Given a unitary matrix $U$ and given a direct sum of Lie algebras all being subalgebras of the Lie algebra of skew-Hermitian operators, e.g. $\mathfrak{su}(n) \oplus \mathfrak{su}(n)$. How do can I check whether or not there exists $g$ in the Lie algebra s.t. $U = e^g$.
#direct sum of Lie algebra
keen sand
hasty patioBOT
Given a unitary matrix $U$ and given a direct sum of Lie algebras all being sub...
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keen sand
a unitary matrix
naive lagoon
formulate your question more precisely, this is difficult to read
what does "lie algebra generated by hermitian" mean? what's the algebra, what's the bracket?
keen sand
I mean any subalgebra of the algebra of all skew-Hermitian operators where the Lie bracket is i times the commutator
keen sand
@naive lagoon I rephrased the question, is it lacking any details anymore?
naive lagoon
If I'm reading this right you're asking how to check whether a unitary matrix is in SU(n) x SU(n)
but then you're effectively asking whether U(n) is "isomorphic" to SU(n) x SU(n)
but you can't argue "up to isomorphism" in this context, when you apply isomorphisms (or even automorphisms) on a lie group of matrices, you get a new group
naive lagoon
<@215155833036734465> I rephrased the question, is it lacking any details anymo...
possibly, I still don't quite understand what you're getting at
if you're trying to formulate your argument up to isomorphism, then you need to reconsider, cuz that won't work
@keen sand
naive lagoon
Given a unitary matrix $U$ and given a direct sum of Lie algebras all being sub...
closest I can think of relative to this is to study the properties of the exponential map
I have no idea what you'll find tho
I haven't thought about lie groups from this angle
keen sand
@naive lagoon let me phrase it differently, Given a generator set of a lie algebra ${g_k}k$ (and I even now that this lie algebra is e.g. $su \oplus su $) is there a way to efficiently check wheter I can write a given unitary matrix as $U =\prod_l \prod_k e^{ \alpha{lk} g_k } $?
naive lagoon
a given unitary matrix
keen sand
yes
lean stagBOT
Oxana
real trellisBOT
@keen sand
real trellisBOT
@naive lagoon The user still needs help with this help request.