Implementing the Casimir elements of a finite dimensional Lie algebra #35590
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Do you have a reference? I don't understand the tensor equation for higher orders, and I'm only familiar with the order-2 case. |
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It is just the definition of being in the center, which you can see on Wikipedia. |
src/sage/categories/finite_dimensional_lie_algebras_with_basis.py
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src/sage/categories/finite_dimensional_lie_algebras_with_basis.py
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| sage: L = LieAlgebra(QQ, cartan_type=['B', 2]) | ||
| sage: U = L.pbw_basis() | ||
| sage: C = L.casimir_element(UEA=U) | ||
| sage: all(g * C == C * g for g in U.algebra_generators()) |
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do we have a test function is_central ?
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Not that I am aware at this level of generality (there is one for semigroup algebras though). It certainly wouldn’t be difficult to add it to the appropriate categories. Although I am slightly worried about ambiguities with “central as an object in category X” and compared to supecenters. Everything is probably okay, but it would need its own PR.
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Thank you. Fixed. |
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There are some glitches in the doc (in the Lie algebra file) to be fixed after looking at the doc. |
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Indeed, good catch. That likely was also breaking the pdf doc. |
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and now the linter is not happy anymore |
That should fix it, although I don't know why it wasn't being caught before... |
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Everything is happy (well, right now the build&tests hasn't finished, but it was passing before and the changes I made should not affect it). |
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Documentation preview for this PR (built with commit b808b91) is ready! 🎉 |
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Thank you. |
📚 Description
The Casimir element is an important element in the center of the universal enveloping algebra of a simple Lie algebra. We provide a generic implementation for all orders (> 1) for all finite dimensional Lie algebras.
📝 Checklist
⌛ Dependencies