All Questions
931 questions with no upvoted or accepted answers
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What is the intersection of all maximal abelian subalgebras of a compact simple Lie algebra, containing a given element?
Let $\mathfrak g$ be a compact simple real Lie algebra and let $x\in\mathfrak g$.
What is the intersection of all maximal abelian subalgebras of $\mathfrak g$ which contain $x$?
For instance, in the ...
2
votes
0
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97
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Non-noetherian Cartier Isomorphism
A result in positive characteristic is that if $R/\mathbb{F}_p$ is a smooth ring, then we have a Cartier isomorphism
$$\Omega_{R}^\bullet\cong H^\bullet(\Omega_R^\bullet)$$
which is essentially ...
- 4,428
2
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145
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Solvability and nilpotency for Banach algebras
Do we have topological counterparts of solvability and nilpotency, which are central concepts for (finite-dimensional) Lie algebras, for infinite dimensional Banach algebras with the commutator ...
- 2,605
2
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135
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Fusion rules for the Lie algebra $\frak{so}_{2n+1}$
For the Lie algebra $\mathfrak{so}_{2n+1}$ where can I find a description of the fusion rules of it fundamental representations? In more detail: For $\pi_i$ and $\pi_j$ two fundamental weights of $\...
- 393
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200
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Element conjugate to a maximal torus
It is well known that any element in a compact connected Lie group is conjugate to an element in a maximal torus. Let G be a Lie group that is not necessarily compact and/or connected. Let $x\in G$ ...
- 59
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236
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Question about adjoint orbits
I am looking for a proof or a reference of the following claim:
Let $G$ be a real connected semi-simple Lie group and let $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ be the Cartan decomposition for its ...
- 139
2
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108
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Invariants of Lie superalgebras
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 ...
- 673
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72
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Dimensions of centralizers in quantum Lie algebras associated to $\mathfrak{sl}_n$
Following ideas of Woronowicz, Lyubaschenko and Sudbery defined in Quantum Lie algebras of type $A_n$ the notion of a quantum Lie algebra $\mathfrak{sl}_n$. Let me focus on the case where $q$ is not a ...
- 7,320
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677
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Derivative of the projection map $\pi: G \times \mathfrak{g} \rightarrow G \times_K \mathfrak{g}$
Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}.$ Let $K$ be a closed subgroup of $G$ with Lie algebra $\mathfrak{k}.$ We define the manifold $$\mathcal{E}:= G \times_K \mathfrak{g}$$
to ...
- 139
2
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69
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Abelian category for $(\mathfrak{g},T)$ modules with nontrival Grothendieck group
Let $G$ be a reductive Lie group over $\mathbb{C}$, and write $\mathfrak{g}$ for its Lie algebra. Let $T\subseteq B\subseteq G$ be a maximal torus and Borel subgroup, where $\operatorname{Lie}B=\...
- 1,077
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171
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Monogenic function fields
Recall that a number field $K$ is said to be monogenic if its ring of integers is of the form $\mathbb{Z}[\alpha]$, or equivalently, if it has a power integral basis. There are many references one can ...
- 103
2
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102
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Submodules of direct sums of Verma modules for the Virasoro algebra
Let $M_1=M(c, h_1)$, $M_2 = M(c, h_2)$ be two Verma modules for the Virasoro algebra. Let $N$ be a submodule of the direct sum $M_1 \oplus M_2$. I am wondering if there is any classification results ...
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What is the "correct" notion of a perfect graded commutative algebra?
My question is rather simplistic. While trying to dualize some statements about rational homotopy algebra of a space I got stuck with the following problem.
We have a notion of a perfect Lie algebra ...
- 1,374
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287
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Frobenius endomorphism is not flat
I am actually going through "Twenty four hours of local cohomology". They discuss the Frobenius endomorphism in Chapter 21. Here is the Exercise 21.6 which I am finding hard to solve:
Find a ...
- 183
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213
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Positive roots and the longest element of the Weyl group
Take $\frak{g}$ a complex semisimple Lie algebra and its Weyl group $W$. Is it true that the number of positive roots of $\frak{g}$ is equal to the length of the longest element of $W$?
- 393
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78
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Examples of curvature-adapted subgroups of semi-Riemannian groups
Let $G$ be a semi-Riemannian group, i.e., a Lie group equipped with a bi-invariant semi-Riemannian metric. I am looking for examples of Lie subgroups that are curvature adapted to $G$.
First, allow me ...
- 985
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115
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Lie derivations of algebra of smooth functions in a symplectic manifold
Let $(M,\omega)$ be a finite-dimensional symplectic manifold. The algebra $C^\infty(M)$ of smooth functions is a Poisson algebra. Derivations $D : C^\infty(M) \to C^\infty(M)$ of the Poisson ...
- 33.3k
2
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95
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Local decomposition of semisimple Lie groups at the identity into the product of centralizer, unstable and stable horospherical subgroups
First let us look at an example. Let $G=\text{SL}(d,\mathbb R)$ and $A$ be its subgroup consisting of diagonal matrices with positive entries.
