All Questions
2,633 questions
10
votes
1
answer
434
views
Viewing exceptional Lie algebras via the classical ones
I've been trying to understand the exceptional Lie algebras through the classical ones that I am more familiar with. In particular I wanted to get a handle on the root spaces and most discussions that ...
- 954
1
vote
1
answer
331
views
On Euler angles decomposition of $\mathrm{SU}(N)$
$\DeclareMathOperator\SU{SU}$I am looking for a (generalized) Euler angles decomposition for $\SU(N)\ (N>1)$ in the following fashion:
$$
\SU(N)\ni m = a\, u \, b
$$
where $a,b$ are independent ...
- 41
3
votes
0
answers
78
views
Set of equivalence classes of a Lie algebra under the action of the automorphism group
I recently became interested in the following question: Given a Lie algebra $\mathfrak{g}$, define two elements $x,y\in\mathfrak{g}$ to be equivalent if there exists an automorphism $\phi\in\...
- 777
3
votes
0
answers
142
views
Conjugacy classes of Cartan subspaces in parahermitian symmetric spaces
Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the ...
- 954
6
votes
1
answer
370
views
Does the isometry group determine the Riemannian metric?
Suppose $G \subset \text{Iso}(M)$ is a Lie group acting smoothly on a (pseudo-)Riemannian manifold $(M, g)$. Then $G$ induces a Lie algebra of Killing vectors on $M$. In this paper by Goenner and ...
- 108k
9
votes
0
answers
1k
views
Ample vector bundles, $H^1=0$ and global generation in characteristic $p$
This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...
- 12.9k
6
votes
4
answers
332
views
How can one show $G/T$ is a coadjoint orbit for a compact Lie group $G$ and $T$ its maximal torus?
$\newcommand{\g}{\mathfrak{g}}$Let $G$ be a compact Lie group and $\g$ its Lie algebra. I came across the the very important result that $G/T$ ($T$ a maximal torus of $G$) can be identified to a ...
- 63.9k
3
votes
1
answer
242
views
Notions of integrability for affine Lie algebras and positive energy representations
Let $\mathfrak{g}$ be a simple (complex) Lie algebra. Given an invariant bilinear form $\kappa : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{C}$, we can form the central extension $\hat{\mathfrak{g}}...
CommunityBot
- 1
5
votes
0
answers
258
views
Reference on "infinite dimensional Lie algebras" from a mathematical physics point of view
It happens that I stumbled on a class of infinite dimensional Lie algebras that are not Kac-Moody algebras and for which I was not really prepared for. I know some general results on infinite ...
- 875
2
votes
1
answer
248
views
What is the Molien series of the SO(2)-invariant ring on the plane (sometimes written C[X]^{SO(2)} )?
Let SO(2) be the group of rotations in the plane. What is the Molien series (sometimes called the Hilbert-Poincare series) of the SO(2)-invariant ring of polynomials?
N.B. The main goal being to ...
- 118k
6
votes
1
answer
771
views
A regular, geometrically reduced but non-smooth curve
Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ?
Of course, the base field should be imperfect.
In Exercise 4.3.22 of Qing Liu's book Algebraic ...
- 149k
2
votes
0
answers
109
views
Integrable modules of the affine Lie algebra $\mathfrak{su}(2)_k$
I am a physicist studying conformal field theory (CFT), so what I state can be not precise.
In physics literature, the affine Lie algebra $\mathfrak{su}(2)_k$ (here $k$ is the level) has only finitely ...
- 121
4
votes
0
answers
197
views
Inequalities between numerical invariants of nonsingular projective Varieties in positive Characteristic
It is well-known that Miyaoka and Yau-type inequalities do not hold in positive characteristic. In "a note on Bogomolov-Gieseker’s inequality in positive characteristic", however, we can ...
