All Questions
2,633 questions
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When is the zero weight space of an irreducible $\frak{sl}_{n+1}$-module non-trivial?
Take the complex semisimple Lie algebra $\frak{sl}_{n+1}$, with space of dominant integral weights $P(\frak{sl}_{n+1})$. For $V(\lambda)$ the irreducible representation corresponding to $\lambda \in P(...
- 969
2
votes
0
answers
387
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The Weyl dimension formula for fundamental weights
The Weyl dimension formula is an equation to calculate the dimension of a simple $\frak{g}$-module $V_{\lambda}$, of highest weight $\lambda$, for $\frak{g}$ a complex semisimple Lie algebra. ...
- 969
3
votes
0
answers
293
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The geometry of the group of automorphisms of a manifold
Given a manifold $M$, the group $Aut(M)$ is made of diffeomorphisms $M\to M$. Since the complete vector fields on $M$ form an infinite dimensional Lie algebra, and each generates a 1 dimensional Lie ...
- 111
2
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0
answers
102
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Classification problem of equivariant vector bundles on affine space
We work over $\mathbb{C}$. Let $\mathfrak{k}$ be a Lie algebra, and let $V,W$ be finite-dimensional $\mathfrak{k}$-modules. Consider the $\mathfrak{k}$-algebra $A:=S^\bullet V$. Write $\mathfrak{m}$...
- 44.7k
5
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1
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792
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Constructing a Kac-Moody group as a quotient of the free product of its root subgroups
The paper "Regular Functions on Certain Infinite-dimensional Groups" by Kac and Peterson describes the construction of a group associated to the datum of a Kac-Moody algebra in a way I haven'...
- 63.9k
2
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0
answers
117
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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 ...
20
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2
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5k
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How to interpret the Sugawara construction from a physical or mathematical viewpoint?
In theoretical physics, the Sugawara theory is a set of formulae and theorems that allow one to construct a stress-energy tensor of a specific type of conformal field theory from a bilinear expression ...
- 4,713
1
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0
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195
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Gross-Hopkins duality
$\DeclareMathOperator\Spf{Spf}\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\Mod{Mod}$One can deduce the invertibility of the Gross-Hopkins dualizing spectrum from purely algebro-geometric ...
- 63.9k
20
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5
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4k
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Equivalent statements of the Riemann hypothesis in the Weil conjectures
In the cohomological incarnation, the Riemann hypothesis part of the Weil conjectures for a smooth proper scheme of finite type over a finite field with $q$ elements says that: the eigenvalues of ...
- 4,713
2
votes
1
answer
361
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Lie algebroid in algebraic geometry
When I did net-surfing at home, I met some geometric backgrounds of Lie algebras and encountered the concept of Lie algebroids. In differential geometry, a Lie algebroid seems to be defined as ...
- 4,008
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2
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670
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Conjugacy classes in the automorphism group of a simple Lie algebra
A lower bound of the number of conjugacy classes in the automorphism group of a simple Lie algebra $\mathfrak{s}$, of finite dimension over an arbitrary field $\mathbb{F}$, can be the size of the ...
3
votes
0
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92
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Closedness of subgroup corresponding to semi-simple real Lie subalgebra
I have an impression (but could be wrong) that I heard that for any semi-simple (real) Lie subalgebra $\mathfrak{k}$ of $\mathfrak{gl}(n,\mathbb{R})$ there exists a connected closed Lie subgroup $K\...
- 63.9k
9
votes
1
answer
356
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Reference request for indecomposable representations of $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$
Is there a good reference on the classification of indecomposable representations of the Lie algebra $\mathfrak{sl}(2)$ over an algebraically closed field of characteristic $p > 0$ (of course, in ...
CommunityBot
- 1
2
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0
answers
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
9
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1
answer
3k
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Lie algebra semisimple if and only if perfect?
If $L$ is a semisimple lie algebra then $L=[L,L]$. Is the opposite true?
- 63.9k
2
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1
answer
118
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The universal envelope U(L) is a PI-algebra iff L is abelian
Let $k$ be a field with $\operatorname{char} k = 0$. Let $L$ be a Lie $k$-algebra. Then the universal envelope $U(L)$ is a PI-algebra iff $L$ is abelian.
Remark:
PI-algebra means polynomial identity ...
