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Graduate student in mathematics.


Mar
26
comment $U(1)$ generators of $SU(2)$
All $U(1)$ subgroups of $SU(2)$ are conjugate in $SU(2)$.
Mar
26
awarded  Informed
Mar
26
comment Is the determinant the “only” group homomorphism from $\mathrm{GL}_n(\mathbb R)$ to $\mathbb R^\times$?
I'm not sure I see how knowing that $SL$ is the commutator of $GL$ tells you that the trace is the unique form that vanishes on commutators. You could probably prove this though using uniqueness of the Killing form on $\mathfrak{sl}_n(\mathbb R)$.
Mar
26
answered Is the determinant the “only” group homomorphism from $\mathrm{GL}_n(\mathbb R)$ to $\mathbb R^\times$?
Mar
26
comment Is the determinant the “only” group homomorphism from $\mathrm{GL}_n(\mathbb R)$ to $\mathbb R^\times$?
a Lie group homomorphism is determined by the corresponding Lie algebra homomorphism. So what are all of the Lie algebra homomorphisms $\mathfrak{gl}_n(\mathbb R) \to \mathbb R$? These are the elements of the dual space of $\mathfrak{gl}_n(\mathbb R)$ that vanish on commutators. Are multiples of the trace the only ones (the trace is the derivative of $\det$)?
Mar
22
comment Prove the Curvature Tensor is a Tensor
@user13223423 if $T$ is a tensor and $D$ is some differential operator, $T = T+D - D$ is a tensor but neither $T+D$ nor $D$ are tensors. Of course, the sum of tensors is again a tensor. With regarding your other question, tensors can be characterized as those linear operators on sections of a vector bundle that are linear over $C^\infty(M$).
Mar
22
revised Prove the Curvature Tensor is a Tensor
added 1 characters in body
Mar
22
comment Prove the Curvature Tensor is a Tensor
Each of those terms is not a tensor on their own. You also had a typo in the definition of $R$, which I edited.
Mar
21
comment Time-dependent flat line bundle
@JoshBurby yep that's correct
Mar
21
comment Example: Maps of Constant Rank - Composition not of Constant Rank
what have you tried? sounds like a homework question.
Mar
20
comment Time-dependent flat line bundle
@JoshBurby see my edit. hope this helps.
Mar
20
revised Time-dependent flat line bundle
added 1097 characters in body
Mar
20
revised determinating the signature of a bilinear form
edited body
Mar
20
comment determinating the signature of a bilinear form
It looks like you have a typo, what should $y_n + 1 - i$ be? Should it be $y_{n+1-i}$ and then the sum should really start at $i=1$ instead of $0$?
Mar
19
answered Time-dependent flat line bundle
Mar
19
comment Time-dependent flat line bundle
I'm guessing you want the connection on $L$ to be that induced from each $\ell_t$?
Mar
18
comment How can I prove that $\mathcal{O}_X(-n-1) \simeq \Lambda^n(T_X)^*$?
Take the top exterior power of the euler sequence: en.wikipedia.org/wiki/Euler_sequence
Mar
18
comment What topology has $Pic(X)$?
In the case that $X$ is a Riemann surface, I think what you're after is called the Jacobi variety.
Mar
18
revised Connection(gauge field) in Fubini-Study metric is pull back of a connection A of line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^{N-1}$
added tag
Mar
18
answered Connection(gauge field) in Fubini-Study metric is pull back of a connection A of line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^{N-1}$