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

Algorithms for Finding the Lie Superalgebra Structure of Regular Super Differential Equations

Xuan Liu

Department of Applied Mathematics Western University

June 8, 2014 Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Contents

Outline

Background

Defining System MONO Expansion

Algorithms and Examples

Super KdV

Discussion and Future Work

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

Applications of Super Differential Equations

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

2. Even non-super (traditional models) can be embeded in a super model that makes determination of analytic solutions easier (e.g. SUSY quantum mechanic; and it has relations to potential symmetries).

3. Even the LHC has eliminated some popular supersymmetry models, supersymmetries are still useful. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Super Objects

Even and Odd

even · even = even even · odd= odd odd · odd = even

Lots of SUPER objects are involved! Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Grassman Algebra

Consider a set of N generators θ1, θ2, . . . , θN , which are as- sumed to have products θi θj such that (i) for all i, j, k = 1, ..., N,

(θi θj )θk = θi (θj θk ); (1)

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

2 θi θj = −θj θi (implies θi = 0); (2)

(iii) each non-zero product

θj1 θj2 ··· θjr

linearly independent of products involving less than r generators. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

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

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

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

NOTE: θ2 = 0 and Q2 = 0. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Determining equations for supersymmetries

1. Reduce to one-parameter Lie super transforamtion about the identity.

2. Apply superprolongation formula to the super DEs.

3. Replace the highest derivatives.

4. Pick the coeffiecients of monomials of the dependent variables and their derivatives.

5. Get the determining equations for supersymmetries.

[See Edgardo Cheb-Terrab’s implementations in Maple.] Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Parametric and Principle Derivatives

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

• Parametric derivatives Par(S) those which can not be obtained as the derivatives of any leading derivative,

• Principle derivatives Prin(S) those are a derivative of some leading derivative. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Regular Super DEs

REGULAR super differential equations have even coef- ficients of their highest derivatives and even parameters and parametric 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.] |{z} odd • For an odd function G(x,θ 1,θ 2), it can be expanded as

gˆ1(x) + fˆ1(x)θ1 + fˆ2(x)θ2 +ˆg 2(x)θ1θ2.

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

Decompose by odd variables MAPLE procedure: MONO • Decompose a super function by its odd variable monomials.

Example

• Even function F (x,θ 1,θ 2),

Fθ1θ1 = 0, Fθ2θ2 = 0, it can be expanded as

f1(x) + g 1(x)θ1 + g 2(x)θ2 + f2(x)θ1θ2. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Decompose by odd variables MAPLE procedure: MONO • Decompose a super function by its odd variable monomials.

Example

• Even function F (x,θ 1,θ 2),

Fθ1θ1 = 0, Fθ2θ2 = 0, it can be expanded as

f1(x) + g 1(x)θ1 + g 2(x)θ2 + f2(x)θ1θ2.

• For an odd function G(x,θ 1,θ 2), it can be expanded as

gˆ1(x) + fˆ1(x)θ1 + fˆ2(x)θ2 +ˆg 2(x)θ1θ2. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Decompose by odd variables MAPLE procedure: MONO • Decompose a super function by its odd variable monomials.

Example

• Even function F (x,θ 1,θ 2),

Fθ1θ1 = 0, Fθ2θ2 = 0, it can be expanded as

f1(x) + g 1(x)θ1 + g 2(x)θ2 + f2(x)θ1θ2.

• For an odd function G(x,θ 1,θ 2), it can be expanded as

gˆ1(x) + fˆ1(x)θ1 + fˆ2(x)θ2 +ˆg 2(x)θ1θ2. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Advantage of MONO Expansion

• A super function can be written more percisely.

• Eliminate the odd independences.

• Change irregular system to regular system.

• Apply MAPLE commmutative commands. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

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 be zero forms the new defining system Sred.

4. Send Sred to the commutative Maple commands rifsimp and initialdata

Output: Return the size of the symmetry group.

Sred: regular, no odd independences, monic, (later, in standard form). Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

A Simple Example

Consider a simple example Qxx = 0. Input: Defining system

 Λ(x, Q) = 0, S = xx (4) −Ξ(x, Q)xx + 2Λ(x, Q)xQ = 0.

