All Questions
Tagged with lie-groups lie-algebras
816 questions
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How can we construct a non-trivial central extension of a Lie group
Let $G$ be a connected and simply connected Lie group with its Lie algebra $\mathfrak{g}$. Assume that $[c]\in H^2 (\mathfrak{g};\mathbb{R})$ is a non-trivial 2-cocycle. Then we can construct a non-...
- 287
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1
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255
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Derivatives and ODEs on Lie groups
I'll keep the question specific to the scenario I'm working with, which is the Lie group SO(3) and its Lie algebra so(3).
Consider two rotation matrices $R_0$ and $R_1$, and define $R_t = \exp_{R_1} ((...
- 123
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Where do root systems arise in mathematics?
One often hears that root systems are ubiquitous in mathematics and physics. The most obvious occurrence of root systems is in the classification of complex simple Lie algebras. Where else do they ...
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3
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Does every nilpotent lie in the span of simple root vectors?
Let $G$ be a reductive group (for simplicity, we can work over $\mathbb{C}$). Let $N \in \operatorname{Lie} G$ be nilpotent. Does there exist a Borel pair $(B,T)$ of $G$ such that $N$ lies in the span ...
- 953
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79
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Extension of a type A Springer fibre
Given a decomposition $p=(p_1,\dots,p_n)$ of $n$, one can associate its corresponding
partial flag variety $$\mathcal{B}_p=\{F=(0=F_0\subset F_1\subset \dots \subset F_n=\mathbb{C}^n) \mid \dim F_i/F_{...
- 1,677
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Does the first fundamental representation of $\frak{sp}_n$ generates all the other fundamental representations
Let $\mathfrak{sp_n}$ be the symplectic Lie algebra, that is, the $C_n$ complex simple Lie algebra. Is it true that the first fundamental, which is to say the vector space, representation $V_1$ of $\...
- 417
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165
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Trivial representation of a maximal torus
Let G be a connected compact Lie group and $T\subset G$ a maximal torus. For an irreducible representation $V_\lambda$ of $G$, the multiplicity of the trivial representation of $T$ in $V_\lambda$ is ...
- 128
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221
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Does every Lie algebra appear as centralizer of an element in a semisimple Lie algebra?
Given a finite dimensional, complex, semisimple (fcss) Lie algebra $\mathfrak{g}$ and an element $x\in\mathfrak{g}$, denote by $\mathfrak{g}^x$ the centralizer of $x$ in $\mathfrak{g}$ i.e. the set $\{...
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Which Lie groups admit finite generation by a set of Lie algebra elements? And what are some known choices of generators which realize this?
Consider a (finite-dimensional) real connected Lie group $G$ with Lie algebra $\frak{g}$. Take a generating set $\mathcal{G} = \{ X_1, \cdots X_n \} $ of $\frak{g}$, i.e. such that any element of $\...
- 143
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Is the adjoint action of $\mathrm{SU}(2)$ on the Schwartz space proper and free?
$\DeclareMathOperator\SU{SU}$Let $t_1, t_2, t_3$ be generators of the Lie algebra $\mathfrak{su}(2)$.
Let us consider a Schwartz space $\mathcal{S}$ defined as
\begin{equation}
\mathcal{S}:= \Bigl\{ \...
- 3,477
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94
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Techniques for computing integrals on $G/K$
Edit: I originally tried writing a simpler question, but to make more clear what I want I wrote the general question below
Let $G$ be a compact group with compact subgroup $K\subseteq G$ (everything ...
- 1,077
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255
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Any "inherent" definition of $\mathrm{SU}(2)$ independent of any matrix representation?
$\DeclareMathOperator\SU{SU}\SU(2)$ is explained in detail here.
However, if I know right, this definition itself is known the "fundamental representation".
I wonder if there is any "...
- 3,477
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1
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Symmetric tensor of highest weight modules for $\mathrm{SU}(2)$
Let $V_i$ be the $(i+1)$-dimensional representation of the special unitary group $\mathrm{SU}(2)$ with the highest weight $i$. Is there any uniform way to compute the irreducible decomposition for the ...
- 951
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If two Lie algebras are isomorphic, under which conditions will their Lie groups also be isomorphic?
Let $G$ and $G'$ be compact connected Lie groups (which are not necessarily simply connected) with Lie algebra $\mathfrak{g}$ and $\mathfrak{h}$. Suppose that the two Lie algebras are isomorphic, ...
- 97
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Conjugacy of Cartan subgroups in $\mathrm{GL}(n)$
$\DeclareMathOperator\SL{SL}$I have probably a very basic question on the structure of semisimple Lie groups. Sorry if it is too elementary.
Let either $G=\SL(n,\mathbb{R})$ or $G=\SL(n,\mathbb{C})$. ...
- 21.8k
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205
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What's an example of an $n$-step nilpotent Lie algebra whose centre is not $g^{(n)}$?
