Tag: lie

calculus and analysis – How to expand Lie characters?

The following involves characters of affine Lie algebras, and I will be using as reference the book on CFT by Francesco et at (here are some screenshots if useful). But hopefully the post will be self-contained.

Let
$$
Theta^k_lambda=sum_{ninmathbb Z}expbig(-2pi i(knz+frac12lambda z-kn^2tau-nlambdatau-lambda^2tau/4k)big)tag{14.176}
$$
and
$$
chi^k_lambda=frac{Theta^{k+2}_{lambda+1}-Theta^{k+2}_{-lambda-1}}{Theta^2_1-Theta^2_{-1}}tag{14.174}
$$

The idea is to expand $chi$ for $lambda=k=1$, $z=0$, in powers of $q=e^{2pi i tau}$. The expected result is
$$
chi^1_1=q^{5/24}(2+2q+6q^2+8q^3+cdots)tag{14.179}
$$

How can I use Mathematica to recover eq.$14.179$ from the other two equations? The naive approach does not quite work, because $chi$ yields an indeterminate form if we take $lambda=1,z=0$ directly. And the sum does not converge for $|mathrm{re}(q)|<1$ (and it cannot be analytically continued), so the expansion around $q=0$ is only asymptotic (not a proper power series in the strict sense).

spells – Can someone under the effect of Join Pasts lie?

Specifically looking at Pathfinder’s Join Pasts, which reads:

With touches to the targets’ foreheads, you bring them into mental communion. The targets can share thoughts and experiences, but not words. When one target attempts to Recall Knowledge, the other can Aid the first target’s skill check, using any Lore skill (even if that Lore wouldn’t normally apply) without having made any preparations to Aid.

Could someone under the effect of this cantrip lie when sharing a thought or an experience? In my game, we’ve used this spell to look at a target’s past experiences and see things through their eyes – but the idea that they could just lie (for instance, by sharing that they grew up impoverished or were stolen from) came up recently.

Show that there is a unique Lie algebra over F of dimension 3 whose derived algebra has dimension 1 and lies in Z(L).

This is an exercise in Humphrey’s ‘Introduction to Lie Algebras and Representation Theory’ (chapter 1.2 number4).

Here is what I’ve done.

Since $(LL)$ has dimension 1, let {$x$} be a basis for $(LL)$. Extend it to a basis {$x, y, z$} for L.Then $(xy)=ax, (yz)=bx$ and $(xz)=cx$. (Not all $a, b, c$ are zeros.) Since ($LL$) lies in $Z(L)$, $0=((xy)z)=a(xz)$ and $0=((xz)y)=c(xy)$. Then we haver four cases where each factor is zero or not zero.

If $(xz)=(xy)=0$, then $(yz)=bx$ should not be zero.

If $a=c=0$, then again, $(xz)=(xy)=0$ and $(yz)=bx$ should not be zero.

Now, here is where I stucked. If ‘$a=0$ and $(xz)neq 0$‘ or ‘$a neq 0$ and $(xz)=0$‘, then how can I get the result “$(yz)=bx$ is not zero.”? Or is there better way to solve this problem?

Thanks in advance.

differential geometry – Transitive action of a lie group in a connected manifold

I am trying to see why is it that if we have a transitive action of a lie group $Grightarrow M$ then if this action is transitive the connected component of the identity $G^0$ also acts transitively on $M$ and that for all $pin M$, $G/G_0 cong G_p/(G_pcap G^0)$.

Since the action is transitive we know that $G/G_pcong M$ and since the map $Grightarrow G_p$ is a submersion we get that the map $Grightarrow M, grightarrow g.p$ is open . I don’t know how useful this is since we only know that $G^0$ is a closed set I don’t think it has to be open since connected components don’t necessarily need to be open, and also I am not sure if this map is even closed or if it is how I could try and prove it.

Any enlightment is appreciated. Thanks in advance.

dnd 5e – Is it ok to lie to players rolling an insight?

A successful Insight check should reveal useful information to players, and an unsuccessful check should emphasize uncertainty

My guiding principles with Insight are:

  • That it reveals information about the immediate situation being
    examined, and not necessarily about the world itself
  • Success suggests a keen understanding of the circumstance, while
    failure indicates poor understanding

I feel it is important to include both elements.

In your specific example, I would present these elements as the bloodthirsty servant being sincere in his belief that he works for the paladin on a successful Insight check. Players should not learn the true state of the world (definitively discovering the relationship between these NPCs just based on evaluating the servant’s claim) from examining what the servant says.

In the case of a failed Insight check I would narrate the outcome as a lack of information, rather than being certain that incorrect information is definitely accurate. The failure of insight just means that they have no particular understanding of meta-information about the servant and the assertion the servant has made. The narration I favor would emphasize that– “he doesn’t seem to be obviously lying to you, but you can’t get a good read on him at all.”

I recommend not forbidding checks because there is no dishonesty to discover. This directly reveals to players the same information they would get on a successful Insight check, but without having to roll. When something seems off to players, or they want to double-check information they receive, they should be encouraged to try to find out via their characters’ ability to examine what they know and perceive.

I suggest not giving false information on a poor roll for similar reasons: players will know they rolled poorly (unless you use a hidden-roll mechanic), and so telling them definitive information pretty clearly marks that information as unreliable. I also advocate not directly lying to players more broadly, but that’s out of scope here.