1. MONO expansion of infinitesimals

 Ξ(x, Q) = f (x) + g (x) ∗ Q, 1 1 (5) Λ(x, Q) = g 2(x) + f2(x) ∗ Q.

2. Subsitution  g + f ∗ Q = 0, 2xx 2xx (6) −f1xx + Q ∗ g 1xx + 2f2x = 0. 3.   g 2xx = 0,  f = 0, 2xx (7) −f1 + 2f2 = 0,  xx x  g 1xx = 0. 4. Send (7) to the commutative Maple commands rifsimp and initialdata Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Maple Output

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

Super Initial Data Mapping

• Define super initial data mapping sID,

sID : {x} ∪ EvenPar(Sred) → F OddPar(Sred) → G (Grassman numbers).

0 n 0 • For x ∈ F , we say that sID is a specification at x if sID(x) = x0. For sID(x) = x0, we mean

0 (sID(x1), sID(x2), ..., sID(xn)) = x .

• This is a well-defined mapping. For any f in Sred, evaluating f is sID(f ) = f (sID(x), sID(Par(Sred)), Prin(Sred)). Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Super Riquier Existence and Uniqueness Theorem • Riquier Bases: the differetial analogs of Gr¨obner bases. • Super Riquier Bases: essentially same as Riquier Bases but with some odd functions which do not depend on odd variables. • Existence and Uniqueness Theorem Theorem Suppose that M is a super Riquier basis with certain ranking . 0 n 0 Fix x ∈ F . Let sID be a specification of initial data for M at x such that sID(f ) is well-defined for all f ∈ M. Then there is an unique solution

0 n u(x) ∈ F[[x − x ]] , if u(x) is even; 0 n u(x) ∈ G[[x − x ]] , if u(x) is odd,

0 i 0 i i to M at x such that Dαu (x ) = sID(δα) for all δα ∈ ParM.

Proof see [C. J. Rust, G. J. Reid, A. D. Wittkof, 1997]. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Finding Structure Constants Theorem

Theorem Suppose that S is a finite defining system with m1 even infinitesimals and m2 odd infinitesimals and Sred is the reduced defining system of S. Both S and Sred have d1 even parametric derivatives (finitely many) and d2 odd parametric derivatives (finitely many). Then under certain order of Par(Sred), the structure constants are uniquely determined. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Finding Structure Constant Algorithm

Input: S, Par(S), EvenInf(S), OddInf(S); Sred, Par(Sred).

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

m m X1 X2 L = Φi ∂ + Ψi ∂ i `1 X`1 `2 Y`2 `1=1 `2=1

and Lj has the same form.

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

m m X1 X2 [L , L ] = A`1 ∂ + B`2 ∂ . (8) i j X`1 Y`2 `1=1 `2=1 Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

3. Compute structure constants relation

d1+d2 X k [Li , Lj ] = cij Lk (9) k=1

m1 d1+d2 m2 d1+d2 X  X k k  X  X k k  = cij Φ` ∂X + cij Ψ` ∂Y . 1 `1 2 `2 `1=1 k=1 `2=1 k=1

4. The equations in (8) and (9) form a linear system with m1 + m2 equations.  d +d P 1 2 ck Φk = A1,  k=1 ij 1 m1 equations ..., (10) d +d  P 1 2 ck Φk = Am1 , k=1 ij m1

 d +d P 1 2 ck Ψk = B1,  k=1 ij 1 m2 equations ..., (11) d +d  P 1 2 ck Ψk = Bm2 , k=1 ij m2

5. Differentiate (10) and (11) w.r.t. Par(S) = {P1, ..Pd1+d2 }.

 d +d P 1 2 ck Pk = C 1,  k=1 ij 1  ...... , (12) ...... ,  d +d  P 1 2 ck Pk = C d1+d2 , k=1 ij d1+d2 Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

