2
votes
0answers
55 views

First Variation of Dyson Series/Magnus Expansion

Given the matrix differential equation $\frac{dU_t}{dt}=A_t U_t$ there are at least two ways to write a formal solution. Both the Dyson series: $U_t = \mathcal{T} e^{\int_{0}^{t} A_t dt}$ and the ...
10
votes
2answers
431 views

Geodesics on $SU(4)$

Are the geodesics of the following metrics on $SU(4)$ known or easy (in a way not known to me!) to find? In the adjoint representation, one can express the Killing form as a matrix and consider it as ...
1
vote
1answer
67 views

Smallest subalgebra of $\mathfrak{su}(4)$ arrising from a control problem on $SU(4)$

What is the smallest subalgebra of $\mathfrak{su}(4)$ containing the span of the set $A = \{A, B_1, B_2\}$ where: $A = i (J^x \sigma_x \otimes \sigma_x + J^y \sigma_y \otimes \sigma_y + J^z \sigma_z ...
3
votes
0answers
303 views

Solving $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ equation

Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$? Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ ...
3
votes
0answers
111 views

Physical relevance of either fundamental identity generalizing Jacobi [closed]

There are two fundamental identities for n-ary generalizations of the Jacobi identity. One fundamental identity is right for Nambu mechanics and such, the other for L_\infty algebras as in CSFT. Which ...
0
votes
1answer
277 views

Symplectic structure on $Sym^kG^{\mathbb{C}} $

Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its complexification. I am looking for a symplectic structure (without use of coordinates) on $$ Sym^kG^{\mathbb{C}}, $$ PS:Here ...
11
votes
1answer
825 views

Is this error in this paper of Langlands fixable?

The FQS criterion for the Virasoro algebra was discovered by Friedan, Qiu and Shenker (1), but the mathematicians found their proof insufficient, so that, FQS (2) and Langlands (3), published in the ...
2
votes
1answer
361 views

A question about flag variety of $SL(n,\mathbb{C})$

We know that the flag variety $SL(2,\mathbb{C})/B$ which $B$ is Borel subgroup, can be identified with $\mathbb{P^1}$, What can we say about $SL(n,\mathbb{C})/B$ which $B$ is Borel subgroup of ...
2
votes
0answers
85 views

The condition of maximality in branching rules of $SO$ group representations

Let the highest weight of a $SO(2n+1)$ representation be given as $(m_1,m_2,...,m_n)$ ($m_1\geq m_2 \geq .. \geq m_n \geq 0$) and the highest weight of a $SO(2n)$ representation be $(s_1,s_2,...,s_n)$ ...
4
votes
1answer
313 views

About using the character formula for $SO(2n)$.

I have known of the following equation for characters of a $SO(2n)$ representation with highest weights $(h_1,...,h_n)$ and for $(t_1,t_2,..,t_n,t_1^{-1},t_2^{-1},..,t_n^{-1})$ being the eigenvalues ...
-1
votes
1answer
253 views

identifying dual of lie algebra of general linear groups

Is there any reference for the following fact? I am looking for a nice and simple proof. Assume that $G=GL(n,C)$, the group of invertible $n\times n$ matrices with complex entries. Why can the dual ...
3
votes
1answer
185 views

Quantized conserved quantities appearing from the Lie-algebra

Hi, consider a simple situation in quantum mechanics: Your Hilbert space is $\mathcal{H}=L^2(\mathbb{R}^3)$ and you use the obvious unitary representation $\pi\colon G=O(3)\times\mathbb{R}^3\to ...
4
votes
2answers
390 views

CFTs corresponding to affine Lie algebras

I want to know how one can write down a CFT such that its conserved currents will satisfy some chosen (affine) Lie algebra $G$. On the few pages leading up to page 192 in here one can see see the ...
2
votes
1answer
240 views

symmetry of generationg function of PDE

We know that for finding the solutions of PDE equations, one of methods is "reduction of PDE", . For nonlinear equation $v_t=(v^{-4/3}v_x)_x+\lambda v$ how can we compute the generators of Lie ...
2
votes
2answers
420 views

smallest simplest $E_8$ -module

What is the smallest simplest(non-trivial) $E_8$ -module ?
10
votes
1answer
834 views

decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$

Do anybody know , why can we write the following decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$ 1) $E_8=(V^{\star}\otimes \wedge^{8}V^{\star})\bigoplus ...
10
votes
1answer
346 views

What are the simple Lie superalgebras of type E?

