All Questions
2,633 questions
7
votes
1
answer
210
views
Universal enveloping algebra of Malcev Lie algebra associated to nilpotent group
Let $G$ be a finitely generated torsion-free nilpotent group. The Malcev completion of $G$ is a nilpotent Lie group $N$ into which $G$ embeds as a lattice. One way to construct this is to take the ...
- 2,368
2
votes
0
answers
81
views
The centralizer and normalizer of products of (SU(n) $\times$ SU(p) $\times$ …) in U(m)
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}
$Consider the special unitary group $\SU(n)$ and the unitary group $\U(m)$.
Below I specify a specfic way to embed $...
- 13k
3
votes
3
answers
1k
views
Nilpotent elements of Lie algebra and unipotent groups
Let $k$ be a field of characteristic 0 (not necessarily algebraically closed), let $G$ be a connected split reductive group over $k$ and let $\mathfrak{g}$ be the Lie algebra of $G$.
Let $X \in \...
- 13k
6
votes
1
answer
312
views
When is a nilpotent Lie algebra isomorphic to the associated graded of its lower central series?
All Lie algebras in this question are finite-dimensional and defined over a field $k$ of characteristic $0$ which I'm happy to take to be $\mathbb{R}$ or $\mathbb{C}$.
$\DeclareMathOperator\gr{gr}$Let ...
- 13k
4
votes
1
answer
101
views
Non-cosemisimple duals of pointed Hopf algebras
I take the following quote from an answer to this question
A Hopf algebra is called pointed if all its simple left (or right)
comodules are one-dimensional. The quantized enveloping algebras and
...
- 12.4k
1
vote
0
answers
188
views
$G$-equivariant modules and Lie algebra cohomology
$\DeclareMathOperator\Id{Id}\DeclareMathOperator\Ad{Ad}$Is there a link between $G$-equivariant modules and Lie algebra cohomology?
Tell me if I'm mistaken:
On one side, if $p:E\longrightarrow M$ is ...
- 13k
3
votes
3
answers
423
views
Are there characteristic-dependent Betti numbers in characteristic not equal to two?
Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\...
- 150
1
vote
2
answers
238
views
Dimensions of $\frak{sl}_n$-representations
The dimension of any irreducible $\frak{sl}_n$-representation $V$ is clearly equal to the dimension of its dual representation $V^*$. Does the dimension of an irreducible $\frak{sl}_n$-representation ...
- 13k
4
votes
2
answers
564
views
Derivation on $SO(3)$
Let
$$u:\mathbb{R}\ni t \mapsto u(t)\in\mathcal{S}, \quad v:\mathbb{R}\ni t \mapsto v(t)\in\mathcal{S}$$
where $\mathcal{S}$ is the unit sphere of $\mathbb{R}^3$.
Consider
\begin{align} R:\ \mathcal{S}...
CommunityBot
- 1
2
votes
1
answer
125
views
Rigidity of Borel Lie algebras
Let $\mathfrak b$ be a Borel subalgebra of dimension $n$ in a real semisimple Lie algebra $\mathfrak g$. I am trying to reconcile two facts about $\mathfrak b$:
$\mathfrak b$ is rigid, that is, the ...
- 63
13
votes
3
answers
3k
views
How to Compute the coordinate ring of flag variety?
Let $G$ be algebraic group over $\mathbb(C)$(semisimple), $B$ be Borel subgroup. consider flag variety $G/B$. It is known that $G/B$ is projective variety. So one consider the homogeneous coordinate ...
- 18.4k
21
votes
2
answers
3k
views
Is every finite-dimensional Lie algebra the Lie algebra of a closed linear Lie group?
This question is closely related to this one.
Ado's theorem states that given a finite-dimensional Lie algebra $\mathfrak g$, there exists a faithful representation $\rho\colon\mathfrak g \to \...
CommunityBot
- 1
4
votes
1
answer
928
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 ...
