξ2i

∂t + Γi∂θ + Λi∂Q , Ξ1j

∂x + Ξ2j∂t + Γj∂θ + Λj

∂Q ]

= ((PO1j1PE3

j1 − PO1

j1PE3i1)θ + 2(PE1i1PE3

j1 − PE1

j1PE3i1) + (PO1i1PO1

j1 − PO1

j1PO1i1))∂x

+6(PE2i1PE3j1 − PE2

j1PE3i1)∂t − (PO1i1PE3

j1 − PO1

j1PE3i1)∂θ.

• Work out the structure constant relation

[Li , Lj ] =4∑

k=1

Ckij Lk

=4∑

k=1

Ckij (Ξ1k

∂x + Ξ2k∂t + Γk∂θ + Λk

∂Q )

=4∑

k=1

Ckij (PO1k1θ + PE1k1 )∂x +

4∑k=1

Ckij PE2k1∂t

+4∑

k=1

Ckij (PE3k1θ − PO1k1 )∂θ +

4∑k=1

Ckij (−3PE3k1Q)∂Q .


• Set an order of parametric derivatives, PE11 � PE21 � PE31 � PO11.

• Set up the linear system as the given order

4∑k=1

Ckij PE1k1 = 2(PE1i1PE3

j1 − PE1

j1PE3i1) + (PO1i1PO1

j1 − PO1

j1PO1i1),

4∑k=1


j1 − PE2

j1PE3i1),

4∑k=1

Ckij PE3k1 = 0,

4∑k=1

Ckij PO1k1 = PO1i1PE3

j1 − PO1

j1PE3i1.


• Provide two copies of initial data to the parametric derivatives

4∑k=1


j1 − PE1

j1PE3i1︸︷︷︸

a1b3−b1a3

) + (PO1i1PO1j1 − PO1

j1PO1i1︸︷︷︸

a4b4−b4a4

),

−→ c113 = 2, c1

44 = 1,

4∑k=1


j1 − PE2

j1PE3i1︸︷︷︸

a2b3−b2a3

),

−→ c223 = 6,

4∑k=1

Ckij PO1k1 = PO1i1PE3

j1 − PO1

j1PE3i1︸︷︷︸

a4b3−b4a3

.

−→ c443 = 1.

• The supercommutator table is

L1 L2 L3 L4L1 0 0 2L1 0L2 0 0 6L2 0L3 −2L1 −6L2 0 −L4L4 0 0 L4 L1

.


Discussion

• The most important advantage of MONO expansion is able tochange the irregular defining system to regular defining system.That is the KEY thing!

• With the help of MONO expansion, we are able to use Mapleto deal with more complicated systems.

• The way of finding structure is algorithmic.


Future Work

• Classification of super models with unspecified functions.

• Exploitation of super structure (mappings, etc.).


Acknowledgement

Thank

PIMS,George Bluman,

Stephen Anco and Alexei Cheviakov,Edgardo Cheb-Terrab,

Greg Reidand the Audience!


ReferencesP. J. Olver.Applications of Lie Groups to Differential Equations.Springer-Verlag, New York-Berlin. 1986.

G. J. Reid, I. G. Lisle, A. Boulton and A. D. Wittkopf.Algorithmic Determinination of Commutation Relations for Lie Symmetry Alge-bras of PDEs.Proc. ISSAC 1992.

M. A. Ayari.Supergroupes de Lie et solutions invariantes pour des equations differentiellesnon-lineaires a valeurs de Grassmann.Ph.D. Thesis, University of Montreal. 1997.

M. A. Ayari and V. Hussin.GLie: A MAPLE Program for Lie Supersymmtries of Grassmann-valued Differ-ential Equations.Comput. Phys. Comm. 100, 157-176, 1997.

I. G. Lisle and G. J. Reid.Symmetry Classification Using Non-commutative Invariant Differential Opera-tors.Comput. Math. 2006.


References

I. G. Lisle, S. L. Huang and G. J. Reid.Structure of Symmetry of PDE: Exploiting Partially Integrated Systems.SNC 2014..

algorithms for finding the lie superalgebra...

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