All Questions
2,633 questions
2
votes
1
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138
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Solvability of derivation Lie algebras of local finite-dimensional commutative algebras
Let $A$ be a finite-dimensional local commutative algebra (with one) over a characteristic zero field $k$. Is it true that the Lie algebra $\operatorname{Der}_k(A)$ of $k$-derivations of $A$ is ...
- 18.4k
4
votes
0
answers
158
views
Relation between two Harish-Chandra homomorphisms
Let $\mathfrak{g}$ be a Lie algebra admitting a triangular decomposition $\frak g = N^-\oplus h\oplus N^+$ and $\gamma$ the classic Harish-Chandra isomorphism defined on the center $\frak Z$ of the ...
- 141
1
vote
0
answers
101
views
Explicit central elements of $\mathcal{U}(\mathfrak{so}(4,1))$
I am interested in finding the central elements of the universal enveloping algebra of the Lie algebra $\mathfrak{so}(4,1)$.
Notation: the 10 generators are $D, J_i, P_i, K_i$ ($i=1,2,3$), satisfying ...
- 163
3
votes
0
answers
223
views
What is a Gelfand-Tsetlin subalgebra?
For context on general Gelfand-Tsetlin theory, see for instance this MO post.
Let's work over $\mathbb{C}$. Fix $n>0$. There is a natural chain of embeddings of the general linear Lie algebras $\...
- 3,318
0
votes
1
answer
206
views
What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?
Let $\mathfrak g$ be a Lie algebra. $\mathfrak g^{(1)}=\mathfrak g$ and $\mathfrak g^{(n+1)}=[\mathfrak g,\mathfrak g^{(n)}]=\mathbb R$-span$\{[X,Y]:X\in\mathfrak g,Y\in\mathfrak g^{(n)}\}$. The ...
- 275
2
votes
0
answers
127
views
Classification of restricted Lie algebras of reductive groups
$\DeclareMathOperator\Lie{Lie}$Let $G/K$ be a reductive group over a field $K$. In characteristic $0$ the Lie algebra is invariant under base change of fields, so to understand $\Lie(G)$ it is enough ...
- 461
7
votes
1
answer
323
views
Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?
In this popular 2019 MO question, user მამუკა ჯიბლაძე asked:
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism between ...
- 492
7
votes
0
answers
1k
views
What's the point of geometric representation theory?
Please forgive the provocative title, what I mean is the following:
One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...
- 157
3
votes
1
answer
206
views
Relation between enveloping algebras and algebras of differential operators
I asked this question on math stack exchange about 3 years ago, but received no answer.
Our base field $\mathsf{k}$ will be algebraically closed of zero characteristic. Let $X$ be an smooth affine ...
- 3,318
3
votes
1
answer
279
views
Wedderburn–Artin like theorem for infinite dimensional Lie algebras?
The Wedderburn–Artin Theorem is one of the cornerstones of the structure theory of (associative) rings.
Wedderburn–Artin Theorem : Let $R$ be a left Artinian ring with zero Jacobson radical. Then $R$ ...
- 3,318
2
votes
1
answer
107
views
Finite dimensional irreducible representations of $\frak{sl}_m$ with non-trivial zero weight spaces
For the special linear algebra $\frak{sl}_{m}$ which finite dimensional irreducible representations $V_{\mu}$ have non-trivial zero weight spaces?
For $\frak{sl}_2$ this is clear: $V_{2k\pi}$ for $\pi$...
- 157
1
vote
0
answers
82
views
Generators of simple Lie groups and finite word length
Let $G$ be a connected simple Lie group with finite center. Let $a=\mathrm{exp}(X)$ be a semisimple element. Then we can decompose the lie algebra of $G$ into the direct sum of the eigenspaces of $\...
- 31
5
votes
1
answer
182
views
Why do most eigenspaces of a Lie algebra automorphism have finitely many orbits?
I'm interested in understanding the following lemma, which Vogan states (Lemma 4.8) in his paper The Local Langlands Conjecture (omitting the "well-known" proof).
Suppose $G$ is a complex ...
- 835
0
votes
0
answers
52
views
Relation between real forms of Lie algebras and root systems on pseudoeuclidean vector spaces
This might be trivial but I cannot see it clearly.
Simple complex Lie algebras are fully classified by the root systems arising from the Cartan subalgebra for which the Euclidean norm is the Cartan-...
- 295
1
vote
2
answers
197
views
What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
- 9,501
2
votes
0
answers
59
views
graded reps of Lie algebras literature
I am currently studying 'advanced' representation theory from a physicist's perspective, including topics like super-Lie algebras. I've come across various gradings (excluding the ℤ2
grading), such as ...