- 2,821
2
votes
0
answers
76
views
Equivalence classes of Hermitian elements in a central simple algebra with an involution of second kind
Let $k_0$ be a field of characteristic 0, $k/k_0$ be a quadratic extension,
and $A/k$ be a central simple algebra over $k$ of dimension $9=3^2$ with an involution of second kind $\sigma$.
Then $^\...
- 14.2k
2
votes
1
answer
199
views
Action of the Casimir on highest weight modules for Kac-Moody algebra
Let $g$ be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let $\{u_j\}$ and $\{u^j\}$ be bases of $g$ dual with respect to a nondegenerate invariant bilinear form $(\cdot|\cdot)$ on $g$, ...
- 550
4
votes
0
answers
465
views
Isometry groups of symmetric spaces
Let $M=G/K$ be a symmetric space where $G=\mathrm{Isom}(M)$ and $K$ is the isotropy at some point $o\in M$. Moreover, let $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ be the Cartan decomposition of $\...
- 201
12
votes
1
answer
2k
views
Relationship between the Witt algebra and vector fields on the circle
I have seen some (apparently contradictory) claims by mathematicians and physicists regarding the existence of an (infinite dimensional) Lie group whose Lie algebra is the Virasoro algebra.
The ...
CommunityBot
- 1
7
votes
0
answers
439
views
Group-like elements of universal enveloping algebra
Suppose $\mathfrak{g}$ is a finite-dimensional Lie algebra over $\mathbb C$. Take $A=U(\mathfrak g[[t]])$, a universal enveloping algebra of $\mathfrak g[[t]]$ over $\mathbb C[[t]]$.
Then we may ...
- 481
1
vote
0
answers
143
views
Why is this operator independent of the choice of basis
I asked this question in MSE but I received no answer
https://math.stackexchange.com/questions/4009524/why-is-the-following-operator-independent-of-the-choice-of-basis/4013636#4013636
Let $G$ be a lie ...
- 139
3
votes
1
answer
284
views
Schubert cells in G/P for reductive G
All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a semisimple complex group. I ...
- 44.7k
11
votes
1
answer
448
views
Rational homology of $\Omega^2 ( \mathbb CP^n \vee S^d)$
What is the rational homology of $\Omega^2 ( \mathbb CP^n \vee S^d)$? Here $\Omega$ denotes based loop space.
- 10.9k
2
votes
0
answers
60
views
Is there always a purely real representative for a metrized solvable Lie group?
Alekseevski proves for Heintze groups (a special class of solvable Lie groups) that any such group admits a (left-invariant) metric which is isometric to a purely real Heintze group (again equipped ...
- 63.9k
13
votes
0
answers
195
views
Relationship between crystal root operators and usual $e_i, f_i$?
Suppose I am working in a symmetrizable Kac–Moody Lie algebra $\mathfrak{g}$. Let $e_1,\dotsc,e_n,f_1,\dotsc,f_n$ denote the usual Chevalley generators of $\mathfrak{g}$. Let $V$ be a highest weight ...
- 820
3
votes
1
answer
182
views
Complete reducibility of integrable modules over symmetrizable Kac-Moody Lie algebras
I am reading the book "Infinite-dimensional Lie algebras" by Victor G Kac. This is a long question regarding my understanding of the following theorem.
In Theorem 10.7 Kac proves the ...
CommunityBot
- 1
6
votes
0
answers
188
views
Word/Loop in $L(\Lambda)$
Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, with Chevalley generators $e_i,f_i$ ($i=1,...,n$).
Let $L(\Lambda)$ denote the irreducible module with highest weight $\Lambda$. Let $v$ denote ...
- 820
4
votes
1
answer
201
views
Is every $M\in \mathfrak{s}\mathfrak{p}_4(F)$ conjugate to an "upper triangular" matrix?
Let $F$ be a field and write $$\mathfrak{s}\mathfrak{p}_4(F)=\left\{\left(\begin{array}{cc} A & B \\ C & -A^T \\ \end{array}\right)\mid A,B,C\in M_2(F), B=B^T, C=C^T\right\}$$ for the ...
