All Questions
2,633 questions
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Explicit $K$-basis of a Lie subalgebra
$\newcommand{\Kbar}{{\overline K}}
\newcommand{\Q}{{\mathbb Q}}
$I consider Example 5.7 of Sansuc's paper Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres of ...
- 14.2k
4
votes
0
answers
184
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Does a dual basis for $U_h(\mathfrak{sl}_2(\mathbb{C}))$ exist?
Let $\mathcal{F}_h(\operatorname{SL}_2(\mathbb{C}))$ be the $\mathbb{C}[[h]]$-algebra generated by $a, b, c, d$ subject to the following relations:
\begin{align*}
& ac = e^{-h}ca, \quad bd = e^{-h}...
- 12.9k
5
votes
0
answers
123
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Algebraic groups and formal group laws in characteristic p
In characteristic zero, there is a well-known equivalence between Lie groups, formal group laws and Lie algebras.
Let $p$ be a prime. The equivalence between Lie groups and Lie algebras has an ...
- 413
7
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3
answers
303
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Enveloping algebra of affine Lie algebra is (not) noetherian
I work over an algebraically closed field of characteristic $0$. Let $\mathfrak{g}$ be a semisimple Lie algebra, $\hat{\mathfrak{g}}=\mathfrak{g}[t,t^{-1}]\oplus\mathbb{C}K\oplus\mathbb{C}D$ the ...
- 12.9k
3
votes
1
answer
129
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Questions about the quotient of extended Weyl group and the isomorphism of extended Weyl group
When I am reading a paper An Algebraic Characterization of the Affine Canonical Basis by Beck, Chari, and Pressley, and I have some questions about some notations.
In the paper, we assume that $\...
- 137
4
votes
1
answer
91
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Lie algebra with finitely generated envelope
If a Lie algebra $\mathfrak g$ is finitely generated, its enveloing algebra $U\mathfrak g$ is finitely generated as an associative algebra. In fact, taking the enveloping algebra of the surjection $\...
- 63.9k
0
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0
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72
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Question about action of exponential of Lie algebras (Faraut and Koranyi's book)
I'm trying to understand a statement on page 49 in the book "Analysis on Symmetric Cones" by Faraut and Koranyi.
The situation is as follows. Let $\Omega$ be a symmetric cone in $V$ and ...
- 633
16
votes
0
answers
188
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Representation theory of Pin groups
I am (still) thinking about branching rules from $\mathfrak{so}(n+m)$ to $\mathfrak{so}(n) \oplus \mathfrak{so}(m)$, using Proctor's paper as the starting point.
Proctor describes this rule for $m = 2$...
- 1,574
2
votes
0
answers
34
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Analytic continuation of a bi-holomorphic automorphism on an irreducible bounded symmetric domain
I need your help.
Let $\Omega \subseteq \mathbb{C}^n$ be a type $IV_n$ Cartan domain, i.e;
$\Omega$ =$\{ z \in \mathbb{C}^n$: $1-2Q(z,\bar z)+|Q(z, z)|^2>0,\qquad Q(z, \bar z)<1 \}$
where $Q(z,...
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votes
0
answers
23
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Existence of a subregular element with abelian centralizer in a quadratic Lie algebra
All Lie algebras here will be finite dimensionnal complex Lie algebra.
We say that such an Lie algebra $\mathfrak{g}$ is quadratic if there exist a skew-symetric, non-degenerate bilinear form or ...
- 188
12
votes
3
answers
802
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The orders of the exceptional Weyl groups
Who first calculated the orders of the Weyl groups $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$? How were these orders calculated?
- 31.5k
2
votes
0
answers
132
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A question about q-binomials at roots of unity
I have a question about a lemma $9.3.6$ in the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. This question comes from page 301, "The restricted specialization&...
- 24.2k
6
votes
1
answer
317
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Which Lie groups are covers of matrix groups?