Let $a\in A$ be an element. We define the stable ...
- 1,565
2
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561
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What is the precise definition of connect semisimple Lie groups "without compact factors" in the literature?
I frequently see in the literature (of homogeneous dynamics, in particular) of the notion "semisimple Lie groups without compact factors" without seeing its precise definition.
I have three (...
- 1,565
2
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165
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Automorphism group of a Lie group $G$ vs that of a covering group $\tilde G$: same or not?
Is it true or false that the Inner (Inn), Outer (Out) and Total (Aut) Automorphism of a Lie group $G$ is the same as the covering group of the Lie group, say $\tilde G$ (regardless of how many types ...
- 10.5k
2
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146
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Koszul differential of the complex $\bigwedge \mathfrak{g}^*$
Let $\mathfrak{g}$ be a finite dimensional Lie algebra. The definition of the Koszul differential is given in the article by Kumar and Vergne on equivariant cohomology page 133 as follow:
Let us now ...
- 139
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55
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Function annihilated by ideal in universal envelope
Let $G$ be a Lie group, $U(G)$ its universal enveloping algebra over $\mathbb C$ and let $J\ne U(G)$ be a left ideal. We consider $U(G)$ as the algebra of left-invariant differential operators on $G$.
...
user130903
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408
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What is a "Lefschetz SL2"?
In the paper "On Minuscule Representations and the Principal SL2" by B.H. Gross (link: here) and some others the terminology "Lefschetz $\operatorname{SL}_2$" is used. I think I am ...
- 2,821
2
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230
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Reference request: the UEA of the LR-algebra of tangent vector fields on a smooth manifold coincides with the derivation ring and the ring of diff ops
Let $\mathcal{M}$ be a smooth real manifold and let $A:= \mathcal{C}\left(\mathcal{M}\right)$ be the real algebra of smooth functions on $\mathcal{M}$.
Recall from McConnell, Robson, Noncommutative ...
- 718
2
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90
views
Lie Groups and Lie algebras related to Jordan algebras
Let $J$ be a Jordan algebra. I knew three relative Lie groups/Lie algebras to $J$.
In the paper "The Capelli Identity, Tube Domains, and the Generalized Laplace Transform"
Jacobson [J] has ...
- 719
2
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answers
96
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Primitive elements in Hopf algebras over the integers
Let $H$ be a Hopf algebra over $\mathbb Z$, and assume that $H$ is cocommutative, graded, generated in degree $1$, and connected (its degree-$0$ part is $\mathbb Z$).
Are there nice, natural ...
- 2,519
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156
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Spherical harmonics, $\frak{sl}_2$, and algebra gradings
Let $S^2$ be the usual $2$-sphere considered as the quotient $S^3/S^1$, and denote by $\operatorname{Pol}(S^2)$ the algebra of polynomial functions on $S^2$. We can decompose $\operatorname{Pol}(S^2)$ ...
- 141
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75
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Explicit example of prime ideal not an intersection of maximal ideals, in universal enveloping algebra
Let $A$ be a $\mathbb k$-algebra. If $A$ is affine commutative, by Nullstellensatz, then every prime ideal of $A$ is an intersection of maximal ideals.
To justify the notion of being primitive in ...
- 485
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81
views
Jordan decomposition for simple Lie algebra
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let us fix a Cartan, a Borel, and generators $x_\alpha$ of negative simple roots. Then $N:=\sum x_\alpha$
is a principal (=regular) nilpotent ...
- 2,751
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155
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Commutators and brackets in nilpotent Lie algebras
Let $\mathfrak{g}$ be a finite-dimensional nilpotent Lie algebra over an algebraically closed field $k$ of characteristic zero. Throughout, let $x,y$ and $z$ be elements of $\mathfrak{g}$. The Baker-...
- 533
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85
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Elements of the Hall basis described via permutations
Good morning,
Suppose that $\mathfrak{g}$ is a free graded Lie algebra generated by the elements $1,\dots, n$, i.e. assume that $\mathfrak{g}_1=\mathrm{span}\{1,\dots, n\}$. Let us focus on the Hall ...
- 277
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76
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Equivalence classes of Hermitian elements in a central simple algebra with an involution of second kind
Let $k_0$ be a field of characteristic 0, $k/k_0$ be a quadratic extension,
and $A/k$ be a central simple algebra over $k$ of dimension $9=3^2$ with an involution of second kind $\sigma$.
Then $^\...
- 14.2k
2
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0
answers
81
views
The centralizer and normalizer of products of (SU(n) $\times$ SU(p) $\times$ …) in U(m)
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}
$Consider the special unitary group $\SU(n)$ and the unitary group $\U(m)$.
Below I specify a specfic way to embed $...
- 10.5k
2
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answers
60
views
Is there always a purely real representative for a metrized solvable Lie group?
