All Questions
2,633 questions
4
votes
1
answer
510
views
Isomorphism classes of nilpotent Lie algebras
I will begin by stating my question, and then write down some related thoughts.
Let $\mathfrak{g}$ be a finite dimensional nilpotent Lie algebra over $\mathbb{C}$. Choose an ideal $\mathfrak{h}$ in $\...
- 1,304
2
votes
2
answers
715
views
relate parabolic subalgebras to gradings?
In "Linear algebraic groups, 2nd ed. T.A.Spinger, Birkhauser" 8.4.5, one finds a characterization of parabolic subgroups via co-characters, as follows:
for simplicity, assume that $k$ is a base field ...
- 1,305
-1
votes
1
answer
240
views
irreducible Classical Lie algebras [closed]
which submodule of FG-module of a lie algebra $L$ will be determined I want to check that how we can find out a classical lie algebra like $D_4$ and $E_6$ are irreducible?
- 367
0
votes
1
answer
122
views
Solving $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$ [closed]
Given an $A \in \mathfrak{su}(n)$, is it always possible to solve for $U,V \in SU(n)$ and $\lambda \in \mathbb{C}$ such that $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$?
Cross posted from ...
- 2,099
3
votes
0
answers
73
views
False optima for control on Lie groups?
Consider the equation
$\frac{d Y_t}{dt} = (A + w(t)B) Y_t$
evolving on a compact semi-simple Lie group $G$ where $\frak{g}$ is the Lie algebra of $G$ and $A,B \in \frak{g}$ and:
$J:G \rightarrow [0,...
- 2,099
5
votes
0
answers
143
views
Polynomials invariant with respect to a nilpotent Lie algebra
Let $\mathfrak{u}$ be a nilpotent Lie algebra and let $\mathbb{C}[\mathfrak{u}]$ be the space of polynomials with the natural coadjoint action of $\mathfrak{u}$.
Can one describe $\mathbb{C}[\...
- 8,597
4
votes
0
answers
72
views
Harmonicity on semisimple groups
I asked this on Math.SE and got no answer, so I'll try my luck here.
Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in ...
- 41
5
votes
1
answer
313
views
Endomorphisms in Category O and Schubert Classes
Let $\mathfrak{g}\supset \mathfrak{b}\supset \mathfrak{h}$ be a complex semisimple Lie algebra, with choice of Borel and Cartan subalgebras, $W$ the Weyl group.
W. Soergel's 'Endomorphismensatz' ...
- 1,201
2
votes
1
answer
317
views
Decomposition of Lorentz-like groups
When studying the Lorentz group $O(1,3)$, one can decompose it into four parts... physicist usually called these
Proper-orthochronuos $\mathscr{L}^{\uparrow}_+$,
Proper-asynchronous $\mathscr{L}^{\...
- 690
1
vote
1
answer
478
views
Irreducible quotient of $U\otimes V$
All modules here are finite dimensional. The field is over complex number. Let $U$ be an irreducible $\mathfrak{sl}_n$-module, and $V$ is a highest weight $\mathfrak{sl}_n$-module. Suppose $U\otimes V$...
- 45
2
votes
0
answers
158
views
Kernel of the Weil homomorphism for compact symmetric spaces
Let $X = G/K$ be a Riemannian symmetric space of compact type and consider the "Weil homomorphism" $$w^\bullet: H^\bullet(BK; \mathbb R) \to H^\bullet(X; \mathbb R),$$ i.e. the map in cohomology ...
5
votes
1
answer
549
views
Maximal dimension of abelian ideals of a Lie algebra and extensions of the ground field
For a Lie algebra $L$ of dimension $n$ over a field ${\mathbb F}$ we denote by $\beta(L)$ the maximal dimension of abelian ideals of $L$. In general, $\beta(L)$ is not preserved under extensions of ...
- 5,878
2
votes
0
answers
157
views
$l$-weights and $l$-character of finite-dimensional highest $l$-weight representation of $L\mathfrak{g}$
I am trying to solve the following problem, which is related to relatively recent results, but I am not sure how to do it.
Problem
In this problem, $\mathfrak{g}=\mathfrak{sl}_{2}$. We study finite-...
- 357
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
4
votes
1
answer
982
views
Orbits of Lie Algebra Actions
It is well known that the image of a free Lie algebra action $\rho:\mathfrak{g}\to\mathfrak{X}^1(M)$ on a manifold $M$ is an integrable distribution of constant rank $(=\rm{dim}\;\mathfrak{g})$. Thus ...