I would like to ask a variation on a question (not yet answered) I previously asked on math.SE, namely:
Which Lie groups are covers of matrix Lie groups? That is, which Lie groups $G$ admit discrete ...
- 16.2k
3
votes
1
answer
231
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Does a representation of the universal cover of a Lie group induce a projective representation of the group itself?
Suppose that $G$ is a connected Lie group, $\tilde{G}$ its universal cover, $p:\tilde{G}\to G$ the covering map. Does a representation $\rho$ of $\tilde{G}$ on a finite-dimensional vector space $V$ ...
- 16.2k
3
votes
1
answer
162
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Compact symmetric spaces and sub-root systems
Given two semisimple complex Lie algebras $\frak{g}$ and $\frak{n}$ such that the root system of $\frak{n}$ arises as a sub-root system of the root system of $\frak{g}$, does this then imply that $\...
- 12.9k
1
vote
0
answers
58
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Linear algebraic group, absolute root system, computing roots
Let $G(F)$ be a reductive linear algebraic group, where $F$ is a local field. Let $T(F)$ be a maximal anisotropic torus of $G$ that splits over a quadratic extension of $F$. Is there an efficient ...
- 11
0
votes
0
answers
124
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Do the following two notions of quantum groups sometimes coincide?
On the one hand there is the notion of quantum groups due to Drinfeld and Jimbo. In there notion a quantum group is defined as a deformation of the universal enveloping algebra of a semisimple Lie ...
- 163
6
votes
1
answer
464
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Adjoint orbits of a finite group of type $G_2$
Let $q=p^\alpha$ be a prime power and $k=\mathbb{F}_q$. Let $G\subseteq \mathrm{GL}_N(k)$ be a simple finite group of Lie type, with root system of type $G_2$, and let $\mathfrak{g}\subseteq \mathfrak{...
CommunityBot
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6
votes
1
answer
154
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Derivations and central extensions of some infinite dimensional simple Lie algebras in characteristic zero
Let $F$ be a field of characteristic zero and consider the polynomial algebra $A=F[x_1,x_2,\ldots,x_n]$ in $n$ indeterminates over $F$. I recall that the derivations of $A$ form a simple Lie algebra $...
- 5,878
5
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0
answers
148
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Tensor product of a Verma module of the highest weight and a Verma module of the lowest weight, $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$
$\DeclareMathOperator\sl{\mathfrak{sl}}\newcommand\hw{\mathrm{hw}}\newcommand\lw{\mathrm{lw}}$Consider $\mathfrak{g}=\sl_2(\mathbb{C})$. Fix $\lambda,\mu\in\mathbb{C}\setminus \frac{1}{2}\mathbb{Z}_{\...
- 51
7
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0
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121
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Cohomology of current algebras in higher dimensions
I am interested in the cohomology of current algebras in $n$ dimensions with coefficients in non-trivial modules. An $n$-dimensional current algebra is a Lie algebra of smooth functions on ${\mathbb R}...
- 1,579
2
votes
1
answer
144
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Paper request: Graev's classification of SU(2,2) irreducible unitary representations
I am interested in Graev's paper in "M. L. Graev:Dokl. Akad. Nauk SSSR,98, 517 (1954); Amer. Math. Soc. Transl.,66, 1 (1968)." in which the irreducible unitary representations of SU(2,2) are ...
- 21.6k
2
votes
0
answers
80
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Does restricting the eigenvalues of Hermitian matrices to the interval $[0,2\pi)$ make the exponential map to the unitary group bijective?
Let $U \in U(n)$ be a generic unitary matrix. Since the unitary group $U(n)$ is compact and connected, I know that the exponential map is surjective, i.e. that every $U \in U(n)$ has the form $U = e^{...
3
votes
1
answer
300
views
Characterization of reductive Klein geometries
In my struggle to understand Cartan/Klein geometries, I have the intuition that reductive Klein geometries are the link to connect the "classical" differential geometry approach with this &...
- 12.9k
2
votes
0
answers
40
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Reconstruction of a Poisson-Lie group structure from a Lie bialgebra $\mathfrak{g}$
Let $(\mathfrak{g}, [,], \delta)$ be a Lie bialgebra where $\delta$ is the cobracket. It is well-known that there exists a simply connected Poisson-Lie group $G$ such that $\mathfrak{g} = \mathrm{Lie}(...
- 63.9k
1
vote
1
answer
102
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Multiplicities and double and triple tensor products of simple $\frak{g}$-modules
Given a complex simple Lie algebra $\frak{g}$ and a simple module $V_{\lambda}$ for some dominant weight $\lambda$. Consider the tensor product decomposition
$$
V_{\lambda} \otimes V_{\lambda} \simeq ...
- 3,054
2
votes
0
answers
49
views
Action of Lusztig braid group operators on locally finite part
Let $\mathbf{U}_q(\mathfrak{g})$
be a Drinfel'd-Jimbo quantum group. The quantum group $\mathbf{U}_q(\mathfrak{g})$ acts on itself by the left adjoint action $ad(u)(x)=u_{(1)}u S(u_{(2)})$, where we ...
- 141
3
votes
0
answers
50
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Root systems of maximally noncomact Cartan subalgebras
Let $G$ be a real reductive Lie group, and $K$ a maximal compact subgroup in $G$. Write $\mathfrak{g}$ for the Lie algebra of $G$, and $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the Cartan ...
- 951
6
votes
2
answers
794
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Tensor algebra and universal enveloping algebra
Let $\mathfrak g$ be a Lie algebra which is not reductive. Let $T(\mathfrak g)$ and $U(\mathfrak g)$ be the tensor algebra and universal enveloping algebra of $\mathfrak g$ respectively. We have a ...
- 492
7
votes
1
answer
262
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A name for the Weyl group of $\frak{so_{2n}}$
For the $A$-series the Weyl group is the symmetric group $S_n$. For the $B$ and $C$ series the Weyl group is the hyperoctahedral group $\mathbb Z_2 \wr S_n$.
A) Does the $D$-series Weyl group $S_n \...
- 10.7k
6
votes
2
answers
1k
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Cartan's Structure Equations VS Cartan's Method of Equivalence
There have been a number of posts on related questions, such as:
Geometric interpretation of Cartan's structure equations,
What is the geometric significance of Cartan's structure equations? ...
- 11
12
votes
1
answer
980
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How are Sheffer polynomials related to Lie theory?
Sheffer polynomials $\{P_n(x)\}$ have generating function $P(x,t) = \sum_{n=0}^{\infty}P_n(x)t^n=A(t)e^{xu(t)}$.
This form reminds me of the Lie group–Lie algebra correspondence. Is there any ...
- 10.5k
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 ...
- 4,713
0
votes
1
answer
85
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Centralisers of elements in a Lie algebra
I am considering the centraliser subgroup of some specific elements in a Lie algebra.
Let $\mathfrak{g}$ be a Lie algebra over characteristic $0$. Let us consider some elements $x:=\{x_1,x_2,\ldots , ...
- 12.9k
4
votes
1
answer
114
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Sum of two positive roots which is not a root: uniqueness of heights of the summands
Consider a (finite reduced irreducible crystallographic) root system $\Phi$ and four positive roots $\alpha,\beta,\gamma,\delta$ such that $\{\alpha,\beta\} \neq \{\gamma,\delta\}$ and $\alpha+\beta=\...
- 550
6
votes
4
answers
415
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Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$
Let $G$ be a connected algebraic group over an algebraically closed field $k$ of characteristic zero (I'm mostly interested in the case of a reductive group).
By the Tannakian formalism, $G(k)$ can be ...
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3
votes
1
answer
85
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Isogeny of compact Lie group with central circle
Suppose $G$ is a connected, compact Lie group and $S^1 \subset G$ is a central subgroup. Can I write $G$ as a quotient of a product group $$G=(S^1 \times H)/Z$$
where the $S^1$ factor maps onto the ...
- 2,368
9
votes
2
answers
378
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Reference for an old result of P. M. Cohn
As it was shown by Malcev, unlike the commutative case, in which every domain can be embedded in a field, there are noncommutative domains that can't be embedded in a division ring.
For noncommutative ...
- 19.4k
3
votes
0
answers
128
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Spectra of Coxeter diagrams and representations of Coxeter groups
Let $S$ be a Coxeter diagram, i.e. an unoriented graph, edges labelled by weights $3,4,5,...$ . This includes (affine) Dynkin diagrams,
Then
Theorem 3.1.3 of Brouwer and Haemers' Spectral Graph ...
- 5,711
1
vote
0
answers
65
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Some details about relationship between central charges and second cohomology group of the Lie algebra
S. Weinberg in his book "The quantum theory of fields" talks about central charge that appear in Lie algebra of a given Lie group. To be more precise, on page 83 in the book, he computes the ...
- 63.9k
3
votes
0
answers
192
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How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
4
votes
1
answer
201
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Lie algebra cohomology and Lie groups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}\DeclareMathOperator\Sp{Sp}$Let $G$ be a complex Lie group and $\mathfrak{g}$ its Lie algebra (which is over $\mathbb{C}$). My first question is:
(...
- 2,368
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0
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42
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Which finite group schemes over positive characteristic have projectivity detected by a set of (generalized) cyclic subgroups?
A finite algebra $B$ over a field $K$ has projectivity detected by a set $S = \{f_i : A \to B \mid i \in I\}$ of maps into $B$ from a fixed algebra $A$ if for any finite module $M$, $M$ is projective ...
- 542
4
votes
0
answers
79
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Closed character formula for the module $L(a\omega_i)$
Let $\mathfrak{g}$ be a complex finite-dimensional simple Lie algebra with a fixed Cartan subalgebra $\mathfrak{h}$. Assume that $\omega_1, \omega_2, \dots, \omega_n\in\mathfrak{h}^{*}$ is the ...
- 41
0
votes
0
answers
69
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A weakening of the definition of positive roots for a root system
Given a root system $\Delta$ a choice of positive/negative roots is a decomposition of the elements of $\Delta$ into two subsets $\Delta^+$ and $\Delta^-$ satisfying
$$\Delta^+ = - \Delta^-\tag{$*$}\...
- 12.9k
0
votes
1
answer
304
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A Lie group whose Lie algebra is equal to (the Lie algebra? of )all functions with fibrewise polynomial growth
Let $M$ be a Riemannian manifold. We denote by $\mathfrak{g}$ the space of all smooth function $f:TM\to \mathbb{R}$ with fibre wise polynomial growth. Is it a Lie algebra wrt the Poisson bracket ...
- 8,098
9
votes
2
answers
317
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Constructing symmetric invariants of the exceptional simple Lie algebra as restrictions
Let $\mathfrak{g}$ be a complex simple Lie algebra with maximal torus $\mathfrak{h}$, Weyl group $W$. The adjoint representation $\operatorname{ad} : \mathfrak{g} \rightarrow \mathfrak{gl(g)}$ extends ...
- 12.9k
2
votes
2
answers
336
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Orthosymplectic superalgebra
Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$
which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
- 12.9k
2
votes
0
answers
34
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Bounding norms of symplectic matrix factorisations and non-separable Hamiltonian flows
Problem setup: Let $e^{hJM}$ be the time-$h$ flow corresponding to the ODE $\dot{x} = JMx$, with $M = \left(\begin{array}{cc}
A & C\\
C^T & B\\
\end{array}\right)$ ...
- 875
0
votes
0
answers
61
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Representation and Laplacian on the Heisenberg group
Let $\pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H=\mathbb C\times \mathbb R$. For $\varphi\in L^2(\mathbb R)$, we have
$$\pi_{\lambda} (z,t)\varphi(\xi)=e^{i\...
- 650