6. Substutite the MONO expansions of EvenInf(S) and OddInf(S) to step 5 and pick the coefficients of odd variable monomials

 d +d P 1 2 ck Pˆ k = Cˆ 1,  k=1 ij 1  ...... , (13) ...... ,  d +d  P 1 2 ck Pˆ k = Cˆ d1+d2 . k=1 ij d1+d2 ˆ ˆ where Par(Sred) = {P1, ..., Pd1+d2 }

7. Set an order of Par(Sred) as ˜ ˜ ˜ ˜ {P1, ..., Pd1 , Pd1+1, ..., Pd1+d2 }, | {z } | {z } even odd and rearrange (13) as the given order,

m +m  P 1 2 ck P˜ k = C˜ 1,  k=1 ij 1  ...... ,  m +m  P 1 2 ck P˜ k = C˜ d1 ,  k=1 ij d1 m +m (14) P 1 2 ck P˜ k = C˜ d1+1.  k=1 ij d +1  1  ...... ,  m +m  P 1 2 ck P˜ k = C˜ d1+d2 . k=1 ij d1+d2 Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

8. Provide two copies of intial data a1, ..ad1+d2 and b1, ..bd1+d2 for the Par(Sred) as the given order and read-off the nonzero struc- ture constants.

k Output: Nonzero structure constants cij and the supercom- mutator table. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Example Start with the defining system of the same example Qxx = 0. Input: S in standard form 1 S = {Λ = 0, Λ = Ξ , Ξ = 0, Λ = 0, Ξ = 0, Ξ = 0}, xx xQ 2 xx QQ QQ xxx xxQ

m1 = 1, m2 = 1, {Ξ, Ξx , ΛQ , Ξxx , Λ, ΞQ , Λx , ΞxQ }, d1 = 4, d2 = 4.

1. Write two supersymmetry vector fields Li , Lj ,

i i j j Li = Ξ ∂x + Λ ∂Q and Lj = Ξ ∂x + Λ ∂Q .

2. Take their Lie superbracket

A z }| { i j j i i j j i [Li , Lj ] = (Ξ Ξx − Ξ Ξx + Λ ΞQ − Λ ΞQ )∂x B z }| { i j j i i j j i +(Ξ Λx − Ξ Λx + Λ ΛQ − Λ ΛQ )∂Q . Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

3. Compute 8 X k [Li , Lj ] = cij Lk k=1 8 8 X k k X k k = ( cij Ξ )∂x + ( cij Λ )∂Q . k=1 k=1 4. The results in step 2 and 4 form the linear system with 1 + 1 equations. 8 X k k i j j i i j j i cij Ξ = Ξ Ξx − Ξ Ξx + Λ ΞQ − Λ ΞQ , (15) k=1 8 X k k i j j i i j j i cij Λ = Ξ Λx − Ξ Λx + Λ ΛQ − Λ ΛQ . (16) k=1 5. Set parametric derivatives in a certain order δ,

{Ξ, Ξx , ΛQ , Ξxx , Λ, ΞQ , Λx , ΞxQ .} 6. Skipped. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

7. Keep differentiating the two equation in step 5 w.r.t. parametric derivatives until we have 8 equations, and modulo them w.r.t parametric derivatives and set them as order δ,

8 X k k i j j i i j j i cij Ξ = Ξ Ξx − Ξ Ξx + Λ ΞQ − Λ ΞQ , k=1 8 X k k i j j j i j j i i j j i cij Ξx = Ξ Ξxx − Ξ Ξxx + ΞQ Λx − ΞQ Λx + Λ ΞxQ − Λ ΞxQ , k=1 8 X k k i j j i 1 i j j i i j j i cij ΛQ = ΞQ Λx − ΞQ Λx + (Ξ Ξxx − Ξ Ξxx ) + ΛQ ΛQ − ΛQ ΛQ , k=1 2 8 X k k i j j i i j j i cij Ξxx = Ξx Ξxx − Ξx Ξxx + 2(Λx ΞxQ − Λx ΞxQ ), k=1 8 X k k i j j i i j j i cij Λ = Ξ Λx − Ξ Λx + Λ ΛQ − Λ ΛQ , k=1 8 X k k i j j i i j j i i j j i cij ΞQ = ΞQ Ξx − ΞQ Ξx + Ξ ΞxQ − Ξ ΞxQ + ΛQ ΞQ − ΛQ ΞQ , k=1 8 X k k i j j i i j j i 1 i j j i cij Λx = Ξx Λx − Ξx Λx + Λx ΛQ − Λx ΛQ + (Λ Ξxx − Λ Ξxx ), k=1 2 8 X k k 1 i j j i i j j i cij ΞxQ = − (Ξxx ΞQ − Ξxx ΞQ ) + ΛQ ΞxQ − ΛQ ΞxQ . k=1 2 Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

