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LieAlgebras[AscendingIdealsBasis] - find a basis for a solvable Lie algebra which defines an ascending chain of ideals
Calling Sequences
AscendingIdealsBasis(Alg)
Parameters
Alg - (optional) Maple name or string, the name of an initialized Lie algebra
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Description
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Every solvable Lie algebra admits a basis [e_1, e_2, ..., e_n] such that the vectors [e_1, e_2, ..., e_k] form an ideal in [e_1, e_2, ..., e_(k + 1)]. AscendingIdealsBasis calculates such a basis.
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Examples
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Example 1.
First we initialize a 5 dimensional Lie algebra.
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| (2.1) |
We can use the command Query/"Solvable" to check that this is a solvable Lie algebra.
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Alg1 >
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| (2.2) |
Now we calculate a basis with the ascending ideals property.
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Alg1 >
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![[_DG([["vector", Alg1, []], [[[1], 1], [[5], -2]]]), _DG([["vector", Alg1, []], [[[2], 1], [[4], -2], [[5], 3]]]), _DG([["vector", Alg1, []], [[[3], 1], [[4], -1]]]), _DG([["vector", Alg1, []], [[[1], 1]]]), _DG([["vector", Alg1, []], [[[2], 1]]])]](/support/helpjp/helpview.aspx?si=6625/file05782/math84.png)
| (2.3) |
The following two commands check, for example, that B[1..3] is an ideal in B[1..4].
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Alg1 >
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![C := BracketOfSubspaces(B[1 .. 3], B[1 .. 4])](/support/helpjp/helpview.aspx?si=6625/file05782/math96.png)
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![[_DG([["vector", Alg1, []], [[[1], -24], [[2], -14], [[3], 26], [[4], 2], [[5], 6]]]), _DG([["vector", Alg1, []], [[[1], 60], [[2], 32], [[3], -60], [[4], -4], [[5], -24]]]), _DG([["vector", Alg1, []], [[[1], -2], [[2], -2], [[3], 4], [[5], -2]]])]](/support/helpjp/helpview.aspx?si=6625/file05782/math99.png)
| (2.4) |
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Alg1 >
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![GetComponents(C, B[1 .. 3], trueorfalse = "on")](/support/helpjp/helpview.aspx?si=6625/file05782/math103.png)
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| (2.5) |
The command Query/"AscendingIdealsBasis" will verify that the basis B has the ascending ideals property.
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Alg1 >
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| (2.6) |
The ascending ideals property becomes apparent if we re-initialize the Lie algebra using the basis B (using the command LieAlgebraData).
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Alg1 >
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![_DG([["LieAlgebra", alg2, [5, table( [ ] )]], [[[1, 4, 1], -24], [[1, 4, 2], -14], [[1, 4, 3], 26], [[1, 5, 1], 10], [[1, 5, 2], 5], [[1, 5, 3], -11], [[2, 4, 1], 60], [[2, 4, 2], 32], [[2, 4, 3], -60], [[2, 5, 1], -10], [[2, 5, 2], -6], [[2, 5, 3], 10], [[3, 4, 1], -2], [[3, 4, 2], -2], [[3, 4, 3], 4], [[3, 5, 1], 7], [[3, 5, 2], 3], [[3, 5, 3], -8], [[4, 5, 1], -22], [[4, 5, 2], -11], [[4, 5, 3], 21]]])](/support/helpjp/helpview.aspx?si=6625/file05782/math137.png)
| (2.7) |
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Alg1 >
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| (2.8) |
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alg2 >
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![[[[,`| `,e1,e2,e3,e4,e5],[,-`---`,-`---`,-`---`,-`---`,-`---`,-`---`],[e1,`| `,0,0,0,-24 e1-14 e2+26 e3,10 e1+5 e2-11 e3],[e2,`| `,0,0,0,60 e1+32 e2-60 e3,-10 e1-6 e2+10 e3],[e3,`| `,0,0,0,-2 e1-2 e2+4 e3,7 e1+3 e2-8 e3],[e4,`| `,24 e1+14 e2-26 e3,-60 e1-32 e2+60 e3,2 e1+2 e2-4 e3,0,-22 e1-11 e2+21 e3],[e5,`| `,-10 e1-5 e2+11 e3,10 e1+6 e2-10 e3,-7 e1-3 e2+8 e3,22 e1+11 e2-21 e3,0]]]](/support/helpjp/helpview.aspx?si=6625/file05782/math151.png)
| (2.9) |
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