Obtaining a Lie algebra from a list of matrices

liealgebra

asked 2025-10-10 01:12:31 +0100

klaaa
713 ●4 ●23 ●55

updated 2025-10-10 01:17:24 +0100

Given a finite list W of nilpotent matrices in Sage. How can I obtain the Lie algebra L generated by those matrices in W? And how can I find the vector space dimension and test whether L is simple using Sage? I googled a bit and found the command:

L=LieAlgebra(associative=[M, N])

here [M,N] is a list of two matrices. It works to give me a Lie algebra, but then nearly no command works to find the dimension of L (I guess it is somehow in the wrong category?)

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answered 2025-10-11 13:03:42 +0100

rburing

11084 ●6 ●81 ●222 https://www.rburing.nl/

At least you can use the interface to GAP:

sage: MS = MatrixSpace(QQ, 2)
sage: m1 = MS([[0, -1], [1, 0]])
sage: m2 = MS([[-1, 4], [3, 2]])
sage: mats = [m1, m2]
sage: gap_L = gap.LieAlgebra(QQ, mats)
sage: gap.Dimension(gap_L)
4
sage: gap.SemiSimpleType(gap_L)
fail

Simple example:

sage: MS = MatrixSpace(QQ, 2)
sage: e = MS([[0, 1], [0, 0]])
sage: f = MS([[0, 0], [1, 0]])
sage: h = MS([[1, 0], [0, -1]])
sage: mats = [e, f, h]
sage: gap_L = gap.LieAlgebra(QQ, mats)
sage: gap.Dimension(gap_L)
3
sage: gap.SemiSimpleType(gap_L)
A1

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Asked: 2025-10-10 01:12:31 +0100

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Last updated: Oct 11

Obtaining a Lie algebra from a list of matrices edit

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