- 1,284
5
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2
answers
407
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What is Pressley and Segal's "basic inner product" for compact simple Lie algebras of types B and C?
In Pressley and Segal's book Loop Groups, they define a "basic inner product" $\langle-,-\rangle$ on a simple Lie algebra to be (minus) the Killing form scaled so that $\langle h_\alpha,h_\...
- 35.5k
2
votes
1
answer
293
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Primitive elements in the universal enveloping algebra of Lie superalgebra
Let $\mathfrak{g}$ be a Lie superalgebra over $\mathbb{C}$. Denote by $U(\mathfrak{g})$ the universal enveloping algebra of $\mathfrak{g}$. We know that there is a natural super Hopf algebra structure ...
- 7,746
21
votes
3
answers
2k
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Invariants for the exceptional complex simple Lie algebra $F_4$
This is an edited version of the original question taking into account the comments below by Bruce. The original formulation was imprecise.
Let $\mathfrak{g}$ denote a complex simple Lie algebra of ...
- 255
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0
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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
votes
1
answer
90
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Non-isomorphic direct products of a solvable and a semisimple Lie algebra
Given a solvable Lie algebra $\frak{a}$ and a semisimple Lie algebra $\frak{g}$ we can take their semidirect product $\frak{a} \rtimes \frak{g}$, with respect to a Lie algebra map $\frak{g} \to \...
- 63.9k
2
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0
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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 $\...
- 13k
6
votes
1
answer
141
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Does the centralizer of a regular element in a semisimple Lie algebra act by polynomials?
Let $\mathfrak g$ be a semisimple Lie algebra over $\mathbb C$, $\rho : \mathfrak g \to \operatorname{End}(V)$ a finite-dimensional irreducible representation and $x \in \mathfrak g$ regular with ...
- 348
1
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0
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133
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Irreducible unitary representations of $\mathrm{SL}(n,\mathbb R)$ from those of $\mathrm{GL}(n,\mathbb R)$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$In the case of a non-Archimedean local field $\mathbb F$, one may reduce the representation theory of $\SL(n,\mathbb F)$ to that of $\GL(n,\...
- 63.9k
0
votes
0
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245
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Decomposition of tensor product of two representations of $so(10,\mathbb{C})$
There exist two 16-dimensional irreducible non-isomorphic representations of $so(10,\mathbb{C})$. Consider the tensor products of each of them with the standard (10-dimensional) representation.
What ...
- 21.8k
1
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1
answer
229
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10-dimensional irreducible representations of $so(10,\mathbb{C})$
Are there 10-dimensional irreducible representations of the Lie algebra $so(10,\mathbb{C})$ which are not isomorphic to the standard representation?
- 32.5k
2
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3
answers
367
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Invariant subbundles of tangent bundle of flag variety
Suppose that $G$ is a complex semisimple Lie group, $P$ a parabolic subgroup of $G$. What are all of the $P$-invariant subspaces of $\mathfrak{g}/\mathfrak{p}$? In various low dimensional examples, I ...
- 26.3k
4
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1
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189
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Equivalence of categories of modules over $U(\mathfrak{g})$ on which $Z(\mathfrak{g})$ acts by central characters
$\newcommand{\g}{\mathfrak{g}}$Setting: $\mathfrak{g}$ is a semisimple complex Lie algebra. Here $\chi_\lambda$ denotes the central character corresponding to the action of $Z(\g)$ on a highest weight ...
- 311
10
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2
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2k
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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 ...
- 35.5k
1
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0
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125
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Is there a classifying space for transitive Lie algebroids? If so, what is it?
Let $M$ be a manifold. The data of a Lie groupoid over $M$ is equivalent to the data of a singular foliation $M=\sqcup\mathcal{F}_i$ and, for each $i$, a map (mod homotopy) $f_i:F_i\to BG_i$ (where $...
- 1,969
11
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2
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1k
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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 ...
- 63.9k
5
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3
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781
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Computation of restricted Lie algebra (co)homology
My question is the following:
Is there a small complex, perhaps analogous to the Chevalley-Eilenberg complex, computing the (co)homology of a restricted Lie algebra over a field of characteristic $...
