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 = fx1;:::; xmg; n independent odd variablesΘ= fθ1;:::;θ ng, q dependent even variables A = fA1;:::; Aqg and p dependent odd variables Q = fQ1;:::; Qpg. 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 g^1(x) + f^1(x)θ1 + f^2(x)θ2 +^g 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 g^1(x) + f^1(x)θ1 + f^2(x)θ2 +^g 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 g^1(x) + f^1(x)θ1 + f^2(x)θ2 +^g 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. 8 > 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 : fxg [ EvenPar(Sred) ! F OddPar(Sred) ! G (Grassman numbers): 0 n 0 • For x 2 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 2 F . Let sID be a specification of initial data for M at x such that sID(f ) is well-defined for all f 2 M. Then there is an unique solution 0 n u(x) 2 F[[x − x ]] ; if u(x) is even; 0 n u(x) 2 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 2 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.