All Questions
2,633 questions
5
votes
2
answers
2k
views
What is the cubic Casimir element of $\mathfrak{sl}_3$?
$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Tr{Tr}$I have been thinking about this for some time but have had no luck. I have found some sources that say higher Casimir elements can be obtained ...
- 4,713
3
votes
1
answer
311
views
Heat kernel of left-invariant metric on 3-sphere
This post concerns the heat equation on $S^3 (\simeq \mathrm{SU}(2))$ endowed with an arbitrary left-invariant metric. We think of $S^3$ as the space of unit quaternions, and its Lie algebra $\...
- 108k
2
votes
1
answer
159
views
Adjoint action on orthogonal complement
Consider a compact Lie algebra $\mathfrak{g} \subset \mathfrak{u}(n)$ and its associated connected, compact Lie group $G$. Let $\mathfrak{g}^{\perp}$ denote $\mathfrak{g}$'s orthogonal complement (as ...
CommunityBot
- 1
8
votes
2
answers
2k
views
Lie algebras to classify Lie groups
What does the classification of Complex Semi-simple Lie algebras buy us in terms of classifying Lie groups? Certainly it classifies complex semi-simple lie groups but can we get any better? I know we ...
- 5,654
1
vote
0
answers
59
views
A certain subalgebra of $\mathfrak{e}_8$ over a p-adic field
Can $\mathfrak{e}_8(\mathrm{k})$ have a maximal subalgebra isomorphic to $\mathfrak{sl}_1(\mathrm{D})\oplus\mathfrak{g}_2(\mathrm{K})$, where $\mathrm{k}$ is a finite extension of some $\mathbb{Q}_p$, ...
- 2,773
2
votes
0
answers
211
views
Is every irreducible representation of $Sp(m)$ a representation of $U(2m)$?
If $Sp(m)$ is the group of linear automorphisms of $\mathbb{C}^{2m}$ which fix a complex symplectic form $\omega$ and a quaternionic structure $j$ on $\mathbb{C}^{2m}$, then it is clear that $Sp(m)$ ...
- 5,215
10
votes
1
answer
339
views
What is the Lie superalgebra generated by permutations?
Consider the group algebra of the symmetric group $\mathbb{C}S_n$. Then there is a corresponding Lie algebra $\mathfrak{L}(S_n)$ defined by
$$[\sigma, \tau] = \sigma\circ\tau - \tau\circ\sigma,$$
...
- 1,996
5
votes
0
answers
373
views
A (possible) Lie algebra extension of the Lie algebra of a foliation
Motivation: The aim of this post is to extend the Lie algebra of a foliation to a bigger Lie algebra. We assume that a manifold $M$ is foliated by compat leaves. The Lie algebra of the foliation is ...
- 366
5
votes
0
answers
250
views
Lie algebras, root systems and qubits
This post is about some concepts I am experimenting with. They are related to the Atiyah problem on configurations. They kind of mix Lie algebras and qubits. Given a compact (say semisimple) Lie group ...
- 5,215
6
votes
0
answers
271
views
Torsion in the Lie algebra cohomology of gl(n,Z)
What is known about the Lie algebra cohomology $H^*(\mathfrak{gl}_n(\mathbb{Z}),\mathbb{Z})$? After passing to $\mathbb{Q}$-coefficients, the question is classical: $H^*(\mathfrak{gl}_n(\mathbb{Q}),\...
- 12.9k
0
votes
1
answer
210
views
Centers of universal enveloping algebra of complex Lie algebras
Let $\mathfrak{g}$ and $\mathfrak{g'}$ be complex Lie algebras such that $\mathfrak{g}$ is a subalgebra of $\mathfrak{g'}$. Let $Z(\mathfrak{g})$ and $Z(\mathfrak{g'})$ be the centers of the universal ...
- 5,878
11
votes
1
answer
381
views
Chromatic representation theory of the symmetric groups?
We know that in characteristic 0, the group ring of the symmetric group $\Sigma_n$ splits via one idempotent for each partition of $n$.
