All Questions
2,633 questions
19
votes
1
answer
977
views
Lang's Jacobian identity: slicker, elementary proof?
In Jeffrey Lang, A Jacobian identity in positive characteristic, J. Commut. Algebra, Volume 7, Number 3 (2015), pp. 393--409, the following result is proven:
Theorem 1. Let $p$ be a prime. Let $\...
- 33.8k
19
votes
1
answer
1k
views
Is there a connected $k$-group scheme $G$ such that $G_{red}$ is not a subgroup?
I've been trying a learn a little more about group schemes by working through a set of exercises on Brian Conrad's website. Exercise 8.3 of http://math.stanford.edu/~conrad/papers/gpschemehw1.pdf ...
- 3,625
19
votes
1
answer
2k
views
Perfectoid approach to resolution of singularities in char $p$
Since perfectoid techniques have built a bridge between char $0$ and char $p$ worlds, it is conceivable that they can be applied to resolution of singularities in char $p$ using their successful ...
- 199
19
votes
2
answers
2k
views
Dual versions of "folding" symmetric ADE Dynkin diagrams?
Start with the Dynkin diagram of an irreducible root system, typically associated with a simple
Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE
diagrams ...
- 52.9k
19
votes
2
answers
3k
views
Bertini theorems for base-point-free linear systems in positive characteristics
Suppose that $X$ is a smooth algebraic variety over an algebraically closed (uncountable if it helps) field of characteristic $p > 0$. Suppose that $L$ is a line bundle, probably ample or at least ...
- 20.5k
18
votes
4
answers
4k
views
Basis-free definition of Casimir element?
Let $V$ be a finite-dimensional vector space and let $\mathfrak g \subset \mathfrak{gl}(V)$ be a representation of a semisimple Lie algebra on $V$. Let $e_1, \dots, e_n$ be a basis for $V$. Let $e_1', ...
- 64k
18
votes
3
answers
1k
views
What happened to the fourth paper in the series "On the classification of primitive ideals for complex classical Lie algebras" by Garfinkle?
In a series of papers in Compositio Math. entitled On the classification of primitive ideals for complex classical Lie algebras I, II and III, Garfinkle describes an algorithm that allows one to ...
- 2,508
18
votes
3
answers
1k
views
Is a retract of a free object free?
I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?
- 1,875
18
votes
5
answers
6k
views
Does a finite-dimensional Lie algebra always exponentiate into a universal covering group
Hi,
I am a theoretical physicist with no formal "pure math" education, so please calibrate my questions accordingly.
Consider a finite-dimensional Lie algebra, A, spanned by its d generators, X_1,.....
- 649
18
votes
3
answers
3k
views
Lifting varieties to characteristic zero.
If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ ...
- 4,015
18
votes
2
answers
1k
views
Is homology finitely generated as an algebra?
If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra?
Is it easier if we impose any of the three conditions: characteristic zero; ...
- 8,727
18
votes
1
answer
1k
views
A linear algebra problem in positive characteristic
Let $A$ be a symmetric square matrix with entries in $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ such that all of its diagonal entries are nonzero. Does there exists always a vector $x$ with all ...
- 4,484
18
votes
3
answers
574
views
Does there exist something like an $H_3$ and $H_4$ (icosahedral) Lie algebra or algebraic group?
The (finite-dimensional) complex simple Lie algebras have been classified by Killing and Cartan a long time ago in the $\mathsf{A}_n,\mathsf{B}_n,\mathsf{C}_n,\mathsf{D}_n$ families and $\mathsf{G}_2,\...
- 32.5k
18
votes
4
answers
4k
views
Is every G-invariant function on a Lie algebra a trace?
I am in the (slow) process of editing my notes on Lie Groups and Quantum Groups (V Serganova, Math 261B, UC Berkeley, Spring 2010. Mostly I can fill in gaps to arguments, but I have found myself ...
- 54.6k
18
votes
3
answers
3k
views
What is a homotopy between $L_\infty$-algebra morphisms
A $L_\infty$-algebra can be defined in many different ways. One common way, that
gives the 'right' kind of morphisms, is that a $L_\infty$-algebra is a graded cocommutative and coassociative ...
- 2,074
18
votes
2
answers
4k
views
What is the physical meaning of a Lie algebra symmetry?
The physical meaning of a Lie group symmetry is clear: for example, if you have a quantum system whose states have values in some Hilbert space $H$, then a Lie group symmetry of the system means that $...
- 118k
18
votes
3
answers
1k
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What is the Zariski closure of the space of semisimple Lie algebras?
Given Leonid Positselski's excellent answer and comments to this question, I expect that the present one is a hard question. Recall that the Lie algebra structures on a (finite-dimensional over $\...