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

8 X k k i j j i i j j i cij Ξ = (Ξ Ξx − Ξ Ξx ) + (Λ ΞQ − Λ ΞQ ) k=1 | {z } | {z } a b −b a 1 2 1 2 a5b6−b5a6 1 1 −→ c12 = 1, c56 = 1, 8 X k k i j j j i j j i i j j i cij Ξx = Ξ Ξxx − Ξ Ξxx + ΞQ Λx − ΞQ Λx + Λ ΞxQ − Λ ΞxQ k=1 | {z } | {z } | {z } a b −b a 1 4 1 4 a6b7−b6a7 a5b8−b5a8 2 2 2 −→ c14 = 1, c67 = 1, c58 = 1, 8 X 1 ck Λk = Ξi Λj − Ξj Λi + (Ξi Ξj − Ξj Ξi ) ij Q Q x Q x 2 xx xx k=1 | {z } | {z } a b −b a a6b7−b6a7 1 4 1 4 3 3 −→ c67 = 1, c14 = 1/2, 8 8 X k k 4 4 X k k 5 5 cij Ξxx = ... −→ c24 = 1, c78 = 2, cij Λ = ... −→ c17 = 1, c53 = 1, k=1 k=1 8 X k k 6 6 6 cij ΞQ = ... −→ c62 = 1, c18 = 1, c36 = 1, k=1 8 X k k 7 7 7 cij Λx = ... −→ c27 = 1, c73 = 1, c54 = 1/2, k=1 8 X k k 8 8 cij ΞxQ = ... −→ c46 = −1/2, c38 = 1. k=1 Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

k Output: Read off the nonzero structure constans cij and get

L1 L2 L3 L4 L5 L6 L7 L8 L1 0 L1 0 L2 + 1/2L3 0 0 L5 L6 L2 −L1 0 0 L4 0 −L6 L7 0 L3 0 0 0 0 −L5 L6 −L7 L8 L4 −L2 − 1/2L3 −L4 0 0 −1/2L7 −1/2L8 0 0 . L5 0 0 L5 1/2L7 0 L1 0 L2 L6 0 L6 −L6 1/2L8 L1 0 L2 + L3 0 L7 −L5 −L7 L7 0 0 L2 + L3 0 2L4 L8 −L6 0 −L8 0 L2 0 2L4 0 Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Finding the Structure for Super KdV

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

1 1 Ξxx = 0, Ξxθ = 0, 1 Ξt = 0, QΞ1 = QΓ − θΛ − θQΞ1 , irregular Λt − Λxxx − aQΛxθ + aθQΛxx = 0, θ x Γ + aQΛ − (3a − 6)Λ = 0, 1 t xQ x ΞQ = 0, 3Ξ1 − Ξ2 = 0, 2 x t Ξx = 0, 2aQΞ1 − aQΓ + aΛ = 0, 2 1 x θ Ξt = 3Ξx , aθQΞ1 − aQΓ + aθΛ + 3Ξ1 − 3Λ + aQΞ = 0, 2 2 x xx xQ θ Ξθ = 0, ΞQ = 0, 1 Γ = 0, Γ = 0, (3a − 6)Λθ + 2aθQΛxQ − aθQΞxx x t 1 1 1 Γ = Λ + 2QΞ1 , −Ξt − 3ΛxxQ + Ξxxx + aQΛθQ + aQΞxθ = 0, θ x 1 ΓQ = 0, (3a − 6)ΛQ − (3a − 6)Γθ + (6a − 12)Ξx = 0, Λx = 0, Λt = 0,

