Adapt interface for orthogonal and symplectic Lie algebras #3391
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lgoettgens
added
topic: LieAlgebras
backport 1.0.x
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Bump @fingolfin |
| such that $f(xv, w) = -f(v, xw)$ for all $v, w \in R^n$. | ||
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| If `gram` is not provided, for $n = 2k$ the form defined by $\begin{matrix} 0 & I_k \\ -I_k & 0 \end{matrix}$ | ||
| is used, and for $n = 2k + 1$ the form defined by $\begin{matrix} 1 & 0 & 0 \\ 0 & 0 I_k \\ 0 & I_k & 0 \end{matrix}$. |
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Why that form in particular? Is that the form you used before?
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No, I used the identity matrix before, which for general users is a bad choice, as eg over QQ the Cartan sub is not split.
This is the form used most often in the literature to construct the B_n and D_n types, eg in Humphreys Introduction to Lie Algebras. I think we should provide just any default here, and thus particular one has, in most cases, nice properties.
| @assert characteristic(R) != 0 || dim == div(n^2 - n, 2) | ||
| s = ["x_$(i)" for i in 1:dim] | ||
| L = lie_algebra(R, n, basis, s; check=false) | ||
| set_attribute!(L, :type => :special_orthogonal, :form => form) |
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Maybe also add a documented form to query the form? Of course that can also come in a future PR.
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@fingolfin do you have a name idea for this function? form can be ambiguous with e.g. the killing form
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invariant_form (if it is invariant :))?
Co-authored-by: Max Horn <[email protected]>
* Small docs improvements * Adapt so interface * Add sp Lie algebra * Adapt to `vec` renaming * Don't assume anything about the kernel structure * Apply suggestions from code review Co-authored-by: Max Horn <[email protected]> --------- Co-authored-by: Max Horn <[email protected]> (cherry picked from commit db3dc1f)
### Backported PRs - [x] #3367 - [x] #3379 - [x] #3369 - [x] #3291 - [x] #3325 - [x] #3350 - [x] #3351 - [x] #3365 - [x] #3366 - [x] #3382 - [x] #3373 - [x] #3341 - [x] #3346 - [x] #3381 - [x] #3385 - [x] #3387 - [x] #3398 - [x] #3399 - [x] #3374 - [x] #3406 - [x] #2823 - [x] #3298 - [x] #3386 - [x] #3412 - [x] #3392 - [x] #3415 - [x] #3394 - [x] #3391
benlorenz
removed
the
backport 1.0.x
As discussed with @fingolfin in some slack thread a long time ago, the function
special_orthogonal_lie_algebrashould be able to handle an arbitrary symmetric bilinear form as input. This is now possible; the previous behavior can be achieved by providingidentity_matrix(ZZ, n)as the form.Furthermore, I added the same interface for the new
symplectic_lie_algebrafunction.