All Questions
2,633 questions
11
votes
1
answer
928
views
Non-algebraic K3 surfaces in characteristic $p$
I have a very naive question.
Recall that over the field of complex numbers, there exist non-algebraic K3 surfaces. Namely, smooth non-projective simply connected compact complex surfaces with ...
- 21.4k
11
votes
3
answers
554
views
Uniform setting for computing orders of algebraic groups over finite quotients of the integers?
A couple of recent questions on MO have involved the characters or the orders of specific finite groups of the form $G(\mathbb{Z}/n\mathbb{Z})$ for a familiar algebraic group $G$ defined over $\mathbb{...
- 52.9k
11
votes
2
answers
1k
views
Cotangent complex of perfect algebra over a perfect field
Let $A$ be a perfect $\kappa$-algebra over a perfect field $\kappa$ of positive characteristic $p$. Then the algebraic (= classical) cotangent complex $L_{A/\kappa}^{\operatorname{alg}}$ is known to ...
- 2,011
11
votes
1
answer
480
views
Uncle of Witt algebra
A Witt algebra W is an infinite-dimensional Lie-algebra defined by the generator relations:
W: $[L_{j},L_{k}]:=(j-k)\cdot L_{j+k}$
And my first thought was: What about the analogous algebra defined by
...
- 4,829
11
votes
2
answers
2k
views
Classification of symplectic resolutions
A. Okounkov said, "symplectic resolutions are Lie algebras of the 21st century." Is there a conjecture on the classification of symplectic resolutions? Do Braverman-Finkelberg-Nakajima Coulomb ...
- 2,974
11
votes
1
answer
810
views
Motivation behind Kac's notation for affine root systems
I'm reading Kac's Infinite Dimensional Lie Algebras. In Chapter 4, he classifies the affine root systems. Bourbaki classified the affine Coxeter groups, but multiple root systems can give the same ...
- 156k
11
votes
2
answers
798
views
Twin categories in representation of Lie algebra
Let $\mathfrak{g}=\mathfrak{n}^-\oplus\mathfrak{h}\oplus\mathfrak{n^+}$ be a triangular decomposition of semisimple Lie algebra. Let $\mathcal{Z}$ be the central of universal envoloping Lie algebra of ...
- 1,457
11
votes
1
answer
571
views
Diagram folding of simple Lie algebras
A non-simply laced simple root system can be constructed from the simply-laced root system by folding the Dynkin diagram and hence the corresponding non-simply-laced Lie algebra can be constructed by ...
- 673
11
votes
2
answers
918
views
On a proposition in Hartshorne's paper "Ample vector bundles on curves"
In Prop. 4.1, p. 87 of the article "Ample vector bundles on curves" (Nagoya Math. J. 43 [1971], 73--89), R. Hartshorne states the following:
Let $A$ be an abelian variety [over an alg. closed field $...
- 3,076
11
votes
2
answers
1k
views
Representations of $\mathrm{SL}(2)$ in characteristic 2
$\DeclareMathOperator\SL{SL}$In characteristic zero one can use the Clebsch-Gordan rule to decompose tensor products of $\SL(2)$-modules. In characteristic $p$, things are more complicated.
I am ...
- 605
11
votes
1
answer
617
views
Is there a unique "natural" action of $\mathsf{SL}_{n+1}$ on $\mathbb{R}^n$?
Context
By acting naturally via $\mathsf{SL}_3$ on $\mathbb{RP}^2=\{[x:y:z]\}$ and by taking the induced action on the affine hyperplane $z=1$ (which we identify with $\mathbb{R}^2$), one can realize ...
- 2,527
11
votes
1
answer
749
views
Realisation of Kac-Moody Lie algebras
I am reading Infinite dimensional lie algebras by Kac. He starts with a $n \times n$ GCM (Generalized Cartan Matrix) $A$ of rank $l$, then he defines the realization associated with the matrix $A$ ...
- 1,269
11
votes
2
answers
882
views
From Weyl groups to Weyl groupoids?
I'm trying to find a framework where the choices in the classical construction of a root system of a semi-simple lie algebra are not needed.
Let $\mathfrak{g}$ be a semisimple lie algebra.
...
- 7,799
11
votes
1
answer
1k
views
How is this (Tannakian) de Rham fundamental group calculated?
$\newcommand{\dR}{\mathrm{dR}}$Edit: I originally asked this question on MSE, but migrated it to MO after a long period of inactivity and a recommendation from another user.
Let $X$ be a complex ...
- 661
11
votes
2
answers
696
views
Bracket of lyndon words?