In characteristic $p$, I believe the analogous statement is that ...
- 52.7k
9
votes
3
answers
4k
views
A terminology issue with the Killing form
I understand the definition of the Killing form $B$ as $$B(X,Y)=\mathrm{Tr}(\mathrm{ad}(X)\mathrm{ad}(Y)).$$
When the Lie group is semisimple the negative of the Killing form can serve as a Riemannian ...
- 63.9k
1
vote
0
answers
165
views
Rotation number for homeomorphisms of a Lie group other than $S^1$
Let $G$ be a Lie group whose Lie algebra is $\mathfrak{g}$ with exponential map $\exp:\mathfrak{g}\to G$.
For what kind of Lie group $G$ the standard process of definition of rotation number ...
- 10.5k
7
votes
2
answers
1k
views
Hochschild Kostant Rosenberg theorem for varieties in positive characteristic?
Is there is a known version of the HKR theorem as proved in say Swan's paper "Hochschild Cohomology of Quasiprojective Varieties" in positive characteristic? I assume something is known about this as ...
- 2,368
7
votes
1
answer
434
views
Do the exceptional root systems arise in the real world?
I am looking for a list of real world examples where the exceptional roots systems $E_6, E_7, E_8, F_4$, and $G_2$, and their associated Lie algebras and Lie groups, arise. To make this question a ...
- 82.7k
3
votes
1
answer
180
views
A filtration on Drinfeld-Jimbo quantum enveloping algebras
For the universal enveloping algebra $U(\frak{g})$ of a Lie algebra $\frak{g}$, one can define in a natural way an increasing $\mathbb{N}_{0}$-filtration. By the Poincaré-Birkhoff–Witt theorem, the ...
4
votes
1
answer
2k
views
Reference request - localisation de g-modules
Does anyone have a link to a copy of Beilinson-Bernstein's "Localisation de g-modules", in which they prove the Beilinson-Bernstein theorem? I can't find it anywhere.
- 2,992
2
votes
0
answers
145
views
The "big bracket" in Lie bialgebras
I am looking for a well-written document such as a survey article or textbook that explores the subject of the "big bracket". This concept is briefly introduced in the appendix of Yvette ...
- 63.9k
4
votes
1
answer
175
views
A pseudo-Riemannian version of a theorem by Fubini
Guido Fubini, ``Sugli spazii che ammettono un gruppo continuo di movimenti,'' Annali di Mat., ser. 3, 8 (1903) 54.: Let $M$ be a Riemannian manifold of dimension $d\ge 3$. Its isometry group cannot be ...
CommunityBot
- 1
2
votes
1
answer
160
views
Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$
Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as ...
- 20.5k
1
vote
0
answers
111
views
Questions on the differential of the Lie logarithm
Let $G$ be a Lie group. Recall the Lie logarithm is well-defined about a neighborhood $U \subset G$ of the identity: $\log:U\to \mathfrak{g}$. I am dealing with a research problem that concerns the ...
- 518
9
votes
2
answers
789
views
Bender-Knuth involutions for symplectic (King) tableaux
First let me recall the combinatorial theory of the characters of $\mathfrak{gl}_m$, a.k.a., Schur polynomials. For a partition $\lambda$, a semistandard Young tableaux of shape $\lambda$ is a filling ...
- 306
1
vote
0
answers
70
views
Minimal $K$-orbit on $\mathfrak{g}$
Let $\mathfrak{g}_0$ be a noncompact simple Lie algebra with Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. Write $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the ...
- 8,597
2
votes
0
answers
105
views
Embeddings of symplectic group into the orthogonal group
Let $\mathfrak{sp}$ denote the complex symplectic Lie algebra and $\mathfrak{so}$ the complex orthogonal one. Do we have an embedding
$$
\mathfrak{sp}_{2n-2} \hookrightarrow \mathfrak{so}_{2n}?
$$
In ...