- 2,821
15
votes
1
answer
911
views
A possible gap in Faltings note to prove the Tate conjecture for finitely generated field over $\mathbb{Q}$
Qeustion:
Given a Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_\ell$ with an ideal $\mathfrak{g}^O$ and a subalgebra $\mathfrak{h}$,
such that $\mathfrak{g}=\mathfrak{g}^O+\mathfrak{h}$.
Now given a ...
- 5,050
2
votes
0
answers
253
views
Künneth formula for algebraic de Rham cohomology
Let $X$ and $Y$ be finite type schemes over a field $k$, and let $H^i(X/k)$ denote the $i$-th algebraic de Rham cohomology group of $X$ over $k$. I'm interested in the extent to which a Künneth ...
- 63.9k
1
vote
1
answer
197
views
Semisimple Lie algebra and convexity
There is any relation between semisimple lie algebras and symmetric cones? I'm saying this because the classification of the euclidean Jordan algebras, dual by Koecher-Vinberg theorem to homogeneous ...
- 954
8
votes
2
answers
619
views
Relationship between $q$-Weyl dimension formula and $q$-analog of weight multiplicity?
$\DeclareMathOperator\dim{dim}$For a dominant (integral) weight $\lambda$ and any (integral) weight $\mu$ of a simple Lie algebra $\mathfrak{g}$, Lusztig's $q$-analog of weight multiplicty $K_{\lambda,...
- 959
3
votes
1
answer
642
views
Decomposition theorem for polarized abelian varieties in positive characteristic
In characteristic zero we have the following decomposition theorem for polarized abelian varieties: it gives an isomorphism between a PPAV and a product of PPAV's of lower dimension and is valid (as ...
CommunityBot
- 1
4
votes
0
answers
108
views
When does the null-cone consist entirely of eigenvectors?
Let $V$ be a finite-dimensional representation of a complex reductive Lie algebra $\mathfrak g$.
For our purposes, we may define the null-cone like this: $v\in V$ belongs to the null-cone if and only ...
- 18.4k
0
votes
0
answers
132
views
Lie algebra action Whittaker model
Let $(\pi, H)$ be an irreducible unitary generic representation of $G=\operatorname{GL}(r,\mathbb{C})$ and let $H^{\infty}$ be its subspace of smooth vectors. Let $W :G\to\mathbb{C} $ be the Whittaker ...
3
votes
1
answer
283
views
Is the theta characteristic attached to an etale double cover of a plane quintic arising from a cubic threefold even in characteristic two?
We work over an algebraically closed field $k$ of characteristic $2$. Let $X$ be a cubic threefold realized as a conic bundle via $f: X \to \mathbb{P}^2$ (after blowing up some line). Let $C \to \...
- 679
13
votes
2
answers
1k
views
Ado's theorem for metric Lie algebras?
Background
Ado's Theorem states that every finite-dimensional Lie algebra over a field of zero characteristic admits a faithful representation.
More precisely, if $\mathfrak{g}$ is a finite-...
- 4,713
2
votes
1
answer
160
views
Is the restriction of the Cartan 3-form on conjugacy classes exact?
Let $G$ be a complex semisimple group and $\mathcal{O} \subset G$ a conjugacy class, i.e. $\mathcal{O} = \{gag^{-1} : g \in G\}$ for some $a \in G$. Let $\Omega$ be the Cartan 3-form on $G$ defined by
...
- 4,519
4
votes
0
answers
409
views
About ℓ-adic and perverse stuff and ℓ-adic cohomology with compact support
Maybe this question is trivial.
We know from this paper at Inv. Math 1976 (DOI link), T. A. Springer constructed representation of the Weyl group $W$
on the cohomology of the Springer fibre. Also, ...
- 63.9k
3
votes
0
answers
203
views
A quantity computed from weights of representations -- Have you seen it?
The following quantity has come up in some work my collaborators and I are doing on equivariant D-modules, and in that particular context it seems to be very significant (i.e. it's the only "...
