Symbolic determinant identity testing and non-commutative ranks of matrix Lie algebras

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Proves that many matrix Lie algebra representations produce singular matrix spaces that have no shrunk subspaces and, except for the zero or adjoint representation, no linear kernel; reduces SDIT for these spaces to PIT on upper‑triangular blocks via computation of a Cartan subalgebra and gives a polytime deterministic algorithm by combining the Lie‑group/Lie‑algebra correspondence (over C) with the De Graaf–Ivanyos–Rónyai Cartan computation; notes proof uses complex fields and leaves extending to other fields and a black‑box SDIT version as open problems.

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" chapters (17)

1. 0:00Outline
2. 1:08Polynomial identity testing
3. 2:48Symbolic determinant identity testing
4. 4:21Matrix space viewpoint
5. 5:32struct subspace
6. 9:08family switch
7. 13:32avi
8. 14:51new examples
9. 19:06Invariant subspace
10. 20:57Composition series
11. 21:55Linear kernel
12. 22:52Lie algebra
13. 25:25adjoint representation
14. 28:18key theorem
15. 28:38cheval level
16. 32:00conclusion
17. 32:48technical issues