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solve_group

  

represent a Lie Algebra of symmetry generators in terms of derived algebras

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

solve_group(G, y(x))

Parameters

G

-

list of symmetry generators

y(x)

-

dependent and independent variables

Description

• 

solve_group receives a list G of infinitesimals corresponding to symmetry generators that generate a finite dimensional Lie Algebra G, and returns a representation of the derived algebras of G.

• 

Derived algebras Gi of G are defined recursively as follows:

  

1 is G;

  

G is the Lie Algebra obtained by taking all possible commutators of 1;

  

in general, Gi+1 is the Lie Algebra obtained by taking all possible commutators of Gi.

• 

Since G is assumed to be finite, there exists a positive integer n with the following properties:

  

(i) Gn+1 = Gn

  

(ii) n is the smallest integer possessing property (i).

• 

solve_group returns a list L of n+1 lists of symmetries with the following properties:

  

The symmetries inside the list L1 form the basis for Gn

  

The symmetries inside the lists L1 and L2 together form the basis for Gn1.

  

In general, the symmetries inside the first n+1i lists of L together form the basis for Gi.

  

In other words, map(op, L[1..n+1-i]) is a basis for Gi.

  

The group G is solvable if Gn is the zero group. If G is solvable then the first element of the returned list L will be the empty list [].

• 

This function is part of the DEtools package, and so it can be used in the form solve_group(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[solve_group](..).

Examples

withDEtools:

Gξx,y,ηx,y

G:=ξx,y,ηx,y

(1)

solve_groupG,yx

,ξx,y,ηx,y

(2)

G200,1,1,0

G20:=0,1,1,0

(3)

XcommutatoropG20,yx

_ξ=0,_η=0

(4)

solve_groupG20,yx

,0,1,1,0

(5)

G210,1,0,y

G21:=0,1,0,y

(6)

XcommutatoropG21,yx

_ξ=0,_η=1

(7)

solve_groupG21,yx

,0,1,0,y

(8)

G1,0,0,1,ⅇy,0

G:=1,0,0,1,ⅇy,0

(9)

XcommutatorG1,G2,yx

_ξ=0,_η=0

(10)

XcommutatorG1,G3,yx

_ξ=0,_η=0

(11)

XcommutatorG2,G3,yx

_ξ=ⅇy,_η=0

(12)

solve_groupG,yx

,ⅇy,0,1,0,0,1

(13)

SL20,1,0,y,0,y2

SL2:=0,1,0,y,0,y2

(14)

XcommutatorSL21,SL22,yx

_ξ=0,_η=1

(15)

XcommutatorSL21,SL23,yx

_ξ=0,_η=2y

(16)

XcommutatorSL22,SL23,yx

_ξ=0,_η=y2

(17)

solve_groupSL2,yx

0,1,0,2y,0,y2

(18)

See Also

canoni

DEtools

DEtools/reduce_order

dsolve,Lie

equinv

eta_k

PDEtools

symgen

Xcommutator

 


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