- 35.5k
2
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1
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499
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A canonical G_m (or G) action on the Slodowy slice
Question
By Slodowy slice I mean a transverse slice at a subregular nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ (in particular I am not intersecting with the nilpotent cone). Consider the ...
- 35.5k
5
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2
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849
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Stabilizers for nilpotent adjoint orbits of semisimple groups
Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$. Suppose that $X\in\frak{g}$ is a nilpotent element (i.e. that $ad_X:\frak{g}\rightarrow\frak{g}$ is ...
- 63.9k
5
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0
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164
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Generalized commutator
A well-known generalization of the commutator for operators is the so-called q-commutator defined as
$$[A,B]_q=AB-qBA.$$
I was wondering if the case where $q$ is not a number but other operator has ...
2
votes
0
answers
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
5
votes
3
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495
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Good even grading and principal Levi type
Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e$ be a nilpotent element in it. In the theory of finite W-algebras one often encounters the following two conditions:
1) $e$ is ...
- 2,821
3
votes
1
answer
459
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Frobenius functor and length of local cohomology
Let $(R,\mathfrak{m})$ be a Noetherian local ring of positive prime characteristic $p$ and let $F$ be the Frobenius functor. Write $d$ for dimension of $R$. Assume that for some $0\leq i< d $ the ...
- 2,821
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1
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273
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'Lie correspondence' for formal power series in non-commuting indeterminates
This is related to an earlier question of mine. I would like an argument or a reference (or a missing hypothesis if needed) for the following.
Let $\mathbb{F}\langle\langle \alpha\rangle\rangle$ and ...
- 2,388
2
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0
answers
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
4
votes
0
answers
215
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Reference request: Radicial morphisms & Jacobson-Bourbaki correspondence
My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two ...
- 485
2
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1
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690
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Restricted universal enveloping algebra of Abelian p-Lie algebra
Question: Let $p$ be a prime. Let $k$ be a commutative ring such that $p=0$ in $k$.
Let $\mathfrak g$ be an abelian $p$-restricted Lie algebra over $k$. In other words, let $\mathfrak g$ be a $k$-...
- 4,492
1
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1
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202
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The identity connected component of centralizers of unipotent orbits
This is, in a way, a follow up question to Unipotent orbits and intersection with Levi and pseudo-Levi subgroups.
I was reading "A generalisation of the Bala–Carter theorem for nilpotent orbits&...
- 1,336
2
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0
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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
5
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1
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143
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PBW basis for the quantized enveloping Lie algebra of $\mathfrak{g}_2$
I would like to know if you have any reference where I can find the canonical PBW basis for $U_q(\mathfrak{g}_2),$ computed using the action of the braid group as defined by Luzstig.
Alternatively I ...
- 8,866
4
votes
1
answer
222
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Spectral sequence from standard/Verma filtration/flag to compute Lie algebra cohomology of tensor product with respect to $\mathfrak{n}$
I'm not sure this question fully qualifies as a research-level math question, but from my (limited) past experience on stackexchanged I feared this question might not get an answer there.
Setting: the ...
- 492
24
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5
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2k
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Lie groups vs Lie monoids
Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie groups, but with the axiomatic existence ...
- 2,369
17
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4
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9k
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Noether's theorem in quantum mechanics
In classical mechanics:
If a Lagrangian $\mathcal{L}$ is preserved by an infinitesimal change in the state space variables $q_i \to q_i + \varepsilon K_i(q)$, this leads to only second order change in ...
- 71
2
votes
3
answers
765
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Computing the index of a Lie algebra: what is known beyond the reductive case?
Recall that an index of a Lie algebra $\mathfrak{g}$ is $\mathrm{ind}\ \mathfrak{g} := \min\limits_{\xi \in \mathfrak{g}^*} \dim \mathrm{Ann}_{\xi}$ where $\mathrm{Ann}_{\xi}=\{h\in\mathfrak{g}| \...
- 35.5k
2
votes
1
answer
141
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An upper bound on the dimension of a subalgebra of $\mathfrak{so}(p,q)$ with non-trivial centre
Let $\mathfrak{so}(p,q)$ be the real definite/indefinite orthogonal Lie algebra, $p,q\ge0$, $p+q=n\in\mathbb{N}$, and $L\subset\mathfrak{so}(p,q)$ a Lie subalgebra with non-trivial centre, $\mathrm{Z}(...
- 63.9k