Let $\mathfrak g$ be a Lie algebra. $\mathfrak g^{(1)}=\mathfrak g$ and $\mathfrak g^{(n+1)}=[\mathfrak g,\mathfrak g^{(n)}]=\mathbb R$-span$\{[X,Y]:X\in\mathfrak g,Y\in\mathfrak g^{(n)}\}$. The ...
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Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?
In this popular 2019 MO question, user მამუკა ჯიბლაძე asked:
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism between ...
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What's the point of geometric representation theory?
Please forgive the provocative title, what I mean is the following:
One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...
- 157
2
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1
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107
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Finite dimensional irreducible representations of $\frak{sl}_m$ with non-trivial zero weight spaces
For the special linear algebra $\frak{sl}_{m}$ which finite dimensional irreducible representations $V_{\mu}$ have non-trivial zero weight spaces?
For $\frak{sl}_2$ this is clear: $V_{2k\pi}$ for $\pi$...
- 157
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Generators of simple Lie groups and finite word length
Let $G$ be a connected simple Lie group with finite center. Let $a=\mathrm{exp}(X)$ be a semisimple element. Then we can decompose the lie algebra of $G$ into the direct sum of the eigenspaces of $\...
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Relation between real forms of Lie algebras and root systems on pseudoeuclidean vector spaces
This might be trivial but I cannot see it clearly.
Simple complex Lie algebras are fully classified by the root systems arising from the Cartan subalgebra for which the Euclidean norm is the Cartan-...
- 295
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2
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976
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The adjoint representation of a Lie group
Let $G$ be a Lie group and $\text{Ad}(G)$ denote its adjoint representation i.e. the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$. The adjoint representation is a real $G$-...
- 131
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1
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Heat kernel of left-invariant metric on 3-sphere
This post concerns the heat equation on $S^3 (\simeq \mathrm{SU}(2))$ endowed with an arbitrary left-invariant metric. We think of $S^3$ as the space of unit quaternions, and its Lie algebra $\...
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210
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Centers of universal enveloping algebra of complex Lie algebras
Let $\mathfrak{g}$ and $\mathfrak{g'}$ be complex Lie algebras such that $\mathfrak{g}$ is a subalgebra of $\mathfrak{g'}$. Let $Z(\mathfrak{g})$ and $Z(\mathfrak{g'})$ be the centers of the universal ...
- 833
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165
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Rotation number for homeomorphisms of a Lie group other than $S^1$
Let $G$ be a Lie group whose Lie algebra is $\mathfrak{g}$ with exponential map $\exp:\mathfrak{g}\to G$.
For what kind of Lie group $G$ the standard process of definition of rotation number ...
- 356
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433
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Do the exceptional root systems arise in the real world?
I am looking for a list of real world examples where the exceptional roots systems $E_6, E_7, E_8, F_4$, and $G_2$, and their associated Lie algebras and Lie groups, arise. To make this question a ...
2
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1
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159
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Adjoint action on orthogonal complement
Consider a compact Lie algebra $\mathfrak{g} \subset \mathfrak{u}(n)$ and its associated connected, compact Lie group $G$. Let $\mathfrak{g}^{\perp}$ denote $\mathfrak{g}$'s orthogonal complement (as ...
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180
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A filtration on Drinfeld-Jimbo quantum enveloping algebras
For the universal enveloping algebra $U(\frak{g})$ of a Lie algebra $\frak{g}$, one can define in a natural way an increasing $\mathbb{N}_{0}$-filtration. By the Poincaré-Birkhoff–Witt theorem, the ...
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110
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Questions on the differential of the Lie logarithm
Let $G$ be a Lie group. Recall the Lie logarithm is well-defined about a neighborhood $U \subset G$ of the identity: $\log:U\to \mathfrak{g}$. I am dealing with a research problem that concerns the ...
- 518
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105
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Embeddings of symplectic group into the orthogonal group
Let $\mathfrak{sp}$ denote the complex symplectic Lie algebra and $\mathfrak{so}$ the complex orthogonal one. Do we have an embedding
$$
\mathfrak{sp}_{2n-2} \hookrightarrow \mathfrak{so}_{2n}?
$$
In ...
- 2,751
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1
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243
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Adjoint action on the universal enveloping algebra and the PBW theorem
Let $\frak{g}$ be a semisimple Lie algebra and $U(\frak{g})$ its universal enveloping algebra. The adjoint action of $\frak{g}$ on itself extends to an action of $\frak{g}$ on $U(\frak{g})$. How does ...
- 157
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Minimal $K$-orbit on $\mathfrak{g}$
Let $\mathfrak{g}_0$ be a noncompact simple Lie algebra with Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. Write $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the ...
- 951
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2
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326
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Lie's third theorem via graded geometry
Lie's third theorem : Given any finite dimensional Lie algebra $\mathfrak{g}$, there exists a Lie group $G$ whose Lie algebra is equal to $\mathfrak{g}$.
In one of the talks, speaker mentions that ...
