All Questions
2,633 questions
3
votes
1
answer
311
views
Linear independence in (graded) Lie algebras
I asked a mixed-up version of this question earlier.
The Lie algebras I have in mind are the homotopy Lie algebras of wedges of finitely many spheres (in dimensions greater than $1$). Thus each ...
- 12.5k
1
vote
2
answers
341
views
Copies of ax+b inside the AN part of an Iwasawa decomposition?
As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book DG, Lie groups and symmetric spaces and Knapp's ...
- 25.8k
4
votes
0
answers
226
views
Explicit description of graded (counital) cofree cocommutative coalgebras
Let $k$ be a field of characteristic $p \neq 2$, and $V = \oplus V_{n}$ be a graded vector space over $k$.
Then, can one compute the graded (counital) cofree cocommutative coalgebra $C(V)$ ...
- 41
2
votes
1
answer
267
views
On Non F-pure ideal and Sharp F-Purity for a pair $(X, \Delta)$ where $K_X+\Delta$ is NOT $\mathbb{Q}$-Cartier
Suppose $(X,\Delta\ge 0)$ is a pair such that $(p^g-1)(K_X+\Delta)$ is an Integral Weil Divisor for some $g>0$ and $X$ is a normal variety. Define $\mathcal{L}_{e,\Delta} = \mathcal{O}_X( (1-p^g)(...
- 165
2
votes
0
answers
606
views
Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces
Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non ...
- 165
2
votes
1
answer
2k
views
What is Extreme/Extremal vector according to some weights
I know this might be a very elementary question. But I could not find the original definition of Extreme(or Extremal)vectors of some weights $\lambda$(fixed the $w\in W$,where $W$ is Weyl group)
I am ...
- 5,515
4
votes
2
answers
694
views
Ample line bundle and Frobenius morphism on smooth toric variety
Let $k$ be an algebraically closed field of $\mathrm{char}(k)=p>0$, $X$ a smooth toric projective variety of $\dim X=n$, $F_X:X\rightarrow X$ the absolute Frobenius morphism of $X$. Then for any $\...
- 85
2
votes
1
answer
528
views
Is there an easy proof of the fact that the intermediate image functor respects weights?
It was proven in BBD (see Corollary 5.3.2) that for an open immersion $j$ the functor $j_{!*}$ preserves weights of mixed sheaves. The proof relies on several previous results; it is especially ...
- 16.9k
7
votes
1
answer
527
views
Nilpotent Lie algebras of vector fields
Let $L$ be a finite-dimensional nilpotent subalgebra of the Lie algebra $W_n$ of all vector fields in $n$ variables (I am interested both in polynomial and formal vector fields). Does there exist a ...
- 2,804
2
votes
1
answer
303
views
Lifting vector fields to its resolution in char $p$
In this paper 4.7, the authors showed that on a normal variety $X$, if there is a tangent vector field on its smooth locus, then it can be lifted as a logarithmic tangent vector field on a log ...
- 656
2
votes
0
answers
258
views
(Co)Homology of groups vs. Lie algebras: polynomial rings
For Lie groups (or algebraic groups over fields) there is a strong relation between the cohomology of the group and the cohomology of its Lie algebra. Some MO-question where this is discussed can be ...
- 17.4k
4
votes
1
answer
578
views
Getting certain modular functions from characters
It is well known that characters of affine Lie algebras have
certain modular properties. For instance, the linear span of all
irreducible characters at a given level must be invariant under a
certain ...
- 694
2
votes
1
answer
154
views
Termination conditions for matrix Lie alebra basis generation via "P. Hall algorithm"
Suppose $g_1,...,g_n\in\mathbb{M}_{d\times d}(\mathbb{C})$ are matrices and we are interested in finding the smallest matrix Lie algebra containing them, that is, the matrix Lie algebra generated by $...
- 123
7
votes
0
answers
315
views
A question on the resolution of parabolic Verma module $M_I(\lambda)$ in BGG category O
I am reading Humphrey's book "Representations of semisimple Lie algebras in the BGG Category O" on Page 189, Proposition 9.6, where he remarked that "Note that if we had developed the full BGG ...
- 71
1
vote
0
answers
187
views
Examples of Lie subalgebras of universal enveloping algebras
I'm looking for non-trivial examples of triples $(\mathfrak g, L, \psi)$, where $\mathfrak g$ and $L$ are finite-dimensional non-abelian Lie algebras over field $\Bbbk$ and $\psi\colon U_{\mathfrak g}\...
- 1,422
2
votes
1
answer
224
views
The orbit $(G\cdot X) \cap \mathfrak{t}$ for $X\in \mathfrak{t}$ singular
This question may be a simple problem for experts. Let $G$ be a connected compact Lie group and $T$ be its maximal torus. Let $\mathfrak{g}$ and $\mathfrak{t}$ be the corresponding Lie algebras. We ...
- 9,019
5
votes
1
answer
822
views
How do you exponentiate a section of the adjoint bundle to get a gauge transformation?