My experiences with running Insight checks this way have been that they help situate players in the game, even if it doesn’t shed much light on the plot. They want more information, and if an Insight check doesn’t provide it they either have to hope for the best and stay wary, or they try to verify claims in other ways (like investigating the claim after the conversation). The opportunity for NPCs to lie to or otherwise deceive the PCs is a part of the adventure, distinct from the players’ dependence on me, the GM, to provide the information necessary for the players play the game at all.

dg.differential geometry – On the orbit of a Frechet Lie group action

Suppose that $G$ is a Fréchet Lie group acting on a Fréchet manifold $X$.
Fix $xin X$ and let $alpha(t)$ be a smooth path in $X$ such that
$$
begin{cases}
alpha(0)=x\
alpha(t)in Gcdot x
end{cases}.
$$
Also denote $rho_{x}:Grightarrow X:gmapsto gcdot x$. Is it true that $alpha'(0)in text{Im}(d_{e}rho_{x})$?

In the finite dimensional setting, this is clearly the case: the orbit $Gcdot x$ is a weakly embedded submanifold of $X$, hence $alpha(t)$ is also smooth as a curve in $Gcdot x$. Consequently $$alpha'(0)in T_{x}(Gcdot x)=text{Im}(d_{e}rho_{x}).$$

In the case that is of interest to me, $G=Diff(M)$ is the space of diffeomorphisms of a compact manifold $M$, and $X$ is the Fréchet space of rank $k$ – distributions $Gamma(Gr_{k}(M))$.

lie groups – Centralizers of semisimple subgroups

If $G$ is a simple Lie group, and $rho: G to GL(V)$ is a representation, then by Shur’s lemma, the group of automorphisms of $rho$ is a reductive subgroup of $GL(V)$. I’m wondering whether this generalizes to the case where $GL(V)$ is replaced by an arbitrary reductive group?

More generally: if $G$ is a semisimple algebraic group (or even reductive), $H$ is a reductive algebraic group, and $rho : G to H$ is a homomorphism, is the centralizer of the image of $rho$ in $H$, $C_{H}(rho(G))$, a reductive subgroup of $H$?

lie groups – Real-world applications for Lie Algebra?

I’m currently trying to get an understanding for Lie Algebra for which I found a lot of useful literatures. However, something that seems to be missing for is the applications of Lie Algebra in real world applications besides maybe the geology. Can anybody state some applications of Lie Algebra (specifically to which kind Lie group they relate to)?

Thank you

differential topology – Equivariant Morse theory for non-compact Lie groups

Let $G$ be a Lie group acting properly on a smooth manifold $M$. The (non-equivariant) definition of a Morse function does not carry over to equivariant functions $M rightarrow mathbb{R}$ (where $mathbb{R}$ has the trivial action) since critical points are no longer isolated: Every orbit of a critical point will consist entirely of critical points.

However, there is an adaption via Morse-Bott functions $f: M rightarrow mathbb{R}$ whose set of critical points is a disjoint union of submanifolds, so called critical submanifolds. Instead of demanding the Hessian at a critical point to be non-degenerate, one requires the Hessian to be non-degenerate on the normal bundle of these submanifolds. We call such submanifolds non-degenerate.

Let us say that a $G$-equivariant map $f : M rightarrow mathbb{R}$ is a $G$-equivariant Morse function if it is a Morse-Bott function whose critical submanifolds are non-degenerate orbits.*

If I understood it correctly, in the late 60’s Wasserman proved the following (see also lemma 4.8 here).

Let $G$ be a compact Lie group. The set of $G$-equivariant Morse functions is dense in the space of all smooth $G$-equivariant maps $C^{infty, G}(M, mathbb{R})$ (equipped with the subspace topology where $C^infty(M,mathbb{R})$ has the strong topology).

This result horribly fails when $G$ is non-compact. Indeed, in this paper Illman and Kankaanrinta prove that if $G$ is a non-compact Lie group acting properly on $M$; and $N$ is any $G$-manifold, then the topology on
$C^{infty, G}(M, N)$ induced from the strong topology on $C^{infty}(M, N)$ is discrete! In their paper, they introduce the so called “strong-weak” topology which sits inbetween the weak and the strong topology on $C^infty(M,N)$. In particular, if $G$ is compact or acts trivial the strong-weak topology is the strong topology, whereas it is the weak topology if $G$ acts cocompactly.

The strong-weak topology seems to be very adequate for the study of equivariant maps, for instance the authors prove that equivariant proper embeddings between proper $G$-manifolds are open in $C^{infty, G}(M,N)$ for the strong-weak topology. Similar results can be found in the linked paper.

Of course now I wonder:

Does Wasserman’s result remain true in the non-compact case if we consider proper Lie group actions and equip $C^{infty, G}(M, mathbb{R})$ with the strong-weak topology?

*(in Wasserman’s paper these are precisely the functions in a space called $mathfrak{M}(M,M)$. His definition of Morse function makes a further technical assumption which I do not need).

dg.differential geometry – characteristic criterion for an invariant curve in Lie Theory

Let be $frac{dy}{dx}=w(x,y)$ an ODE, and let be $(hat{x},hat{y}) = (hat{x}(x,y;t),hat{y}(x,y;t))$ a symmetry of a one parameter Lie Group. In the book “Symmetry Methods for Differential Equations” by Peter E.Hydon it is given the next criterion in order to know if a solution cuvre $C:y=f(x)$ is invariant:

enter image description here

I would like to know why the vector $(1,y'(x))$ tangent to C at $(x,y)$ must be parallel to the tangent vector to the orbit curve at $(x,y)$ that is $(xi(x,y),eta(x,y))$.
I dont understand why they have to be paralell.