Here is a simple question regarding the standard Lyndon basis for the free Lie Algebra. Suppose I take two lyndon words $m$ and $n$ and their standard bracketings $B(m)$ and $B(n)$ as elements in the ...
- 727
11
votes
1
answer
1k
views
PBW theorem over a Q-algebra, without freeness or flatness
Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-Lie algebra, which is not necessarily free as a $k$-module. Let $S\left(L\right)$ denote the symmetric algebra of $L$ (over $k$), constructed ...
- 33.8k
11
votes
1
answer
1k
views
Baker–Campbell–Hausdorff formula: prime divisors of denominators
Consider the Baker–Campbell–Hausdorff formula (Wikipedia page):
$$Z(X,Y) := X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] + \dotsb$$
Many sources, including the Wikipedia ...
- 7,320
11
votes
2
answers
2k
views
Inverse of Baker-Campbell-Hausdorff
This should be quite simple to answer. I have a situation in which I must have an explicit expression for the inverse of Baker-Campbell-Hausdorff. More precisely:
I have two power series $P_1(X,Y)$, $...
- 7,442
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
11
votes
1
answer
675
views
Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)
Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in Goldman-...
- 113
11
votes
1
answer
312
views
Homotopes of simple Lie algebras
Let $\mathfrak{g}$ be a complex simple Lie algebra with bracket $[x,y]$. For which $z\in \mathfrak{g}$ does the formula
$$
\mu(x,y)=ad (z)([x,y])=[z,[x,y]]
$$
define another Lie bracket on the same ...
- 12.1k
11
votes
2
answers
553
views
A list of all irreducible 4-dimensional real representations
I need a reference to a complete list of all
faithful real 4-dimensional irreducible representations
of real Lie algebras.
The list itself is not very hard to obtain.
Using the Levi decomposition,
it'...
- 9,237
11
votes
2
answers
7k
views
How to find Casimir operators?
Given a general Lie algebra, is there a general procedure to find all its Casimir operator?
- 111
11
votes
1
answer
334
views
An identity in Lie algebras over fields of positive characteristic
Let $L$ be a Lie algebra over a field of characteristic $p>0$ and $D$ a derivation of $L$. For every $x\in L$ denote by $\mathrm{ad} x$ the adjoint map $\mathrm{ad}x: L \rightarrow L, a\mapsto [x,...
- 630
11
votes
1
answer
554
views
Cohomological Proof of Serre Relations for a Symmetrizable Kac-Moody Algebra
In 'Infinite Dimensional Lie Algebras, 3rd edition', Kac mentions at the top of p. 170 that 'a simple cohomological proof of Theorem 9.11 was found by O. Mathieu (unpublished)'.
Does anyone know how ...
- 329
11
votes
2
answers
593
views
Is there an analog of Clifford Theorem in the setting of Lie algebras?
A classical theorem of Clifford states that if G is a finite group and K a field, then every irreducible right KG-module is a completely reducible right KN-module, where N is any normal subgroup of G. ...
- 5,878
11
votes
2
answers
863
views
Valuations and separable extensions
Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...
- 19.6k
11
votes
1
answer
483
views
3/4-Lie superalgebras: how much of a theory can one develop?
Let $\mathfrak{s} = \mathfrak{s}_0 \oplus \mathfrak{s}_1$ be a real Lie superalgebra. (The ground field does not matter much, but at least one formula will not work as written if the characteristic ...
- 33.3k
11
votes
2
answers
525
views
What subalgebras of $\mathfrak{so}(2n)$ or $\mathfrak{sp}(2n)$ are orthogonal to the centre of $\mathfrak{gl}(n)$?
Consider the Lie algebra inclusions $\mathfrak{gl}(n) \subset \mathfrak{so}(2n)$ and $\mathfrak{gl}(n) \subset \mathfrak{sp}(2n)$. Let $\mathfrak{c} \subset \mathfrak{gl}(n)$ denote the centre. ...
- 54.6k
11
votes
1
answer
490
views
Free graded Lie algebras
Let $R$ be a graded commutative unital ring and $M$ a graded $R$-module (all gradings are over $\mathbb{Z}$). I'm looking for a reference for the following statement:
If $M$ is $R$-free, then the ...
- 985
11
votes
2
answers
2k
views
History of Jordan Canonical Form?
Can anyone suggest a reference that discusses the history of the Jordan canonical form? In particular, I am interested in:
When and how was it first stated? (I understand it was independently stated ...
- 3,761
11
votes
1
answer
615
views
Do Richardson varieties have rational singularities in arbitrary characteristic?
The title basically asks the question. I'll review the relevant terminology and explain what I have and haven't found in the literature.
Let $G$ be a reductive group. Let $v \leq w$ be elements of ...
- 156k
11
votes
1
answer
439
views
Lie subalgebras of $\chi^{\infty}(M)$ of codimension $n = \dim M$
For a connected $n$-manifold $M$, the Lie algebra of all smooth vector fields is denoted by $\chi^{\infty}(M)$. For a point $p\in M$ we define $L_{p}=\{X\in \chi^{\infty}(M)\mid X(p)=0\}$. Of course ...
- 366
11
votes
0
answers
183
views
Are algebras with rational structures dense in varieties of real Lie nilpotent algebras?
One says that a real nilpotent Lie algebra has $\mathbb Q$-structure if it has a basis with rational structure constants. It is well known that there are nilpotent Lie algebras without $\mathbb Q$-...
- 723
11
votes
0
answers
437
views
A rather strange algebra
Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector ...
- 3,318
11
votes
0
answers
202
views
Quiver and relations for blocks of category $\mathcal{O}$
In Vybornov - Perverse sheaves, Koszul IC-modules, and the quiver for the category $\mathscr O$ an algorithm is presented to calculate quiver and relations for blocks of category $\mathcal{O}$ .
...
- 26.5k
11
votes
0
answers
192
views
The $\frak{sl}_2$-representation on a symplectic manifold
Any symplectic manifold $(M,\omega)$ carries a representation of $\frak{sl}_2$: Define the maps
$$
L: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~~~~ \Lambda: \Omega^\bullet \to \Omega^{\bullet}, ~~~~~~...
- 643
11
votes
0
answers
662
views
How to define (and compute) the Cartan-Killing form of the group of volume-preserving diffeomorphisms?
This question was raised a while ago in a blog post by Terry Tao on the Euler-Arnold equation and he called it "quite tricky".
Has anyone in the meantime tried to formulate this question precisely, ...
- 1,675
11
votes
0
answers
432
views
Connection between Gelfand-Tsetlin basis and SSYT basis in Schur module
Consider an $n$-dimensional complex vector space $V$ with a chosen basis $e_1,\ldots,e_n$. This basis defines a Cartan decompostion of $GL(V)\cong GL_n$ and for an (integral dominant) highest weight $\...
- 3,513
11
votes
0
answers
576
views
What's known about the mod 2 reduction of the level l Jacobi modular equation?
Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...
- 5,422
11
votes
0
answers
1k
views
Do the Standard Conjectures imply parts of the "Weil II" Riemann Hypothesis?
It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...
- 1,764
10
votes
5
answers
4k
views
Low dimensional nilpotent Lie algebras
This is a reference request question. I would like to know more on the structure of low dimensional nilpotent lie algebras. I heard that up to dimension 6 there are only finitely many isomorphism ...
- 2,479
10
votes
6
answers
2k
views
Proofs in the same vein as Ax-Grothendieck
I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...
- 23.4k
10
votes
3
answers
2k
views
Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?
Background/motivation
It is a classical fact that we have a natural isomorphism $Sym^n (V^*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is ...
- 14.8k
10
votes
2
answers
1k
views
Abstract Jordan Decomposition different from usual Jordan Decomposition
It's known that if $L\subset gl(V)$, with $V$ finite dimensional, is a semisimple Lie algebra, then the abstract and usual Jordan decompositions in $L$ coincide. Is it possible to provide a counter-...
- 349
10
votes
4
answers
2k
views
Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$
One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,
$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$
the ring of ...
- 3,541
10
votes
3
answers
755
views
Invariants of exterior powers
Let $\mathfrak{g}$ be the Lie algebra of $GL(n,\mathbb{R})$. Let $\theta(X) = - X^T$ be the Cartan involution on $\mathfrak{g}$; it induces decomposition as $\mathfrak{g} = \mathfrak{k} \oplus \...
- 601
10
votes
3
answers
2k
views
nth term in the Baker-Campbell-Hausdorff formula
I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...
- 245
10
votes
2
answers
2k
views
Chevalley Groups over an arbitrary ring.
My question is simply about the Chevalley groups over rings. In many books, including Carter's book on "Simple groups of Lie types", the groups are considered over fields. I have checked the ...
- 2,508
10
votes
4
answers
1k
views
Longest Element of an Affine Weyl Group
I know that the Weyl groups of affine Lie algebras don't have a longest element, but are there any good substitutes for w_0. In particular, is there any good substitute for a reduced decomposition of ...
- 5,548