Λθ = 0, 1 ΛQ = −2Ξx + Γθ .

where Ξ1 = Ξ1(x, t, θ), Ξ2 = Ξ2(t), Γ = Γ(t, θ), and Λ = Λ(x, t, θ, Q). Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Necessary MONO Expansion

• The MONO expansion for all four infinitesimals is

 Ξ1   g g f   f  11 12 12  θ  11 2 g g f f  Ξ  =  21 22 22  Q +  21  .    f f g     g   Γ   31 32 32  θQ  31  f f g g Λ 41 42 42 (x,t) 41 (x,t)

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

(g 11)x = (g 11)t = g 12 = f12 = 0, (f11)x = 2f31, (f11)t = 0;

g 21 = g 22 = f22 = 0, (f21)x = 0, (f21)t = 6f31; (f31)x = (f31)t = f32 = g 32 = 0, g 31 = −g 11;

f41 = g 42 = 0, f42 = −3f31, g 41 = 0,

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

• Hence we have

1 2 Ξ = g 11θ + f11, Ξ = f21, Γ = f31θ − g 11, Λ = −3f31Q. • Work out the Lie superbracket

1i 2i i i 1j 2j j j [Li , Lj ] = [Ξ ∂x + Ξ ∂t + Γ ∂θ + Λ ∂Q , Ξ ∂x + Ξ ∂t + Γ ∂θ + Λ ∂Q ] j j j i i j j i i j j i = ((g 11f31 − g 11f31)θ + 2(f11f31 − f11f31) + (g 11g 11 − g 11g 11))∂x i j j i i j j i +6(f21f31 − f21f31)∂t − (g 11f31 − g 11f31)∂θ . • Work out the structure constant relation

4 X k [Li , Lj ] = Cij Lk k=1 4 X k 1k 2k k k = Cij (Ξ ∂x + Ξ ∂t + Γ ∂θ + Λ ∂Q ) k=1 4 4 X k k k X k k = Cij (g 11θ + f11)∂x + Cij f21∂t k=1 k=1 4 4 X k k k X k k + Cij (f31θ − g 11)∂θ + Cij (−3f31Q)∂Q . k=1 k=1 Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

• Set an order of parametric derivatives, {f11, f21, f31, g 11}.

• Set up the linear system as the given order

4 X k k i j j i i j j i Cij f11 = 2(f11f31 − f11f31) + (g 11g 11 − g 11g 11), k=1 4 X k k i j j i Cij f21 = 6(f21f31 − f21f31), k=1 4 X k k Cij f31 = 0, k=1 4 X k k i j j i Cij g 11 = g 11f31 − g 11f31. k=1 Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

• Provide two copies of initial data to the parametric derivatives

4 X k k i j j i i j j i Cij f11 = 2(f11f31 − f11f31) + (g 11g 11 − g 11g 11), k=1 | {z } | {z } a1b3−b1a3 a4b4−b4a4 1 1 −→ c13 = 2, c44 = 1, 4 X k k i j j i Cij f21 = 6(f21f31 − f21f31), k=1 | {z } a2b3−b2a3 2 −→ c23 = 6, 4 X k k i j j i Cij g 11 = g 11f31 − g 11f31 . k=1 | {z } a4b3−b4a3 4 −→ c43 = 1. • The supercommutator table is

L1 L2 L3 L4 L1 0 0 2L1 0 L2 0 0 6L2 0 . L3 −2L1 −6L2 0 −L4 L4 0 0 L4 L1 Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Discussion

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

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

• The way of finding structure is algorithmic. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Future Work

• Classification of super models with unspecified functions.

• Exploitation of super structure (mappings, etc.). Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

Acknowledgement

Thank CMS, Organizers, Greg Reid and the Audience! Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

References P. 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 differentielles non-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. Outline Background Defining System Algorithms and Examples Super KdV Discussion and Future Work Acknowledgement and References

References

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