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LieAlgebras[Series] - find the derived series, lower central series, or upper central series of a Lie algebra or a Lie subalgebra

Calling Sequences

     Series(AlgName, keyword)

     Series(S, keyword)

Parameters

     AlgName    - (optional) the name of a Lie algebra 𝔤

     keyword    - a string, one of "Derived", "Lower", "Upper"

     S          - a list of vectors defining a basis for a Lie subalgebra of a Lie algebra 𝔤

 

Description

Examples

Description

• 

The derived series of a Lie algebra 𝔤 is the sequence of ideals Dk𝔤𝔤 defined inductively by D0𝔤 = 𝔤 and Dk+1𝔤 = Dk𝔤 , Dk𝔤. See BracketOfSubspaces for the definition of the Lie bracket A, B of 2 subspaces A, B 𝔤. Note that Dk+1𝔤  Dk𝔤 . The derived series terminates when Dk+1𝔤 = 0 or Dk+1𝔤 = Dk𝔤. The Lie algebra 𝔤  is solvable if  Dk+1𝔤 = 0.

• 

The lower central series of a Lie algebra 𝔤 is a sequence of ideals defined inductively by L0𝔤 = 𝔤 and Lk+1𝔤 = 𝔤 , Lk𝔤. Note that Lk+1𝔤  Lk𝔤 . The lower central series terminates when Lk+1𝔤 = 0 or Lk+1𝔤 = Lk𝔤. The Lie algebra 𝔤 is nilpotent if  Lk+1𝔤 = 0 .

• 

If h 𝔤 is an ideal, then the generalized center is GCh = {x 𝔤 | x, y h for all y  𝔤}. The upper central series of a Lie algebra 𝔤is a sequence of ideals Ck𝔤 defined inductively by C0𝔤 = GC0 and Ck+1𝔤 =GCCk𝔤. Note that Ck𝔤  Ck+1𝔤 . The upper central series terminates when Ck+1𝔤 = 𝔤 or Ck+1𝔤 = Ck𝔤.

• 

Series(AlgName, keyword) calculates the series defined by the keyword for the Lie algebra AlgName. If the first argument AlgName is omitted, then the appropriate series of the current Lie algebra is found.

• 

Series(S, keyword) calculates the series defined by the keyword for the Lie subalgebra S (viewed as a Lie algebra in its own right).

• 

Series returns a list of list of vectors L = A1, A2, ... where Ak is a basis for the k1 term in the appropriate series. The list L ends with Am if [i] Am1 = Am; or [ii] in case of the derived and lower series if Am =; or [iii] in the case of the upper series Am= 𝔤 .

• 

The dimensions of the subalgebras in these series can be easily computed with the Maple map and nops commands.

• 

The command Series is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Series(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Series(...).

Examples

with(DifferentialGeometry): with(LieAlgebras):

 

Example 1.

First we initialize a Lie algebra and display the multiplication table.

L1 := _DG([["LieAlgebra", Alg1, [5]], [[[2, 3, 1], 1], [[2, 5, 3], 1], [[4, 5, 4], 1]]]);

L1:=e2,e3=e1,e2,e5=e3,e4,e5=e4

(2.1)

DGsetup(L1):

 

The derived series:

Alg1 > 

DS := Series("Derived"); map(nops, DS);

DS:=e1,e2,e3,e4,e5,e1,e3,e4,

5,3,0

(2.2)

 

The lower central series:

Alg1 > 

LS := Series("Lower"); map(nops, LS);

LS:=e1,e2,e3,e4,e5,e1,e3,e4,e1,e4,e4,e4

5,3,2,1,1

(2.3)

 

The upper central series:

Alg1 > 

US := Series("Upper"); map(nops, US);

US:=e1,e3,e1,e3,e2,e1,e1,e3,e2

1,2,3,3

(2.4)

 

Example 2.

We compute the different series for the subalgebra S = e1, e2, e3, e4.

Alg1 > 

S1 := [e1, e2, e3, e4]:

 

The derived series:

Alg1 > 

DS := Series(S1, "Derived"); map(nops, DS);

DS:=e1,e2,e3,e4,e1,

4,1,0

(2.5)

 

The lower central series:

Alg1 > 

LS := Series(S1, "Lower"); map(nops, LS);

LS:=e1,e2,e3,e4,e1,

4,1,0

(2.6)

 

The upper central series:

Alg1 > 

US := Series(S1, "Upper"); map(nops, US);

US:=e4,e1,e2,e1,e3,e4

2,4

(2.7)

See Also

DifferentialGeometry

LieAlgebras

BracketOfSubspaces

Center

Centralizer

GeneralizedCenter

Query[Nilpotent]

Query[Solvable]