LieAlgebras[Series] - find the derived series, lower central series, 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
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Description
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The derived series of a Lie algebra is the sequence of ideals D^k(g) in g defined inductively by D^0(g) = g and D^(k + 1)(g) = [D^k(g), D^k(g)]. Note that D^(k + 1)(g) is contained in D^k(g).
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The lower central series of a Lie algebra is a sequence of ideals L^k(g) in g defined inductively by L^0(g) = g and L^(k + 1)(g) = [g, L^k(g)]. Note that L^(k + 1)(g) is contained in L^k(g).
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The upper central series of a Lie algebra is a sequence of ideals C^k(g) in g defined inductively by C^0(g) = GeneralizedCenter(0) and C^(k + 1)(g) = GeneralizedCenter(C^k(g)). Note that C^(k)(g) is contained in C^(k + 1)(g). If h is an ideal of the Lie algebra g, then GeneralizedCenter(h) is the ideal of vectors x in g such that [x,y] in h for all y in g.
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LieAlgebraSeries(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.
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LieAlgebraSeries(S, keyword) calculates the series defined by the keyword for the Lie subalgebra S (viewed as a Lie algebra in its own right).
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LieAlgebraSeries returns a list of list of vectors L = [A_1, A_2, ...] where A_i is a basis for the (i - 1) term in the appropriate series. The list L with A_m terminates if [i] A_(m - 1) = A_m; or [ii] in case of the derived and lower series if A_m = []; or [iii] in the case of the upper series A_1 = [] or A_m = all of g.
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The dimensions of the subalgebras in these series can be easily computed with the Maple map and nops commands.
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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(...).
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Examples
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Example 1.
First we initialize a Lie algebra and display the multiplication table.
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The derived series:
Alg1 >
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The lower central series:
Alg1 >
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| (2.3) |
The upper central series:
Alg1 >
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Example 2.
We compute the different series for the subalgebra S = [e1, e2, e3, e4].
Alg1 >
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The derived series:
Alg1 >
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The lower central series:
Alg1 >
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The upper central series:
Alg1 >
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| (2.7) |
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