Suppose $E$ is a vector bundle with structure group $G$ and let $P = F(E)$ be the frame bundle. Let $\mathfrak{g}_P$ denote the associated bundle to the adjoint representation of $G$ on its Lie ...
- 1,474
10
votes
2
answers
393
views
Counting points on varieties of low codimension
The graduate students here at MIT have been thinking about questions like the following: Over $\mathbb{F}\_q$, how many symmetric matrices are there with nonzero determinant and $0$'s on the diagonal? ...
- 156k
0
votes
3
answers
171
views
Does there exist an $A$ and $\mathfrak{su}(2) \subset B$ such that $\mathfrak{su}(3) \simeq A \otimes B$
I'm in the middle of trying to prove something at the moment and am looking for a decomposition of the Lie algebra $\mathfrak{su}(3)$ into a tensor product of some algebra $A$, and another $B$ ...
- 1,479
0
votes
2
answers
758
views
What is a Module over a Lie algebroid?
Let $\alpha: \mathfrak g_A \to T_{A/k}$ be a Lie algebroid over a $k$-algebra $A$. Numerous facts about and its universal enveloping algebra comes from the theory of ring differential operators on $A$....
- 1
3
votes
2
answers
237
views
Reconstructing a Lie group Banach representation from the Lie algebra rep. on analytic vectors
Dear all,
I have some difficulties with the following assertion in the book of Kirillov.
Let $G$ be a connected Lie group, and T a given (!) representation of G on a Banach space V.
Let $V^\omega$ ...
- 399
1
vote
1
answer
161
views
A question about G-Manifolds
I am looking for a clear reason for following fact:Is there any reference ?
Why a $G$-invariant differential form $\omega$ on a homogeneous $G$-manifold $M=G/H$ is uniquely determined by its value at ...
user21574
4
votes
0
answers
741
views
The normalizer a maximal compact subgroup of a semi-simple Lie group
Let $G$ be a semi-simple real Lie group such that $|\pi_0(G)|<\infty$ and let $K$ be a maximal compact subgroup of $G$.
Q1: How does one prove that $N_G(K)=K$?
So I know a nice (and low-tech) ...
- 7,609
0
votes
1
answer
274
views
when $g^*$ is invariant under $Ad(G)$?
Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra.
Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under $...
user21574
3
votes
1
answer
194
views
Zero-divisors in a graded Lie algebra
Let $\mathfrak{g}$ be positively graded Lie algebra over $\mathbb{Q}$, concentrated in even degrees.
Question: If $\mathfrak{g}$ is not free, must there exist linearly independent elements $a,b\in\...
- 35.9k
1
vote
2
answers
369
views
Gerstenhaber bracket out of $L_\infty$ algebras
Given a Lie algebra g, with $Ug$ being its universal enveloping algebra, one can construct a cochain complex $d: Ug^n \rightarrow Ug^{n+1}$, and a Gerstenhaber bracket on $\oplus_n Ug^n$ so that $\...
- 41
3
votes
0
answers
154
views
Homology of derivations of Differential Graded Lie algebra
Let $(L,d)$ be a Differential Graded Lie Algebra ($L=\bigoplus L_i$ and $d:L_i \to L_{i-1}$ satisfying the graded Leibniz rule).
On the algebra $\mathrm{Der}L$ of derivations of $L$ define a grading ...
- 131
12
votes
1
answer
796
views
Lie's third theorem via differential graded algebras?
Dennis Sullivan, "Infinitesimal computations in topology", Publ. IHES: At the end of section 8, he writes, among other things, roughly the following.
Let $\mathfrak{g}$ be a (finite-dimensional, real)...
- 20.9k
-2
votes
3
answers
1k
views
sl(2)-modules... [closed]
I'm trying to learn some Lie algebra without much knowledge of representation theory. While being asked to prove some things about an sl(2)-module, why can one assume that the module is irreducible ...
- 13
8
votes
1
answer
852
views
Symmetrization map for universal enveloppings. What are Harish-Chandra images of symmetrization (poisson-center of S(gl_n) ) ?
For any Lie algebra symmetrization maps Poisson center of S(g) to center of U(g).
Consider g=gl_n, the Poisson center of S(gl_n) is isomorphic to algebra of symmetric polynomials in eigenvalues.
...
- 24.9k
2
votes
1
answer
737
views
Schur Weyl duality for sl_n representations
Consider a finite dimensional vector space $V$ and the general linear group $GL(V)$ acting on it. Both $GL(V)$ and the symmetric group $S_d$ act on the tensor product of $d$ copies of $V$, and by Weyl ...
- 596
7
votes
1
answer
799
views
Liftability of Enriques Surfaces (from char. p to zero)
Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with ...
- 839
1
vote
0
answers
108
views
General quantum highest-weights dimension formulas
The formulas hold modulo typos :-)
It is well known (tl;dr fun fact: not well enough for me, I forgot where I saw it so I guess-computed it from the data in the Hayashi paper; promptly I found it in ...