- 3,079
4
votes
0
answers
284
views
modularity of elliptic curves over function fields in positive characteristic
Let $F$ be a global function field over a finite field, and $E$ be an elliptic curve over $F$. Much like the number field case, it is natural to study the Galois representation on the Tate module of $...
- 685
4
votes
0
answers
130
views
7D simple Lie algebras over $\mathbb{F}_3$
Up to isomorphism, what are all the seven-dimensional simple Lie algebras over the field with three elements?
- 63.9k
7
votes
0
answers
297
views
Associativity of the Campbell-Baker-Hausdorff operation on a Banach-Lie algebra
Let $(\mathfrak{g}, [\cdot,\cdot]_\mathfrak{g}, \Vert \cdot \Vert_\mathfrak{g})$ be an infinite-dimensional Banach-Lie algebra, and let us define for any $a,b \in \mathfrak{g}$ the series
$$~ Z^\...
- 171
12
votes
4
answers
869
views
Breaking up the free Lie algebra into GL irreps
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$The free Lie algebra $\L(V)$ generated by an $r$-dimensional vector space $V$ is, in the
language of https://en.wikipedia.org/wiki/...
- 4,713
1
vote
1
answer
298
views
Degenerate representation
Let $\rho : \mathbb{R}^n\to \mathfrak{so}(2m)$ be a faithful representation of the commutative Lie algebra $\mathbb{R}^n$ into the Lie algebra of skew-symmetric matrices. There is an orthonormal basis ...
- 2,902
4
votes
0
answers
128
views
Real Representation ring of $U(n)$ and the adjoint representation
I have two questions:
It is well known that the complex representation ring $R(U(n))=\mathbb{Z}[\lambda_1,\cdots,\lambda_n,\lambda_n^{-1}]$, where $\lambda_1$ is the natural representation of $U(n)$ ...
CommunityBot
- 1
17
votes
2
answers
2k
views
Is every Lie subgroup of GL(V) isomorphic to a (maybe another) closed subgroup of GL(V)?
I am gathering material for an exposition and I note that some texts (e.g. Ise and Takeuchi, "Lie Groups I & II", Stillwell, "Naive Lie Theory", Hall, "Lie Groups, Lie Algebras, and ...
- 12.9k
3
votes
1
answer
195
views
A representation of $\frak{sl}_n$ as partial derivatives on polynomials
As is known to all, the Lie algebra $\frak{sl}_2$ admits a very nice representation on
$$
\mathbb{K}[X,Y]
$$
the polynomials in two variables, given by
$$
E \mapsto X\frac{\partial }{\partial Y}, ~~ F ...
- 1,144
4
votes
1
answer
230
views
Are isotypic components of $S(\mathfrak{g})$ finite-dimensional?
Let $\mathfrak{g}$ be a complex simple Lie algebra. Let $S(\mathfrak{g})$ be the algebra of polynomial functions on $\mathfrak{g}$, viewed as a $\mathfrak{g}$-representation. Are the isotypic ...
- 954
2
votes
0
answers
218
views
Borel-Weil-Bott theorem for wonderful compactification in characteristic p
Are there any known results for a Borel-Weil-Bott theorem for the wonderful compactifications over characteristic $p$ (i.e., theorems that classify the cohomologies of all line bundles on a wonderful ...
- 41
0
votes
0
answers
71
views
Non-proper orthant automorphisms
Given a real vector space $V$ of dimension $d$, it is known that the automorphism Lie group of the nonnegative orthant $\mathcal{O}^+_d$ can be described just as $$\mathrm{Aut}(\mathcal{O}^+_d)=\...
- 954
3
votes
1
answer
316
views
Complete $2$-step solvable Lie algebras
A Lie algebra is complete if its center is zero and all its derivations are inner. I would like to study a class of Lie algebras, in particular
Let $C$ be the class of finite dimensional $2$-step ...
- 63.9k
2
votes
0
answers
129
views
Kac-Peterson modular forms and shifted theta functions
Let $\Lambda$ be the root lattice corresponding to an ADE root system $R$ of rank $n$. With the ADE assumption, the weight lattice is simply the dual lattice $\Lambda^{\vee}$. Given any weight vector $...