- 41
1
vote
0
answers
84
views
Extra-Lorentzian Kac-Moody algebras
My question is about Kac-Moody (KM) algebras of finite rank with symmetrized Cartan matrices $B = C A$ ($A$ is Cartan matrix) of signatures $(-,-,+,...,+)$, $(-,-,-,+,...,+)$, etc. i.e. with $2$,...
- 371
3
votes
0
answers
440
views
Decomposition of a representation of SU(N) into representations of SU(N-1)
Let $\omega_k$ be the highest weight of the $k$-th antisymmetric representation of $\mathfrak{su}(N)$. Consider an irreducible representation of $\mathfrak{su}(N)$, characterized by its highest ...
- 630
3
votes
1
answer
273
views
Reduction of antisymmetric complex matrices
Let $E=\mathfrak{so}(n,\mathbb{C})$ be the Lie algebra of antisymmetric complex matrices. We consider the action of the complex orthogonal group $SO(n,\mathbb{C})$ on $E$ by conjugation. Is there a ...
- 4,123
4
votes
3
answers
340
views
Invariant symmetric bilinear forms and H^4 of BG
I am reading this paper of Teleman and Woodward.
On page 4, they say that $H^4(BG;\mathbb{R})$ can be identified with the space of invariant symmetric bilinear forms on $\mathfrak{g}_k$. Why is this ...
- 21k
1
vote
1
answer
700
views
CM liftings of abelian varieties and liftings of Frobenius
It is well-known that if $A$ is an ordinary abelian variety over a finite perfect field $ k$ of characteristic $ p>0$ and $ W=W(k)$ is the ring of Witt vectors over $ k$, then the canonical ...
- 395
4
votes
0
answers
814
views
Adjunction Formula for Weil Divisors on a Normal Variety X
Let $X$ be a normal variety over an algebraically closed field $k$ of characteristic $p>0$ and $S$ be a prime Weil divisor on $X$ which is normal too. Now if $K_X+S$ is NOT $\mathbb{Q}$-Cartier, ...
- 165
7
votes
0
answers
167
views
How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?
Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...
- 9,019
3
votes
2
answers
378
views
how many injective homomorphism between two lie algebra sl2 and sp6 up to conjugate by Sp6?
how many injective homomorphism between two lie algebra $sl_2 $and $sp_6$ up to conjugate by$Sp_6$ ?
- 709
2
votes
1
answer
190
views
Kostant's theorem about U(g) being free over Z(g) and a corollary of it
Hello,
$g$ is a complex semisimple Lie algebra.
There is the result that $U(g)$ is free over $Z(g)$.
There is another result: If $E$ is a finite dimensional representation of $g$, then $Hom(E,U(g)^{...
- 5,562
6
votes
0
answers
161
views
LS paths construction
Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on $W/W_{...
- 61
0
votes
2
answers
2k
views
non discrete valuation ring [closed]
Hi,
I am looking for examples of non-discrete valuation rings. Could you help me?
Thanks
- 141
0
votes
0
answers
216
views
Cayley graphs for finitely presented Lie algebras
I have seen that an important tool of finitely presented groups consists in writing down its Cayley graph with respect to a given set of generators, and then try to extract data like the coarse ...
0
votes
1
answer
81
views
Centralizer of the derived algebra in a non-perfect Lie algebra
Is there a non-perfect Lie algebra for which the centralizer of the derived algebra is trivial?
0
votes
0
answers
152
views
A solvable Lie algebra
Consider the Lie algebra $\mathfrak{g}$ span by $x_1,\ldots,x_n,y_1,\ldots,y_n$ with the commutator
$\left[y_i,x_i\right]=y_i, \left[x_i,x_{i+1}\right]=y_i, \left[y_i,x_{i+1}\right]=-y_i,\quad 1\leq ...
- 9
2
votes
0
answers
246
views
Non-semisimple Lie algebra tensors
Let $\mathfrak{L}$ be a non-semisimple Lie algebra. Let $T_i$ be its generators. As usual, define the structure constants $C_{ij}^k$ by $[T_i,T_j]=C_{ij}^kT_k$ and the metric tensor $g_{jm}$ by $g_{jm}...
- 4,829
0
votes
0
answers
71
views
Curves in $\mathfrak{su}(n)$ with specific property
Consider a curve $\gamma_s= U_s^{\dagger} b U_s$ in $\mathfrak{su}(n)$ where $U_s$ is a smooth curve on $SU(n)$ (starting at $U_0 = \mathbb{I}$) and nonzero $b\in \mathfrak{su}(n)$ and $s \in[0,T]$ ...