- 20.2k
4
votes
1
answer
428
views
p-torsion in the Picard group of a regular projective curve
Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$.
If $C$ is smooth, then the connected component ${\rm Pic}^0_C$ of the Picard scheme of ...
- 19.6k
2
votes
0
answers
230
views
Lie bracket on the unshifted tangent complex?
My problem is as follows: if $X$ is a derived scheme, or derived stack, or any kind of a space where tangent complex makes sense, I guess there should be a lie bracket on its tangent complex, ...
- 121
13
votes
1
answer
411
views
Chiral homology for the Virasoro algebra and/or affine Lie algebra
I want to understand what concrete analytical objects are found in chiral homology of higher degree of a vertex algera (-module) $M$. More precisely: I can obtain conformal blocks on a surface $\Sigma$...
- 63.9k
8
votes
1
answer
943
views
Automorphisms over finite field that do not lift to an automorphism in characteristic zero
My main question is the following: is there an automorphism of the affine space $\mathbb{A}^n$ (automorphism of an algebraic variety) defined over a finite field which does not lift to an automorphism ...
- 7,722
145
votes
14
answers
50k
views
Why study Lie algebras?
I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...
- 3
12
votes
1
answer
392
views
Non-conjugate subgroups that are conjugate in complexification
In trying to come up with a counter-example in my line of research, I would like to find an example as follows:
$G$ is a semisimple Lie group with complexification $G^{\mathbb{C}}$. $H_1, H_2 \...
- 63.9k
3
votes
0
answers
197
views
How to write down an explicit equation of given degree yielding a smooth hypersurface in a projective space?
Let F be a field of positive characteristic $p$ and let $d,n$ be two positive integers.
Can we explicitly write down an equation defining a smooth hypersurface $X_d⊂\mathbb P^n_F$ of degree d ?
This ...
- 417
3
votes
1
answer
282
views
Peter–Weyl decomposition for compact Lie groups with isomorphic Lie algebras
Let $G$ and $H$ be two compact Lie groups with isomorphic Lie algebras $\frak{h} \simeq \frak{g}$, but which are non-isomorphic as topological spaces. From the isomorphism assumption it (should) ...
- 118k
1
vote
1
answer
96
views
Small deformations of maximally symmetric 3-spaces
I am looking for all 'small' deformations of the three 3-dimensional Riemannian spaces with maximal symmetry, the pseudo-sphere, Eucidean space and the sphere. By a theorem by Fubini (1903) the ...
CommunityBot
- 1
4
votes
1
answer
226
views
Maximal quotients of the enveloping algebra of a simple Lie algebra
Let $\mathfrak{g}$ be a simple Lie algebra with Cartan subalgebra $\mathfrak{h}$, $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak g$, $Z(\mathfrak g)$ the center of $U(\mathfrak{g})$, ...
- 2,992
1
vote
0
answers
118
views
Kazhdan-Lusztig Conjecture over non-algebraically closed field
Let $G$ be a split connected semi-simple (or reductive) algebraic group over a (non-archimedean) field $k$ of characteristic zero. Denote by $\mathfrak{g}=\mathrm{Lie}(G)$ the semi-simple (or ...
- 473
16
votes
0
answers
756
views
Is there a "natural" proof of the equality $4^2=2^4$?
This question, or rather any answer that it might receive, would probably belong to the realm of Awfully sophisticated proof for simple facts. Still, I claim that I have quite serious motivation for ...
- 18.4k
5
votes
1
answer
346
views
Restricting representations to a principal $\mathfrak{sl}(2)$
Let $\mathfrak{g}$ be a semi-simple Lie algebra over $\mathbb{C}$ with simply connected group $G$ and suppose that
$$\mathfrak{g} = \bigoplus_i\mathfrak{g}_i$$ is a $\mathbb{Z}$- or $\mathbb{Z}/n\...
- 13k
5
votes
0
answers
154
views
One-parameter family of algebra structures: characterizing trivial deformation as adjoint 2-cocycle being a 2-coboundary
$\DeclareMathOperator\C{\mathbf{C}}$Motivation: this post discusses a simple criterion for a 1-parameter family of $n$-dimensional complex algebras to be a "trivial deformation", i.e., be ...
- 63.9k
32
votes
2
answers
1k
views
A question about subspace in ${\bigwedge}^2({\mathbb R}^n)$
Let $E$ be a linear subspace of ${\bigwedge}^2({\mathbb R}^n)$. What is the minimal dimension of $E$ that guarantees $E$ contains a nonzero element of the form $X\wedge Y$, with $X, Y\in{\mathbb R}^n$?...
- 79.9k
7
votes
1
answer
574
views
Does Aut(G) → Out(G) always split for a compact, connected Lie group G?
The outer automorphism group of a topological group $G$ is constructed by the short exact sequence
$$
1\longrightarrow \operatorname{Inn}(G) \longrightarrow \operatorname{Aut}(G) \longrightarrow \...
- 13k
1
vote
0
answers
172
views
How to interpret divided powers in Kostant Z Form when passing to a field of characteristic p > 0?
Let $G = GL_n(\mathbb{K})$, where $\mathbb{K}$ is a field of characteristic $p > 0$. Let $\mathfrak{g} = \mathfrak{gl}_n(\mathbb{C})$. Let $e_{ij}$ denote the elementary matrices which are a basis ...
- 239
3
votes
1
answer
632
views
A formula for the dual Coxeter number
Let $\Phi$ be the root system of a finite dimensional simple Lie algebra $\mathfrak g$, with dual Coxeter number $h^\vee$.
Let $\alpha_0\in \Phi$ be a long root (if all the roots have the same length, ...
- 13k
10
votes
1
answer
322
views
$\pi_{2p^kn - 1}(P^{2n+1}(p^r))$ contains a $\mathbb{Z}_{p^{r+1}}$ summand
I am reading Neisendorfer's paper Samelson products and exponents of homotopy groups and related papers. I am stuck on theorem 14.1 on page 21, which says that there exists a $\mathbb{Z}_{p^{r+1}}$ ...
- 1,263
1
vote
2
answers
223
views
Computing kernel in the category $\mathcal{O}$
Let $\mathfrak{g}=\mathfrak{gl}_3$ over $\mathbb{C}$ with positive roots
\begin{equation*}
\Phi_+=\{\alpha_1=(1,-1,0),\alpha_2=(1,0,-1),\alpha_3=(0,1,-1)\}.
\end{equation*}
Consider the morphism
\...
- 1,368
6
votes
0
answers
173
views
Equivalence between $\mathcal{D}_\lambda$ modules and $\mathcal{D}_{0}$ modules
Fix $G$ a finite dimensional reductive group and $\lambda$ a weight. Apparently the category of $\mathcal{D}_\lambda$ modules on $G/B$ is equivalent to the category of $\mathcal{D}_0$ modules on $G'/B'...
- 5,711
3
votes
0
answers
81
views
The Loday-Quillen-Tsygan theorem for topological (Fréchet) algebras
In "Additive K-theory" by Tsygan and Feigin, Section 0.4, a statement is given which seems to generalize (cohomological version of) the well-known Loday-Quillen-Tsygan theorem
$$H_{\text{CE}}...
CommunityBot
- 1
-4
votes
2
answers
421
views
Would it be simpler, pedagogically speaking, if textbook writers introduced root systems as an example of a quandle?
I could never, for the life of me, recall the definition of a root system in Lie theory. It probably doesn't help that I've never taken a course on Lie Theory - the algebra, or the groups, or the ...
- 53.3k
2
votes
1
answer
152
views
Notation on supergeometry — parity
I know that given a manifold $M$ and its corresponding tangent bundle $TM$ we can call $\Pi TM$ the space of forms parametrized (via charts) by $\{x_i\}_{i=1,\dotsc,n}$ and its corresponding cotangent ...
- 63.9k