Background Simple finite dimensional Lie superalgebras over $\Bbb C$ have been classified. There are the Cartan type superalgebras (algebras of purely odd vector fields), two strange families P(n) ...
6
votes
4answers
455 views

level 2,3 characters of affine su(2)

Does anyone know where I can find an explicit formula to compute the level 2 or level 3 characters of affine $su(2)$? I have found several sources that give a formula to compute the level 1 ...
1
vote
1answer
510 views

very very basic question on semi-simple Lie algebras

I have a very basic question on Lie algebras. I'm doing particle physics, and a lot of emphasis seems to be placed on the weight diagrams of simple Lie algebras. But these simple Lie algebras are ...
1
vote
2answers
581 views

Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal ?

Thank you Cristi Stoica for your answer to the previous post of this question. Your hint is to the point I think. We should look at the requirements to construct the corresponding root system. My ...
0
votes
1answer
996 views

Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?

Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?
3
votes
3answers
1k views

Is the Lie algebra-valued curvature two-form on a principal bundle P the curvature of a vector bundle over P?

I am an analyst struggling through some geometry used in physics. Some background: For some Lie group $G$, let $P$ be a principal $G$-bundle over the smooth manifold $M$. Let $\omega$ be a connection ...
2
votes
1answer
221 views

Characterization of the weight orbit in the projective space via second order Casimir.

This is the spin-off of the question I previously asked. First, let me remind you some notation from that question: $G_0$ - compact, simply connected Lie group giving rise (by complexification) ...
3
votes
1answer
125 views

What's the name of an algebra? is it isomorphic to $w_\infty \times w_\infty$?

Just as we know, $w_\infty$:=span {${z^\alpha }\partial _z^\beta|\alpha,\beta\in\mathbb{Z}, \beta\geq0$ }. But, what's the name of the following algebra, span {$\{{z^{\alpha_1}}{y^{\alpha_2}}\partial ...
3
votes
1answer
679 views

highest weight orbit characterization (reformulated and extended )

Edit 1: I think that the question was not stated clearly enough so modified it a little Edit 2: I thought over the physics that lies behind this question which led me to reformulation of the original ...
6
votes
2answers
479 views

Killing form vs its counterpart in a given represenation

Let $\mathfrak{g}$ be a semi-simple Lie algebra and let $\phi:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ be its finite-dimensional complex irreducible representation. You can define two non-degenerate ...
21
votes
3answers
999 views

What was Casimir's precise role in describing the center of the universal enveloping algebra of a semisimple Lie algebra?

This question is prompted by a recent MO question on explicit computations of Weyl group invariants for certain exceptional simple Lie algebras: 37602. Like some others who started graduate study in ...
3
votes
0answers
202 views

Characters of Kac-Moody algebra from orbifold

In a Wess–Zumino–Witten model on some Lie group G, the character of a particular integrable representation is the same as the specialized character from the corresponding Kac-Moody algebra. Suppose ...
4
votes
2answers
492 views

Under what conditions will a unitary matrix fix a subspace which does not diagonalize the generating Hamiltonian?

Hello, this is my first post here. I hope that it is not too vague; I will be as precise as I can, but I have more of a meta-problem so please forgive me if this is inappropriate. My question is ...
76
votes
3answers
6k views

Has the Lie group E8 really been detected experimentally?

A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced, "Quantum ...
10
votes
2answers
2k views

Noether's Theorem in Quantum Mechanics

In classical mechanics: If a Lagrangian L is preserved by an infinitesimal change in the state space variables qi -> qi + εKi(q) leads to only second order change in the Lagrangian: $$ 0 = ...
29
votes
8answers
5k views

What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic ...