All Questions
2,633 questions
3
votes
2
answers
365
views
Does there exist a canonical "degree" filtration on quantum groups?
For any lie algebra $\mathfrak g$, there is a natural filtration on $U(\mathfrak g)$ by "degree": the filtered piece $U^{\leq n}(\mathfrak g)$ is just the image in $U(\mathfrak g)$ of $\bigoplus_{k=0}^...
- 18.7k
1
vote
1
answer
91
views
Smallest subalgebra of $\mathfrak{su}(4)$ arrising from a control problem on $SU(4)$
What is the smallest subalgebra of $\mathfrak{su}(4)$ containing the span of the set $A = \{A, B_1, B_2\}$ where:
$A = i (J^x \sigma_x \otimes \sigma_x + J^y \sigma_y \otimes \sigma_y + J^z \sigma_z \...
- 2,099
5
votes
0
answers
287
views
Nori fundamental group and etale fundamental group in positive characteristic
Let $X$ be a smooth projective surface over an algebraically closed field of char $p > 0$. Suppose that $\pi_{1}^{et}(X) = \{1\}$. Can Nori fundamental group scheme of $X$ be non-trivial?
- 163
3
votes
1
answer
805
views
Finite connected groups over a perfect field of characteristic p
In 14.4 of "Introduction to Affine Group Schemes" it is proved (!) that if $A$ represents a finite connected group scheme over a perfect field $k$ of characteristic $p$ then $A$ has the form $k[X_{1}, ...
- 163
4
votes
0
answers
85
views
Homological dimension of Joseph quotients
Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ not isomorphic to $sl(n)$.
Let $\mathcal O$ be the minimal nilpotent orbit in $\mathfrak g^*$. Joseph proved that there exists unique two-...
- 7,037
7
votes
0
answers
236
views
Invariant theory of $SL_2$ over a field of positive characteristic
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $W$ be a finite dimensional $SL_2$-module over $k$. Let $V$ be the natural representation of $SL_2$.
What can be said - in ...
- 605
6
votes
1
answer
184
views
Embedding of F(4) in OSp(8|4)?
Is the superconformal algebra in five dimensions, $F(4)$, a subalgebra of the (maximal) six-dimensional superconformal algebra $OSp(8|4)$?
1
vote
0
answers
123
views
How to define the determinant of a morphism between graded Lie algebras?
I have the following question. Suppose $g_1$ and $g_2$ are two finite dimensional, nilpotent, stratified Lie algebras and $A:g_1\to g_2$ is a morphism of the graded Lie algebra. I wonder whether there ...
- 1,881
6
votes
1
answer
2k
views
How to calculate partition function of a QFT by summing over irreducible representations of the symmetry group?
By definition computing the partition function of a QFT amounts to doing a Feynman Path Integral exactly. At a schematic level I can see why this can become a question of summing/integrating over ...
- 3,541
2
votes
0
answers
172
views
Springer Isomorphisms for Adjoint Simple Exceptional Groups
I'm trying to understand explicitly a construction of Springer isomorphisms for adjoint exceptional groups given by Bardsley and Richardson. Their construction is as follows. Let $G$ be an adjoint ...
- 2,902
4
votes
1
answer
458
views
What is known on finite dimensional nilpotent Lie algebras with maximal index ?
The index of a Lie algebra is
$\mathrm{ind}(\mathfrak{g})=\mathrm{min}_{\lambda \in \mathfrak{g}^{*}} \mathrm{dim} \mathfrak{g}^{\lambda}$, where $\mathfrak{g}^{\lambda} = \lbrace x\in \mathfrak{g} \...
- 41
1
vote
1
answer
366
views
Resolutions of Lie algebras
We have a good notion of dgc algebra resolutions of commutative algebras.
Is there an explicit construction of a dg Lie algebra resolution of a Lie algebra?
- 3,880
5
votes
1
answer
728
views
Convexity radius of a Lie Group
Is there a nice formula/method to find the convexity radius of a matrix Lie group (the manifold can be noncompact) ?
Edited based on comments:
Definition : Convexity Radius (Berger - Panoramic View ...
- 207
12
votes
1
answer
709
views
Cartan involution for finite W-algebras
Does anybody know if there is an analog of the Cartan (anti)involution for W-algebra
associated to a nilpotent element e, which is principal in some Levi subalgebra
of semi-simple Lie algebra g? ...
- 7,037
2
votes
0
answers
562
views
Complex Finite Dimensional Representation of GL(N,C)
What are all the complex finite dimensional linear representation of $GL(N,\mathbb{C})$?
We already know all the complex finite dimensional linear representation of SU(N).
- 3,804
6
votes
1
answer
318
views
Littelmann path operators for an arbitrary positive root
Looking at a few of Littelmann's papers, he seems to only apply root operators $f_\alpha$ for $\alpha$ a simple root. However, the definition seems to make perfect sense for $\alpha$ any positive ...
- 71
1
vote
1
answer
156
views
tensor product of two irreducibles having same maximal weight
Is there any explicit decomposition of tensor product of two finite dimensional irreducible modules of simple lie algebras whose highest weights are same?
- 73
6
votes
1
answer
635
views
Groups of Hodge type, hodge structure on Lie algebra
Hi,
Let $W$ be a real algebraic group, and $G$ the associated complex group. Then $W$ is of Hodge type if there is a $\mathbb{C}^*$ action on $G$ such that $U(1)$ preserves $W$ and the action of $-1$ ...
- 91
0
votes
1
answer
230
views
Name for ideal generated by Lie subalgebra
Let $\mathfrak{m}$ be a Lie sub-algebra of the Lie algebra $\mathfrak{g}$. Is there a name for the smallest ideal of $\mathfrak{g}$ containing $\mathfrak{m}$? It certainly exists and coincides with ...
- 2,479
3
votes
0
answers
70
views
Attainability of Global Optima In Optimal Control
Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation:
$\frac{d x(t)}{dt} = F(x,u)$
one can consider the ...
- 2,099
1
vote
0
answers
96
views
Largest dimensional Lie subgroup of $SU(N)$ [duplicate]
What is the largest (Lie) subgroup of $SU(n)$ in the sense of its dimension.
I am aware of this potential duplicate subgroup of SU(N) with maximal manifold dimension , however, the title of this ...
- 2,099
3
votes
2
answers
733
views
If the morphism of root data induced by an isogeny of a reductive group is a Frobenius, is then the isogeny itself a Frobenius?
Let $G$ be a reductive (or just semisimple) algebraic group over an algebraically closed field $k$ of characteristic $p > 0$, let $T$ be a maximal Torus and let $f:G \rightarrow G$ be an isogeny. ...
- 5,243
2
votes
0
answers
80
views
Vanishing of finite difference operators by composition under a cyclic condition
Consider $n$ finite difference operators $D_1, \ldots, D_n$
acting on real-valued functions $f_1 (y), \ldots , f_n (y)$
of a variable $y$, with the following properties:
(i) $D_i f_i (y) = 0$ for ...
5
votes
1
answer
366
views
Differential of a nilpotent or semisimple element
Let $G$ be an algebraic connected subgroup of $GL_n(\mathbb{C})$ and let $\chi : G \to \mathbb{C}^*$ a character. Consider $d_e\chi : \mathfrak{g} \to \mathbb{C}$ the differential of $\chi$ at the ...
- 671
1
vote
1
answer
225
views
non-locally simple $\mathcal{g}$-modules
I'm interested in an example of a simple $\mathcal{g}$-module $M$ over some locally simple Lie algebra say $\mathcal{g}\simeq gl(\infty)$ such that $M$ is not isomorphic to a direct limit of simple ...
- 501
8
votes
1
answer
520
views
The exceptional Lie algebra $\mathfrak{g}_2$ and binary cubics
How is the exceptional 14-dimensional Lie algebra $\mathfrak{g}_2$ related to the covariant algebra for the binary cubic?
Here are some details on this question. This algebra is generated by 4 forms,...
- 1,277
9
votes
1
answer
777
views
Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0
In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.
Is there any interpretation of these operators in ...
- 139
3
votes
1
answer
201
views
Jacobi identity for circular permutations
Let $\left(g_i\right)$ be a sequence of $N$ elements of a Lie algebra. Let $s$ be a cyclic permutation of $N$ elements of order $N$: $(1,2,...,N)\to(2,...,N,1)$.
Let
$c_N=\sum\limits_{k=1}^N[g_{s^k(...
- 2,205
5
votes
0
answers
387
views
Are these empirical discoveries about the Serre Swinnerton-Dyer ring of prime level modular power series actual theorems?
In this question Joel Bellaiche constructed an algebra, M, of modular forms for
gamma_0 (N) in finite characteristic (which he called p, but I'll call ell) and asked to know its structure. Matt ...
- 5,422
7
votes
1
answer
1k
views
Explicit convergence of Baker-Campbell-Hausdorff
Let g be a finite dimensional simple Lie algebra over C. The Baker-Campbell-Hausdorff series defines a (multivariable) analytic function from a neighborhood of 0 in g \times g \to g. What is the ...
- 71
5
votes
1
answer
620
views
Commutator formulas in a universal enveloping algebra
I am interested in finding formulas for commutators of symmetrized monomials in a universal enveloping algebra. Let $C(x_1,\ldots, x_n)= (1/n!)\sum x_{\sigma(1)}\cdots x_{\sigma(n)}$ where the sum ...
- 51
4
votes
2
answers
340
views
Lie algebra cohomology over non-fields
This is probably a very elementary question. I'm trying to get an explicit description of the cochain complex and coboundary maps for Lie algebra cohomology over $\mathbb{Z}$, and more generally, over ...
- 7,320
8
votes
1
answer
775
views
Is an irreducible holomorphic symplectic manifold a simple Lie algebra?
The tangent bundle of a hyper-Kahler manifold gives a quadratic Lie algebra in the derived category. Can this be regarded as a simple Lie algebra according to Vogel's definition?
A point of view that ...
- 9,132
2
votes
1
answer
1k
views
Several question on Affine Lie algebra
These questions might be elementary for I just started to learn affine Kac-Moody algebra.
It is well known that if we consider finite dimensional Lie algebra, we have the folloing projection:
$R(\...
- 5,515
2
votes
1
answer
304
views
Connected extensions of finite by connected algebraic groups
Let $1\to H\to E\to G\to 1$ be a short exact sequence of algebraic groups defined over an algebraically closed field $k$ of characteristic $p$. Suppose $H$ is a finite group, and $G$ and $E$ are ...
- 631
0
votes
1
answer
994
views
faithful adjoint representation
When $n>3$ is even, how can I show that $PGL(n,\mathbb{R})$ has a faithful adjoint representation? Of course when n is even, $PGL(n,\mathbb{R})$ is not connected.
- 159
2
votes
1
answer
232
views
An innocent looking subgroup of $U(n)$
Consider the Lie subalgebra of $\mathfrak{u}(n)$ given by $L = \{A \in \mathfrak{u}(n): \sum_{j=1}^n A_{ij} = 0 \text{ for all } i \in [n]\}$. What is its dimension? What does the corresponding Lie ...
- 4,466
2
votes
1
answer
286
views
Stabilizers and Quotients of a Nilpotent Lie Algebra
Let $\frak{g}$ be a finite-dimensional complex nilpotent Lie algebra. What conditions on $\xi\in\frak{g}$ ensure the existence of a (canonical or non-canonical) surjective morphism $\frak{g}\...
- 4,920
2
votes
0
answers
244
views
Descent theory of line bundles on abelian varieties under isogenies (in char p>0)
I have a couple of questions regarding the descent theory of line bundles on abelian varieties under isogenies in positive characteristic.
Let $X$ be an abelian variety and $L\in Pic(X)$ a line ...
- 614
2
votes
1
answer
251
views
Spectral sequence in Lie algebra
The exact sequence $0\rightarrow sl(A)\rightarrow gl(A)\rightarrow A/[A,A]\rightarrow 0$ gives rise to a spectral sequence in homology, I want to know the details for this spectral sequence at the ...
- 713
4
votes
0
answers
169
views
Automorphisms of Nilmanifolds
Let $\mathfrak{g}$ be an n-dimensional, rational, nilpotent Lie algebra with simply connected that lie group $G$. It is stated in some papers that if $A$ is an automorphism of $\mathfrak{g}$ which is ...
- 419
1
vote
2
answers
1k
views
homotopy between solutions of Maurer-Cartan equation
If $S_0, S_1$ are two solutions of Maurer-Cartan equation $dS+\frac{1}{2}{S,S}=0$ for a dg-Lie algebra $g$, do we have a suitable concept of homotopy between $S_0$ and $S_1$?
- 1,499
3
votes
0
answers
742
views
Explicit description of O^{cris}_n in Fontaine/Messing
Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 ...
2
votes
2
answers
605
views
Resolution of a free lie algebra as a module over its universal enveloping algebra.
Let $L=L(V)$ be a free Lie algebra on a vector space $V$ and $A=T(V)$ the tensor algebra. Make $L$ into a module over $A$ consistent with the formula $a\cdot \alpha=[a,\alpha]$ for $a\in V$ and $\...
- 1,355
0
votes
2
answers
386
views
Zariski closures of one parameter additive maps in positive characteristic
Suppose we are in characteristic $p$, and that the field, $K$, that we are working over is imperfect. We have a map $\Theta: K \to K^{\delta}$ where each coordinate function $\Theta_i$ is an additive ...
- 75
1
vote
1
answer
151
views
hamilton type Lie algebras
If n be positive integer and for an n-tuple of positive integers m=(m1,...,mn) then p(n,m) is graded and filtered subalgebra of W(n,m).p(n,m) is called non-alternating hamilton lie algebra over GF(2). ...
- 367
3
votes
0
answers
235
views
The fundamental in the tensor square of a complex representation of $SO(N)$
I would like to figure out whether there is an irreducible complex (in the sense non-self-conjugate) representation of a group $SO(N)$, $N>2$, whose tensor square contains the fundamental ...
- 173
7
votes
1
answer
718
views
Ways to characterize supersingular primes?
I've read the definition, and it basically says p is a supersingular prime iff
the fundamental domain of a group generated by \Gamma(p) and a matrix ((0, 1), (-p, 0)) is rational.
And there's a ...
- 15.1k
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
2
votes
1
answer
307
views
On a Strongly F-regular Pair (X, \Delta)
Let $X$ be a normal projective variety over a field of characteristic $p>0$ and $(X, \Delta\geq 0)$ be a pair such that $K_X+\Delta$ is $\mathbb{Q}$-Cartier whose index is not divisible by $p$. ...
- 165