- 2,099
1
vote
1
answer
980
views
Cartan-Weyl basis and irreducible representations
I am just starting to look at this concept in particle physics, and wondering if someone could clear something up for me. It concerns irreducible representations and the Cartan-Weyl basis. (I'm ...
- 21
0
votes
0
answers
177
views
Connection between Lie algebras and fusion rings
Example: Take the irreps of $SU(2)$: $0,1/2,1,...$ (Spin notation.) The quantum dimensions are $1,q+1/q,q^2+q+1+1/q+1/q^2,...$. At $q=(-1)^{1/5}$ this evaluates to $1,\phi,0,...$ and you get the ...
- 4,829
2
votes
0
answers
222
views
Computing maximal ideals of a Lie algebra
Would you know an algorithm (or an automatic method) that computes all maximal ideals $J$ of a given Lie algebra?
Or an algorithm that computes all maximal ideals $J$ containing a given minimal ideal ...
- 73
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
2
votes
1
answer
443
views
Can this Lie group written as a direct product?
Let $G=G_1.G_2$ be a Lie Subgroup of $SO(k) \times SO(2) \subset SO(k,2)$, where
$G_1=SU(k/2) \subset SO(k)$ and $G_2$ is a Lie subgroup of $SO(k) \times SO(2)$ isomorphic to $SO(2)$.
Let $G_1 \cap ...
- 121
1
vote
1
answer
574
views
very very basic question on semi-simple Lie algebras
I have a very basic question on Lie algebras. I'm doing particle physics, and a lot of emphasis seems to be placed on the weight diagrams of simple Lie algebras. But these simple Lie algebras are ...
- 21
1
vote
0
answers
71
views
Low-dimensional classical r-matrices
Let $g= gl_2$. Suppose that $r \in g \otimes g$ satisfies the following properties:
(1) $r_{12} + r_{21} \in g \otimes g$ is $g$-invariant, $r_{12} = r$, $r_{21} = \tau \ r_{12}$.
(2) $[r_{12}, r_{...
- 6,211
5
votes
3
answers
757
views
Realizing higher level Fock spaces
Let $\mathfrak{g}$ = $\mathfrak{gl}_{\infty}$.
To each positive integer $k$ one can associate the level $k$ Fock space $\mathcal{F}_{k}$.
For a dominant weight $\lambda$ of level $k$, one can ...
- 485
4
votes
0
answers
209
views
Partial differential Equation over characteristic p
I want some references on partial differential equations over characteristic $p$. If we have a first order partial differential equation, how can we check whether there exists polynomials or rational ...
3
votes
1
answer
386
views
Modules which are direct sum of weight spaces.
For a semisimple Lie algebra $\mathfrak{g}$, a highest weight module $V(\lambda)$ with highest weight weight $\lambda$ has the property that every submodule $W$ of $V(\lambda)$ is a direct sum of the ...
- 6,211
2
votes
0
answers
90
views
Singularities of the Quantum propagator (baby version)
Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$:
$V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$
...
- 2,099
3
votes
1
answer
359
views
Lie algebras with a one-dimensional maximal subalgebra
Let L be a Lie algebra with a one-dimensional maximal subalgebra. Is the following true?
Over a perfect field of characteristic 0 or p > 3, every such finite-dimensional Lie algebra is either 2-...
- 310
0
votes
0
answers
257
views
Basis for Witt algebra in general format
The Witt algebra $W(n,m)$ is defined as the set of element ${∑f_jD_j such that f_j∈A(n,m)}$ with usual Lie bracket. I am a bit confused about basis for $W(n,m)$? What is the meaning of $"f_jD_j"$ ? I ...
- 367
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
0
votes
1
answer
129
views
About Kahler curvature operator
I have problems on how to consider the Kahler curvature operator. I know that one can consider the Riemannian curvature operator $R$ as a linear transformation from $\mathfrak{so}(n,\mathbb{R})$ to $\...
- 351
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
2
votes
0
answers
138
views
Exploiting a finite presentation of a Lie algebra
In "my youth" I computed a finite presenation of the Poisson algebra on $S^2$ (finite presentation as a Lie algebra).
In what ways might this be useful? Does this allow you to extract information ...
-2
votes
1
answer
172
views
Action of automorphism group on Lie algebra [closed]
I want to know whether an automorphism group of a simple Lie algebra over $GF(2)$, acts transitively on non-zero elements of Lie algebra or not? How can I check this property?
- 367
6
votes
2
answers
507
views
Concerning the dimension of a complex variety modulo a prime
Let V be a complex affine variety given as the vanishing set of a set of polynomials with integral coefficients. I have 3 questions.
1)
Under what assumption will the dimension of V over C remain ...
- 63