- 54.6k
18
votes
1
answer
2k
views
Independence of $\ell$ of Betti numbers
When $X$ is a smooth proper variety over $\mathbb F_q$, we know by Deligne's theory of weights that the dimension of $H^i_{\operatorname{\acute et}}(\bar X, \mathbb Q_\ell)$ does not depend on $\ell$. ...
- 28.8k
18
votes
1
answer
727
views
(Dis)similarity between groups and Lie algebras
There are many questions which sound similar or the same for groups and Lie algebras. Some (very few, actually) of those questions have identical solutions and answers. Some have identical answers but ...
- 2,804
18
votes
1
answer
686
views
$8 \times 31 = 8 \times 31$?
The Lie algebra $\mathfrak{e}_8$ has (at least) two ways to be written as a direct sum of $31$ Cartan subalgebras.
First, Thompson and Smith showed that the (compact or complex) Lie group $\mathrm{E}...
- 54.6k
18
votes
0
answers
987
views
"Special" meanders
One of the open problems in combinatorics is enumeration of meanders.
Here on MO I only could find them under the heading not-especially-famous-long-open-problems-which-anyone-can-understand.
Since my ...
- 18.4k
17
votes
2
answers
3k
views
Why is one interested in the mod p reduction of modular curves and Shimura varieties?
Why is one interested in the mod p reduction of modular curves and Shimura varieties?
From an article I learned that this can be used to prove the Eichler-Shimura relation which in turn proves the ...
user19475
17
votes
4
answers
2k
views
What are supersingular varieties?
For varieties over a field of characteristic $p$, I saw people talking about supersingular varieties.
I wanted to ask "why are supersingular varieties interesting". However, as I don't want to ask an ...
- 2,040
17
votes
4
answers
3k
views
Folding by Automorphisms
Background reading: John Stembridge's webpage.
The idea is that when you want to prove a theorem for all root systems, sometimes it suffices to prove the result for the simply laced case, and then ...
- 8,866
17
votes
4
answers
9k
views
Noether's theorem in quantum mechanics
In classical mechanics:
If a Lagrangian $\mathcal{L}$ is preserved by an infinitesimal change in the state space variables $q_i \to q_i + \varepsilon K_i(q)$, this leads to only second order change in ...
- 22.8k
17
votes
2
answers
831
views
Relationship between "different" quantum deformations
This is a generic question, a good answer to it may be a reference to a corresponding paper\textbook, but any useful comments would be okay too.
Let $\mathfrak{g}$ be a (simple) Lie algebra and $U_q(\...
- 765
17
votes
2
answers
1k
views
A short proof for simple connectedness of the projective line
The Riemann-Hurwitz formula implies that the projective line $\mathbb{P}^1_K$ over any algebraically closed field $K$ is simply connected (i.e., $\pi_1^{et}(\mathbb{P}^1_K) = 1$; equivalently, if $\...
- 3,362
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 ...
- 1,161
17
votes
2
answers
890
views
Counting degrees of freedom in Lie algebra structure constants (aka why are there any nontrivial Lie algebras of dim >5?)
This is a question about the true number of constraints imposed by the Jacobi identity on the structure constants of a Lie algebra.
For an $n$-dimensional Lie algebra, there are $\frac{n^2(n-1)}{2}$ ...
- 285
17
votes
2
answers
947
views
Are the unipotent and nilpotent varieties isomorphic in bad characteristics?
In characteristic 0 or good prime characteristic, there are standard ways to relate the unipotent variety $\mathcal{U}$ of a simple algebraic group $G$ and the nilpotent variety $\mathcal{N}$ of its ...
- 52.9k
17
votes
1
answer
2k
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So, did Poincaré prove PBW or not?
This seems to be a question whose answer depends on whom you ask. Maybe we can come up with a final answer?
It is known that Poincaré, at least, invented something that can be called Poincaré-...
- 33.8k
17
votes
1
answer
4k
views
Gap in an argument in Fulton & Harris?
I'm reading through the two chapters in Fulton and Harris on the representation theory of $\mathfrak{sl}(3,\mathbb{C})$, in preparation for lecturing on them this week. I'll use F&H's notation, ...
- 2,713
17
votes
2
answers
2k
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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 ...
17
votes
1
answer
541
views
Splitting the injection that you get from the Poincaré-Birkhoff-Witt theorem
Let $\mathfrak g$ be a Lie algebra over a field of characteristic zero, with universal enveloping algebra $U\mathfrak g$. By the Poincaré-Birkhoff-Witt theorem one knows that $i:\mathfrak g \to U\...
- 40.3k
17
votes
2
answers
2k
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Examples of representations of quantum groups
I am trying to learn some basic theory of quantum groups $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a simple Lie algebra, say $sl_n(\mathbb{C})$. As far as I heard the finite dimensional ...
