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derivedSeries -- compute the derived series of a set of vector fields

Synopsis

• Usage:
L=derivedSeries(k,M)
L=derivedSeries(k,m)
• Inputs:
• k, an integer, a nonnegative integer
• M, , of vector fields
• m, , of vector fields
• Outputs:
• L, a list, containing m or M, and the first k terms of the derived series. The types of the terms of the list agrees with the input type.

Description

Compute the first few terms of the derived series of a set of vector fields. For an object $n$, if we define $n^{(0)}=n$ and $n^{(i+1)}=bracket(n^{(i)},n^{(i)})$, then $n^{(i)}$ is the $i$th term of the derived series, and will be in the $i$th index position of the returned list.

As with bracket, this function computes different things, depending on the type of the parameter. See differences between certain bracketing functions for more information.

The following Lie algebra is generated by a finite-dimensional representation of gl_2.

 i1 : R=QQ[a,b,c]; i2 : D=derlog(a*c-b^2) o2 = image | 2b a 0 0 | | c 0 b a | | 0 -c 2c 2b | 3 o2 : R-module, submodule of R

The derived series (when provided with a Matrix) stabilizes as a representation of sl_2:

 i3 : ds=derivedSeries(3,gens D) o3 = {| 2b a 0 0 |, | 2b -2b 0 -2a 0 0 |, | 0 0 0 -4b 4b 0 -2a 2a 0 | c 0 b a | | c -c 0 0 a -a | | 0 0 0 -2c 2c 0 0 0 0 | 0 -c 2c 2b | | 0 0 0 2c 2b -2b | | 0 0 0 0 0 0 2c -2c 0 ------------------------------------------------------------------------ 0 2a -2a 0 0 0 |, | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8b -8b 0 0 0 -2a 0 0 0 2a 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4c -4c 0 0 0 -4b -2c 2c 0 4b 0 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ------------------------------------------------------------------------ 0 -8b 8b 0 0 0 0 0 0 0 0 0 0 0 0 0 -8a 8a 0 0 0 0 0 0 0 -8b 8b 0 0 0 0 -4c 4c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4a -4a 0 0 0 0 -4c 4c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8c -8c 0 8b -8b 0 0 0 0 0 0 0 0 0 ------------------------------------------------------------------------ 0 0 0 0 0 8b -8b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8a -8a 0 0 4a 0 0 0 4c -4c 0 0 0 0 -4a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8b 0 0 0 0 0 0 0 0 0 -8b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -8c 8c 0 ------------------------------------------------------------------------ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |} -4a 4a 0 0 4a -4a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | -8b 8b 0 0 8b -8b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | o3 : List i4 : apply(ds,f->gens trim image f) o4 = {| 2b a 0 0 |, | 2b a 0 |, | 2b a 0 |, | 2b a 0 |} | c 0 b a | | c 0 a | | c 0 a | | c 0 a | | 0 -c 2c 2b | | 0 -c 2b | | 0 -c 2b | | 0 -c 2b | o4 : List

The derived series (when provided with a Module) stabilizes immediately with a linear part isomorphic to sl_2, and a higher-order part equal to ideal (a,b,c)*D.

 i5 : derivedSeries(3,D) o5 = {image | 2b a 0 0 |, image | 2b a 0 0 0 0 |, image | 2b a | c 0 b a | | c 0 a bc b2 0 | | c 0 | 0 -c 2c 2b | | 0 -c 2b 2c2 2bc b2-ac | | 0 -c ------------------------------------------------------------------------ 0 0 0 0 |, image | 2b a 0 0 0 0 |} a bc b2 0 | | c 0 a bc b2 0 | 2b 2c2 2bc b2-ac | | 0 -c 2b 2c2 2bc b2-ac | o5 : List i6 : trim((image commutator(gens D))+(ideal (a,b,c))*D) o6 = image | 2b a 0 0 0 0 | | c 0 a bc b2 0 | | 0 -c 2b 2c2 2bc b2-ac | 3 o6 : R-module, submodule of R

This Lie algebra is generated by a finite-dimensional representation of t_2, a solvable Borel subalgebra of gl_2. Consequently, the linear part disappears:

 i7 : D=derlog(a*(a*c-b^2)) o7 = image | a 0 0 | | 0 b a | | -c 2c 2b | 3 o7 : R-module, submodule of R i8 : derivedSeries(3,gens D) o8 = {| a 0 0 |, | 0 0 0 |, 0, 0} | 0 b a | | 0 a -a | | -c 2c 2b | | 0 2b -2b | o8 : List i9 : derivedSeries(3,D) o9 = {image | a 0 0 |, image | 0 ac 0 ab 0 a2 0 |, image | | 0 b a | | a 0 bc 0 b2 0 0 | | | -c 2c 2b | | 2b -c2 2c2 -bc 2bc -ac b2-ac | | ------------------------------------------------------------------------ 2ab 0 a2 0 0 0 2ac2 0 0 |, image | 2ab2+a2c a2b -ac 2b2+ac 0 ab 0 a2 bc2 ac2 bc3 | | 0 0 -4bc 6bc -ac 2ac b2-ac 2ab 0 2bc2 2c4 | | -4b2c+ac2 -abc ------------------------------------------------------------------------ 0 0 a3 0 0 0 0 6abc2 0 a2c2 0 0 a2c ab2 0 a2b 0 0 a3 ac3 2b2c2-ac3 0 abc2 b3c 2abc 2abc -a2c 2a2c b3-abc ab2-a2c 2a2b -4bc3 2bc3 -ac3 2ac3 2ac3 ------------------------------------------------------------------------ 0 0 0 2ac5 0 |} 0 b4 ac4 bc5 bc7 | b2c2-ac3 2abc2 2bc4 0 2c8 | o9 : List