- 1,701
20
votes
6
answers
4k
views
Polynomial invariants of the exceptional Weyl groups
Let $\mathfrak{g}$ be a simple complex Lie algebra, and let $\mathfrak{h} \subset \mathfrak{g}$ be a fixed Cartan subalgebra. Let $W$ be the Weyl group associated to $\mathfrak{g}$. Let $S(\mathfrak{h}...
- 1,446
10
votes
1
answer
1k
views
Are there "reasonable" criteria for existence/non-existence of Levi factors or their conjugacy in prime characteristic?
Classical theorems attributed to Levi, Mal'cev, Harish-Chandra for a finite
dimensional Lie algebra over a field of characteristic 0 state that it has a Levi decomposition (semisimple subalgebra plus ...
- 2,821
4
votes
0
answers
147
views
Is the homogeneous coordinate ring of a flag variety a UFD?
I was wondering if $G$ is a semisimple complex algebraic group, then is the homogeneous coordinate ring of a flag variety a UFD or not?
- 201
1
vote
0
answers
100
views
Extension of an involution on $G$ to an involution on $G_\mathbb{C}$
I asked this question on MSE https://math.stackexchange.com/questions/4475382/extension-of-an-involution-on-g-to-an-involution-on-g-mathbbc but didn't receive any answer so far. My question is the ...
- 139
1
vote
0
answers
83
views
Right-invariant metrics on the unitary groups and embeddings in the complexification
Let $G = SU(n)$ and $G_c = SL(n, \mathbb{C})$. Let $g$ be a right-invariant metric on $G$ and let $g_k$ be the Killing metric on $G_c$.
Define the map $p$ from $G_c$ to $G$ which maps $h \in G_c$ to
$$...
- 5,215
4
votes
0
answers
296
views
de Rham Witt complex vs. de Rham complex of the Witt ring
I am reading the paper "Revisiting the de Rham-Witt complex" by Bhatt-Lurie-Mathew and I am a bit confused about the difference between $W\Omega_R^*$ and $\hat{\Omega}^*_{W(R)}$.
Let $\...
- 218
2
votes
0
answers
82
views
Question on a remark in Speh's paper
I am reading Birgit Speh's paper entitled "Unitary representations of Gl(n,R) with nontrivial (g,K)-cohomology" in Invent. Math. 71 (1983), no. 3, 443–465. In Remark 1.2.2.(b), it says that &...
- 1,121
4
votes
0
answers
204
views
Explicit description of wonderful compactification for PGL_3
Let $k$ be an algebraically closed field of positive characteristics. Let $X$ be the wonderful compactification of $PGL_3$ (see for example section 6 of "Frobenius Splitting Methods in Geometry ...
- 163
0
votes
0
answers
202
views
What is the importance of Cartan decomposition of a semi-simple Lie algebra?
I just started learning about Cartan decomposition of semi-simple Lie algebras, and I'm curious to know what are their applications in studying semi-simple Lie algebras. My guess was that it might be ...
- 139
2
votes
0
answers
190
views
Root systems and subroot systems
Given the root system $E_{6}$ with basis $\alpha_{1},\dotsc,\alpha_{6}$. How would I find all subroot systems of $E_{6}$ (up to Weyl equivalence) where I can write the basis of each subroot system in ...
- 21
2
votes
0
answers
147
views
Automorphism groups of "reductive" Lie algebras in positive characteristic
I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras.
Let $G$ be a reductive group ...
- 63.9k
3
votes
2
answers
585
views
Multiplication of extreme vector
This question might be elementary and standard.
Standard Notions: Let $g$ be a semisimple Lie algebra. Let $\pi=({\alpha_{1},....\alpha_{n}})$
be simple roots
$P^{+}(\pi)=\Sigma\mathbb{N}\alpha _{i}$
...
- 1,446