- 2,751
0
votes
1
answer
243
views
Adjoint action on the universal enveloping algebra and the PBW theorem
Let $\frak{g}$ be a semisimple Lie algebra and $U(\frak{g})$ its universal enveloping algebra. The adjoint action of $\frak{g}$ on itself extends to an action of $\frak{g}$ on $U(\frak{g})$. How does ...
- 3,545
1
vote
1
answer
158
views
Commuting time dependent vector fields and pullback invariance
Let $X_t, Y_t \in C^\infty(\mathbb{R}; \mathfrak{X}^\infty(M))$ be (smooth, or something else if it's necessary) time dependent vector fields.
Is there some analogue of the following fact in finite ...
- 2,479
0
votes
1
answer
210
views
A question regarding Hermite polynomials and exponential operators $\exp[e^{x^2/2}p(\frac{d}{dx})e^{-x^2/2}]f(x)$
Is it possible to express $$\exp\left[\mathrm{e}^{x^2/2}p\left(\frac{d}{dx}\right)\mathrm{e}^{-x^2/2}\right] f(x)$$ as an integral transform or something similar? $p(x)$ is a polynomial.
$$\exp\left[\...
- 6,696
6
votes
1
answer
393
views
An alternative form of the Kazhdan-Lusztig conjecture
Fix a complex semisimple Lie algebra $\mathfrak{g}$. Denote by $W$ the corresponding Weyl group, with length function $\ell$ and Bruhat order $\leq$. Let $\lambda$ be an integral anti-dominant weight. ...
- 3,054
5
votes
1
answer
160
views
Derived subalgebra of a restricted Lie algebra
Let $L$ be a restricted Lie algebra over a field of characteristic $p>0$. It is well known that the commutator subalgebra $[L,L]$ is not necessarily restricted (that is, closed under the $p$-map).
...
- 12.9k
5
votes
1
answer
346
views
Analogues of Sullivan Theory at a prime for coformality
In rational homotopy theory, one can study the rational homotopy and cohomology categories via an algebraic structure, respectively the rational Lie Algebra model and the Sullivan cdga model.
If I am ...
- 5,859
4
votes
0
answers
92
views
Lie bracket of general unipotent matrices
Let $k$ be a field (of characteristic $0$). Let
$$
X:=\begin{pmatrix}1&x_{1,2}&x_{1,3}&\cdots&x_{1,n}\\ &1&x_{2,3}&\cdots&x_{2,n}\\ &&1&\cdots&x_{3,n}\\ ...
- 12.9k
20
votes
0
answers
408
views
Ado's theorem and the reduction to positive characteristic
The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case?
The ...
- 2,934
8
votes
2
answers
684
views
Slodowy slice intersecting a given orbit "minimally"?
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Is it true that for any $X\in\mathfrak{g}$, there exists an $\mathfrak{sl}_2$-triple $(e,h,f)$ in $\mathfrak{g}$ such that
We have $X\in e+Z_{\...
- 1,856
5
votes
0
answers
169
views
Reference request: Whitehead's lemma for semisimple Lie algebras
Let $\mathfrak{g}$ be a semisimple Lie algebra and let $V$ be a representation. One formulation of Whitehead's lemma is that the Lie algebra cohomology of $V$ is given by
$$H^{\bullet}(\mathfrak{g}, V)...
- 12.9k
1
vote
0
answers
67
views
Poisson bracket on $T^*T\mathrm{SU}(1,1)$
Consider the cotangent bundle of the tangent bundle $T^*TG$ of a Lie group $G$. Denote its the Lie algebra by $\mathfrak{g}$. By left translations, we have the trivialization $T^*G \cong G \times \...
- 829
1
vote
0
answers
88
views
Bad primes of twists of modular curves $X_E^{-1}(p)$
I am interested in a good reference for reading about primes of bad reduction for the modular curve $X_E^{-}(N)$, which is a twist of $X(N)$ parametrizing all elliptic curves $E'$ whose $N$-torsion ...