- 6,023
4
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92
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Lie bracket of general unipotent matrices
Let $k$ be a field (of characteristic $0$). Let
$$
X:=\begin{pmatrix}1&x_{1,2}&x_{1,3}&\cdots&x_{1,n}\\ &1&x_{2,3}&\cdots&x_{2,n}\\ &&1&\cdots&x_{3,n}\\ ...
- 449
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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 ...
- 356
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393
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An alternative form of the Kazhdan-Lusztig conjecture
Fix a complex semisimple Lie algebra $\mathfrak{g}$. Denote by $W$ the corresponding Weyl group, with length function $\ell$ and Bruhat order $\leq$. Let $\lambda$ be an integral anti-dominant weight. ...
- 1,391
1
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1
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157
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Commuting time dependent vector fields and pullback invariance
Let $X_t, Y_t \in C^\infty(\mathbb{R}; \mathfrak{X}^\infty(M))$ be (smooth, or something else if it's necessary) time dependent vector fields.
Is there some analogue of the following fact in finite ...
- 193
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67
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Poisson bracket on $T^*T\mathrm{SU}(1,1)$
Consider the cotangent bundle of the tangent bundle $T^*TG$ of a Lie group $G$. Denote its the Lie algebra by $\mathfrak{g}$. By left translations, we have the trivialization $T^*G \cong G \times \...
- 829
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What do the Carnot groups act on?
My question is in some sense a less ambitious version of the following MO question where the answer was inconclusive.
A Carnot group of step $N$ can be identified within the tensor algebra, modulo ...
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What is the exponential map from the Lie algebra $\mathfrak{sl}(2,\mathbb{C})\ltimes_\textrm{ad}\mathfrak{sl}(2,\mathbb{C})$ to its Lie group?
$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\Exp{Exp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\sl{\mathfrak{sl}}$Let $G:=\SL(2, C) \ltimes_{\Ad} \SL(2,C)$, where $\...
0
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105
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Concrete examples of quantum duality principle
Let $G$ be a Poisson Lie group, $\mathfrak{g}$ be a Lie algebra of $G$, $G^*$ be a dual of $G$, $\mathscr{C}(G^*)$ be a Poisson algebra of $G^*$, and $U_h(\mathfrak{g})$ be a quantized universal ...
- 255
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71
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Integrating homomorphisms of Borel subalgebras
Let $G$ be a connected simple complex Lie group and $\mathfrak{g}$ be its Lie algebra. Let us fix a root decomposition, let $\mathfrak{b}_\pm$, $\mathfrak{n}_+$ and $\mathfrak{h}$ be the corresponding ...
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105
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Weyl group action on the Lie algebra [duplicate]
Let $W$ be the Weyl group of a complex semisimple Lie algebra $\frak{g}$. Certainly $W$ acts on the root system $R$ of $\frak{g}$ but can it be made to act on $\frak{g}$ or on the universal enveloping ...
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67
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A branching law involving 2-power exterior representations
Let $K=SU(n)$.
We take a maximal torus $T$ in $K$ and fixed a simple root system with fundamental weights $\eta_1,\dots,\eta_{n-1}$.
For $\mu$ a dominant weight of $K$, we denote by $(\tau_\mu,V_\mu)$ ...
- 2,446
3
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0
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139
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Root space inner products and the partial order on roots
For a root system $R$ and a choice of positive roots $R^+$ it is a standard fact (see, e.g., Bourbaki, "Lie Groups and Lie Algebras," Theorem 1 of Section 1.3 of Chapter VI) that
if $(\...
2
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0
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180
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Howe duality vs first fundamental theorem in invariant theory
I'm working on Howe duality, and R. Howe proved that the Howe duality of $\mathrm{GL}_n$ is equivalent to the first fundamental theorem (FFT) in invariant theory. So, Howe duality gives a ...
5
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0
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146
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Is every linear Lie group of bounded geometry?
$\newcommand\norm[1]{\lVert#1\rVert}$Given any point $p$ of a smooth Riemannian manifold $M$ there exists $r\in (0,\infty]$ such that the Riemannian exponential is a diffeomorphism in the geodesic ...
- 293
4
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0
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115
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Examples of non-equivariant momentum maps
What are examples of non-equivariant momentum maps?
Off the top of my hat, I know about the following examples:
the action of translations of a symplectic vector space (yielding the Heisenberg group ...
- 5,824
3
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1
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140
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Asymptotics of Haar moments on general Lie groups
I am trying to understand the asymptotics of Haar Moments on general compact Lie groups (in particular, subgroups of $\mathrm{SU}(n)$). I have learned that closed form formulae for these moments are ...
- 179
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2
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642
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Is it true $\left\|\log(RS)\right\|≤\left\|\log(R)+\log(S)\right\|$ for all $R,S \in \mathrm{SO}(3)$, where $\|\cdot\|$ is the Frobenius norm?
$\DeclareMathOperator\SO{SO}$I asked this initially in math stack exchange, but thought to ask it here since it is more advanced and related to my research topic. I study optimization on Lie groups ...
- 518