- 21
0
votes
0
answers
246
views
What does the set of all fundamental coweights look like?
Let $\Phi$ be an irreducible root system in a Euclidean vector space $V$. Let $W$ denote its Weyl group. Choose a base $\Delta=\{\alpha_1,...,\alpha_r\}$ for $\Phi$. Then $\Delta$ is a basis for $V$. ...
- 2,751
2
votes
0
answers
120
views
Is there an explicit description of a gauge transformation $F$ such that $U_{\hbar}(\mathfrak{g})$ and $(U(\mathfrak{g})[[\hbar]])_F$ are isomorphic?
Let $\mathfrak{g}$ be a semisimple Lie algebra, let $t$ be its canonical 2-tensor, and let $\Phi_{KZ}$ be a Drinfeld associator.When $R_{KZ}=e^{\hbar t/2}$, $(U(\mathfrak{g})[[\hbar]],\Phi_{KZ},R_{KZ})...
- 255
3
votes
2
answers
979
views
The adjoint representation of a Lie group
Let $G$ be a Lie group and $\text{Ad}(G)$ denote its adjoint representation i.e. the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$. The adjoint representation is a real $G$-...
- 131
2
votes
0
answers
102
views
Category O for (Yangian) toroidal Lie algebras?
Suppose throughout that $g$ is a finite-dimensional simple Lie algebra over $\mathbb{C}$ and let us denote:
$$g_{[2]} := g \otimes_{\mathbb{C}} \mathbb{C}[v^{\pm 1}, t^{\pm 1}]$$
$$g_{[2]}^+ := g \...
- 1,516
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
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 $\...
- 161
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
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}),\...
- 4,487
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 ...
- 833
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
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 ...
- 64k
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 ...
- 366
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 ...
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 ...
- 179
17
votes
2
answers
2k
views
How to think of algebraic geometry in characteristic p?
How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's ...
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 ...
- 442
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 ...
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
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 ...
- 412
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}?
$$
In ...
- 2,751
0
votes
1
answer
210
views
A question regarding Hermite polynomials and exponential operators $\exp[e^{x^2/2}p(\frac{d}{dx})e^{-x^2/2}]f(x)$
Is it possible to express $$\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[\...
- 350
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 ...
- 157
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 ...
- 951
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).
...
- 630
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 ...
- 2,152
3
votes
2
answers
331
views
Lie's third theorem via graded geometry
Lie's third theorem : Given any finite dimensional Lie algebra $\mathfrak{g}$, there exists a Lie group $G$ whose Lie algebra is equal to $\mathfrak{g}$.
In one of the talks, speaker mentions that ...
- 6,023
4
votes
0
answers
92
views
Lie bracket of general unipotent matrices
Let $k$ be a field (of characteristic $0$). Let
$$
X:=\begin{pmatrix}1&x_{1,2}&x_{1,3}&\cdots&x_{1,n}\\ &1&x_{2,3}&\cdots&x_{2,n}\\ &&1&\cdots&x_{3,n}\\ ...
- 449
0
votes
1
answer
304
views
A Lie group whose Lie algebra is equal to (the Lie algebra? of )all functions with fibrewise polynomial growth
Let $M$ be a Riemannian manifold. We denote by $\mathfrak{g}$ the space of all smooth function $f:TM\to \mathbb{R}$ with fibre wise polynomial growth. Is it a Lie algebra wrt the Poisson bracket ...
- 366
6
votes
1
answer
393
views
An alternative form of the Kazhdan-Lusztig conjecture
Fix a complex semisimple Lie algebra $\mathfrak{g}$. Denote by $W$ the corresponding Weyl group, with length function $\ell$ and Bruhat order $\leq$. Let $\lambda$ be an integral anti-dominant weight. ...
- 1,391
1
vote
1
answer
158
views
Commuting time dependent vector fields and pullback invariance
Let $X_t, Y_t \in C^\infty(\mathbb{R}; \mathfrak{X}^\infty(M))$ be (smooth, or something else if it's necessary) time dependent vector fields.
Is there some analogue of the following fact in finite ...
- 193
20
votes
0
answers
408
views
Ado's theorem and the reduction to positive characteristic
The synopsis: proofs of Ado theorem in positive characteristic are simple, and in characteristic $0$ are difficult. Can one infer the characteristic $0$ case from the positive characteristic case?
The ...
- 2,934
5
votes
0
answers
169
views
Reference request: Whitehead's lemma for semisimple Lie algebras
Let $\mathfrak{g}$ be a semisimple Lie algebra and let $V$ be a representation. One formulation of Whitehead's lemma is that the Lie algebra cohomology of $V$ is given by
$$H^{\bullet}(\mathfrak{g}, V)...
- 491
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