- 21.8k
17
votes
1
answer
502
views
Irreducibility of root-height generating polynomial
The height $ht(\alpha)$ of a positive root $\alpha$ in a (finite, crystallographic) root system $\Phi$ is $\sum_{i=1}^n c_i$ where $\alpha = \sum_{i=1}^n c_i \alpha_i$ is its decomposition as a sum of ...
- 2,753
17
votes
2
answers
835
views
Generators of the cohomology of a Lie algebra
Fix a characteristic zero ground field. One can easily check that if $\mathfrak g$ is a simple Lie algebra, then the trilinear map map $\omega$ given by $$\omega(x,y,z)=B([x,y],z),$$ with $B$ the ...
- 47.3k
17
votes
1
answer
2k
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What is the recent development of D-module and representation theory of Kac-Moody algebra?
I just started to collect the papers of this field and know little things. So if I make stupid mistake, please correct me.
It seems that there are several approaches to localize Kac-Moody algebra(in ...
- 5,515
17
votes
2
answers
1k
views
Higher level analogs of Nicolas-Serre theory
NICOLAS-SERRE THEORY
Let $F \in Z/2[[x]]$ be $x+x^9+x^{25}+...$, the exponents being the odd squares, and $V$ be the space spanned by the $F^k$ with $k$ odd. Nicolas and Serre define formal Hecke ...
- 5,422
17
votes
0
answers
1k
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Relations in a certain Lie algebra
Let ${\mathfrak g}$ be the (real) Lie algebra generated by infinitely many generators $D_i, E_i$ with $i=1,2,3,\dots$ subject to the following relations for any natural numbers $i,j$:
\begin{gather*}
[...
- 114k
17
votes
0
answers
1k
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Katz--Mazur for abelian varieties
Over $\mathbb Z$, there is a smooth DM stack $A_g$ classifying abelian varieties.
Over $\mathbb Z[\frac 1N]$, there is finite etale cover $A_g(N)_{\mathbb Z[\frac 1N]}\to A_g\otimes\mathbb Z[\frac 1N]...
- 18.7k
17
votes
0
answers
553
views
Lie algebras vs. graph complexes
A ribbon graph is a graph in which every vertex has valence at least three and is equipped with a cyclic ordering of its adjacent half edges. The ribbon graph complex $\mathcal{G}_*$ is the chain ...
- 2,035
17
votes
0
answers
917
views
Combinatorial identity involving the Coxeter numbers of root systems
The setup is:
$R$ = irreducible (reduced) root system;
$D$ = connected Dynkin diagram of $R$, with nodes numbered $1,2,...,r$;
$\hat D$ = extended Dynkin diagram, nodes numbered $0,1,2,...,r$;
$\...
- 2,449
16
votes
3
answers
4k
views
How does one write the "gothic" letters ($\mathfrak{g}$) in handwriting?
Most mathematical notation is designed with handwriting in mind in the first place, and typography must then try to follow, not always very successfully. However there is a particular type of notation ...
16
votes
5
answers
2k
views
About the intrinsic definition of the Weyl group of complex semisimple Lie algebras
It may be a easy question for experts.
The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h}$...
- 9,019
16
votes
4
answers
6k
views
How many three dimensional real Lie algebras are there?
The main point of the question is: I would like to know whether there are only finitely many, countable infinitely many or even uncountable many isomorphism classes of $3$-dimensional real lie ...
- 161
16
votes
4
answers
3k
views
Decompose tensor product of type $G_2$ Lie algebras.
Let $G$ be a semisimple Lie algebra over $\mathbb{C}$. Let $V(\lambda)$ be the irreducible highest weight module for $G$ with highest weight $\lambda$. If $G$ is of type A, we can decompose $V(\lambda)...
- 6,211
16
votes
3
answers
4k
views
Generators of invariant polynomials of semisimple Lie algebra
Suppose $\mathfrak{g}$ is a complex semi-simple Lie algebra. By a theorem of Chevalley, we know that $S(\mathfrak{g})^\mathfrak{g}$, i.e. the $\mathfrak{g}$ invariant polynomials, is generated by $l$ ...
- 587
16
votes
3
answers
2k
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On Category O in positive characteristic
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$. In the case that $k$ has characteristic 0, there has been intensive study of the BGG category O of representations of ...
- 3,637
16
votes
2
answers
952
views
Is there a simple system that has $\text{SU}(3)$ symmetry?
The buckle at the end of a belt has $\text{SU}(2)$ symmetry, if the rotations around the three coordinate axes are taken as generators. See, for example, the paper by Hart, Francis and Kauffman, ...
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