- 637
4
votes
0
answers
277
views
Representations of $\mathrm{sl}(3,\mathbb{C})$ and Catalan-like paths
Background on representations of $\mathrm{sl}(3,\mathbb{C})$
In Chapter 6 of Brian C. Hall's book "Lie Groups, Lie Algebras, and Representations", he constructs the irreducible ...
- 484
5
votes
2
answers
669
views
classification for coadjoint orbits of lower or upper triangular matrices
Is there any classification for coadjoint orbits of lower or upper triangular matrices in general case $n\times n$. Is there any reference?
- 1,446
2
votes
1
answer
170
views
Automorphism of positive characteristic field
Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$.
I am interested in ...
- 50.7k
5
votes
0
answers
146
views
What do the Carnot groups act on?
My question is in some sense a less ambitious version of the following MO question where the answer was inconclusive.
A Carnot group of step $N$ can be identified within the tensor algebra, modulo ...
- 63.9k
4
votes
0
answers
279
views
What is the exponential map from the Lie algebra $\mathfrak{sl}(2,\mathbb{C})\ltimes_\textrm{ad}\mathfrak{sl}(2,\mathbb{C})$ to its Lie group?
$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\Exp{Exp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\sl{\mathfrak{sl}}$Let $G:=\SL(2, C) \ltimes_{\Ad} \SL(2,C)$, where $\...
- 3,065
0
votes
0
answers
105
views
Concrete examples of quantum duality principle
Let $G$ be a Poisson Lie group, $\mathfrak{g}$ be a Lie algebra of $G$, $G^*$ be a dual of $G$, $\mathscr{C}(G^*)$ be a Poisson algebra of $G^*$, and $U_h(\mathfrak{g})$ be a quantized universal ...
- 255
0
votes
0
answers
71
views
Integrating homomorphisms of Borel subalgebras
Let $G$ be a connected simple complex Lie group and $\mathfrak{g}$ be its Lie algebra. Let us fix a root decomposition, let $\mathfrak{b}_\pm$, $\mathfrak{n}_+$ and $\mathfrak{h}$ be the corresponding ...
3
votes
0
answers
58
views
Locally finite positive energy modules generated by singular vectors at positive levels?
This is question is about whether or not certain modules for an affine Lie algebra are generated by their singular vectors. I begin with some background.
Backround on affine Lie algebras. Let $\...
CommunityBot
- 1
0
votes
0
answers
138
views
Is $[n]_q!$ invertible in $\mathbb C [[h]]\ $?
Consider the Hopf algebra $U_h (sl_2 (\mathbb C))$ over the ring $\mathbb C [[h]]$ generated by $E, F, H$ and relations $:$
$$[H, E] = 2 E,\ \ [H, F] = - 2 F,\ \ [E, F] = \frac {q^H - q^{-H}} {q - q^{-...
- 41
4
votes
0
answers
128
views
Tannakian reconstruction and the distribution algebra
$\DeclareMathOperator\Dist{Dist}\DeclareMathOperator\Lie{Lie}\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\End{End}$Let $G$ be an affine group scheme over a commutative ring $k$ (I am mainly ...
- 12.9k
1
vote
0
answers
105
views
Weyl group action on the Lie algebra [duplicate]
Let $W$ be the Weyl group of a complex semisimple Lie algebra $\frak{g}$. Certainly $W$ acts on the root system $R$ of $\frak{g}$ but can it be made to act on $\frak{g}$ or on the universal enveloping ...
24
votes
2
answers
1k
views
Lie groups generated by finitely many Lie algebra elements
Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. A standard fact is that $G$ is generated by $\exp(\mathfrak{g})$, i.e. every $g \in G$ can be written as $g=\exp(x_1)\cdots\exp(x_n)$ ...
- 33.3k
2
votes
0
answers
68
views
A branching law involving 2-power exterior representations
Let $K=SU(n)$.
We take a maximal torus $T$ in $K$ and fixed a simple root system with fundamental weights $\eta_1,\dots,\eta_{n-1}$.
For $\mu$ a dominant weight of $K$, we denote by $(\tau_\mu,V_\mu)$ ...
- 2,446