- 4,829
4
votes
2
answers
339
views
Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular?
Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular?
- 16.9k
0
votes
0
answers
163
views
Generalized weight space
In their paper Lepowsky and Mcmollum sketch theory of weights in a more general setting. Here is their definition of a weight space:
If $A$ is a subset of $\mathfrak g$ and $\lambda$ is a function ...
- 1
0
votes
1
answer
680
views
Higher order Approximation of Lie groups [closed]
Maybe the following is trivial or folklore, but I can't find any concrete proof of
the theorem, that higher order derivatives of Lie groups don't give any new information
above what is coded in its ...
- 137
12
votes
2
answers
829
views
Matrices into path algebras
I was thinking about quivers recently, and the following idea came to me.
Let ei,j denote the matrix unit in Mn for 1 ≤ i,j ≤ n. Let Γ denote the complete quiver on vertices {1, …,...
- 1,756
4
votes
0
answers
304
views
Lie algebra cohomology
Let $\mathfrak{g}$ be a simple complex Lie algebra of $rank(\mathfrak{g})\geq2$ and dimension $d$. Fix a (non-zero) invariant bilinear form $(\cdot,\cdot)$ on $\mathfrak{g}$ and let $\{x_i\}_{1\leq i\...
- 41
3
votes
1
answer
465
views
How can one find generators of basic differential forms on homogeneous spaces?
Dear all,
In short, my problem is that I would like to have a better control of the 1-forms on a homogeneous space. Contrary to the group case, the module of differential form is not trivialisable. ...
- 399
10
votes
2
answers
1k
views
Does a universal Frobenius map exist?
For any prime p, one has the Frobenius homomorphism Fp defined on rings of characteristic p.
Is there any kind of object, say U, with a "universal Frobenius map" F such that for any prime p and any ...
- 2,399
4
votes
1
answer
325
views
About the term "tangential derivation" on a free Lie algebra.
Let $\mathcal{lie}_n$ be the free Lie algebra generated by $n$ elements $x_1,\ldots, x_n$. A derivation $u\in \text{Der}(\mathcal{lie}_n)$ is called tangential if there exist $a_i\in \mathcal{lie}_n, ...
- 9,019
2
votes
0
answers
47
views
$TSU(n)$ completely integrable with 3 $SU(2)$ invariant functions?
Consider the Lie group $SU(n)$ endowed with the standard bi-invariant metric. Then $SU(n)$ can be viewed as a symmetric space of $K_{n,n} := SU(n) \times SU(n)$.
Define $M := K_{n,n} /SU(n)$. Using ...
- 501
1
vote
0
answers
217
views
Representations of $\mathfrak{so}(3)$ ($\mathfrak{so}(2,1)$) and $SO(3)$ ($SO(2,1)$)
(Apologies if this question is too basic!) I have explicit 5-dimensional real representations of $\mathfrak{so}(3)$ and $\mathfrak{so}(2,1)$, and I want to know whether it's necessarily true that the ...
- 818
1
vote
2
answers
467
views
Explanation of $y = x \exp(\triangle)$ for a Lie Group
Let $M$ be a non-compact matrix Lie group and $T_e M$ its lie algebra.
Consider a point $x \in M $ and $ \triangle \in T_e M$.
To move from $x$ to a point $y \in M$ along $\triangle$, below group ...
- 207
2
votes
1
answer
332
views
Ample bundle under Frobenius morphism
Let $k$ be an algebraically closed field of char($k$)=p>0, $X$ a smooth projective variety over $k$, $F:X\rightarrow X^{(1)}$ the relative Frobenius morphism. Let $E$ be an ample vector bundle on $X$. ...
- 85
1
vote
1
answer
326
views
A Criterion for Reductivity of Lie Subgroups
Let $G$ be a connected, simply-connected, complex, semisimple Lie group. Suppose that $H$ is a Zariski-closed subgroup of $G$ with reductive Lie algebra $\frak{h}$. Under what conditions may one ...
- 4,920
2
votes
2
answers
2k
views
Reference of primitive root mod p
Can any body give me a reference of the result about primitive root mod p for a class of prime number p.
The result that I am looking for is something along this line:
$2$ is a primitive root mod $p$...
- 73
-1
votes
2
answers
806
views
The lie algebra of the orthogonal group of an arbitrary space time metric
Let X ad Y be two vectors in R4, and define the inner product of X and Y as:
(X*Y) = gikXiYk (summation convention for repeated indicies)
Then we consider the 4x4 matrix g whose components are gik. ...
- 251
6
votes
2
answers
1k
views
Jordan decomposition in a classical group
Let $\mathfrak{g} \subset \mathfrak{gl}_n$ be one of the classical real or complex semisimple Lie algebras. If $g \in \mathfrak{g}$, then $g$ has a Jordan decomposition $g = g_s + g_n$ with $g_s$ ...
- 2,713
0
votes
1
answer
130
views
equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution
I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...
- 2,099