Alekseevski proves for Heintze groups (a special class of solvable Lie groups) that any such group admits a (left-invariant) metric which is isometric to a purely real Heintze group (again equipped ...
- 21
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230
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Lie bracket on the unshifted tangent complex?
My problem is as follows: if $X$ is a derived scheme, or derived stack, or any kind of a space where tangent complex makes sense, I guess there should be a lie bracket on its tangent complex, ...
- 121
2
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108
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A tri-grading on the de Rham complex of a Lie group?
The tangent space of a (compact) Lie group $G$ is given by its Lie algebra. Assuming for convenience that $G$ is connected, its Lie algebra $\frak{g}$ decomposes into a Cartan part, as well as ...
- 203
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Simple modules for universal enveloping algebras and Weyl algebras
Let $A$ be the universal enveloping algebra of a fintie dimensional Lie algebra (simple if needed) or the Weyl algebra.
Question: Are there recent survey articles about the (possibly infinite ...
- 26.5k
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102
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lie algebra bundle and underlying vector bundle
Let $G$ be a connected reductive group over a field $k$. Let $E$ be a $G$-bundle, then we can form the adjoint bundle $ad(E)$ which is a Lie algebra bundle over $k$.
As a vector bundle it is trivial, ...
- 3,472
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291
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Automorphisms group of complex and real simple Lie algebras
$\DeclareMathOperator{\Inn}{\operatorname{Inn}}\DeclareMathOperator{\Aut}{\operatorname{Aut}}\DeclareMathOperator{\Out}{\operatorname{Out}}\DeclareMathOperator{\g}{\mathfrak{g}}$According to Wikipedia,...
- 2,147
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61
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4-dimensional Lie subalgebras of $o(4,\mathbb{C})$ and of $o(3,\mathbb{C})\ltimes\mathbb{C}^3$
I would like to know all complex 4-dimensional Lie subalgebras of $o(4,\mathbb{C})$ and of $o(3,\mathbb{C})\ltimes\mathbb{C}^3$.
- 427
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answers
55
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Free resolutions of universal enveloping algebras for simple, finite dimensional Lie algebras
I'm currently studying Anick's resolution on the context of universal enveloping algebras for certain Lie algebras, namely some of the smallest cases: $A_1,A_2,A_3,B_2,G_2$, and so on.
What are some ...
2
votes
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answers
175
views
Jordan normal form in a reductive group
Let $G$ be a connected complex reductive group. Given an element $g \in G$, there is a canonical decomposition $g = s u$ such that $s$ is semisimple, $u$ is unipotent and $s$ and $u$ commute. ...
- 491
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100
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Composing equal characteristic and mixed characteristic deformations
Suppose $k$ is an algebraically closed field of characteristic $p>0$ and $A$ is a $k$-point on a moduli space of certain objects over $k$. Suppose, moreover, that $A$ deforms to a $k[[t]]$-point $B$...
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Proof of parametrisation of $\hat{\mathfrak{g}}$-intertwiners of induced modules of affine Lie algebras
I am not 100% sure that this question belongs here, since I think the topic does but the specific problem I have might not. Feel free to tell me so if this is the case. It concerns the proof of ...
- 223
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175
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Question on the center of universal enveloping algebra
Let $G$ be a unitary group $U(n)$ and $G’$ is a subgroup $U(m)$. (i.e. $m\le n$)
Let $\mathfrak{g},\mathfrak{g}'$ be the complexified Lie algebra of $G,G’$ and $\mathfrak{z},\mathfrak{z}’$ be the ...
- 1,759
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38
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Classification of involutions on $G_{2}$-homogeneous spaces
Are you aware of a systematic classification of involutions on $G_{2}$-homogeneous spaces?
- 2,630
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578
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Lyndon basis of free Lie algebras
Let $A = \{a,b,c,d\}$ be a set of totally ordered alphabets, a Lyndon word over $A$ is a word $w$ in $A^*$ such that if $w=uv$ is a factorization of $w$ into non-empty subwords, then $u<v$ in ...
- 1,269
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155
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which operators are "really truly positive"?
Let's say that an operator G on a Hilbert space $\mathcal{H}$ is "really truly positive" iff
$\Vert\exp(-tG) \exp(-tG^*)\Vert_{op}<1$ for all $t>0$
How can we characterize the set of operators ...
- 121
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126
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Imaginary roots in $\widetilde{E}_8$
Consider the root system of a Kac-Moody algebra. Denote by $\alpha_i$ the simple root associated with node $i$
by for $i \in \{1, \ldots, n-1\}$ and by $\beta$ the simple root associated with $n$.
...
- 6,211
2
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0
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109
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Nilpotent vs unstable elements in $U(\mathfrak{sl}_2)$
$\def\sl{\mathfrak{sl}}\DeclareMathOperator\ad{ad}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\End{End}\DeclareMathOperator\PSL{PSL}$We consider the Lie algebra $\sl_2$ with its usual basis $\{ e,...
