# Tagged Questions

**7**

votes

**1**answer

306 views

### Examples of Richardson orbit closures not having a symplectic resolution?

This is a follow-up to a recent question asked by Peter Crooks here. The answer by Ben Webster includes a helpful link to the corrected arXiv version of Baohua Fu's 2003 Invent. Math. paper ...

**8**

votes

**4**answers

487 views

### Computing $\int_0^T e^{itA}Be^{-itA} dt$ without an infinite series

I'm hoping to compute the following integral: $\int_0^T e^{itA}Be^{-itA} dt$ where $iA, iB$ are traceless anti-Hermitian matrices (i.e. $\mathfrak{su}(n)$). I have found the following form for the ...

**1**

vote

**0**answers

74 views

### Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions
$\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...

**2**

votes

**1**answer

198 views

### R-linear representations of sl(2,C)

Is there some good reference for the classification of finite-dimensional ${\mathbb R}$-linear (as opposed to ${\mathbb C}$-linear) representations of $\mathfrak{sl}_2{\mathbb C}$?
Equivalently, what ...

**1**

vote

**1**answer

101 views

### 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 ...

**2**

votes

**0**answers

308 views

### Solving $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ equation

Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$?
Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ ...

**3**

votes

**1**answer

255 views

### Special Riemannian connections?

Assume that $E$ is a bundle of Lie Algebras. Let $g$ be an invariant metric on $E$, that is for all $p\in M$, $$g_p([x,y],z)+g_p(y,[x,z])=0,$$ where $x,y,z\in E_p$ are arbitrary.
Is there a ...

**5**

votes

**3**answers

762 views

### How many three dimensional real Lie algebras are there?

The main point of the question is: I would like to know whether there are only finitely many, countable infinitely many or even uncountable many isomorphism classes of $3$-dimensional real lie ...

**0**

votes

**0**answers

46 views

### Special family of Metrics on Transitive Lie Algebroids?

Let $\rho:E\longrightarrow TM$ is a transitive Lie Algebroid, then $L=ker\rho$ is bundle of lie algebras. Suppose $\Gamma:TM\longrightarrow E$ be a linear splitting. Define
$$\nabla_X ...

**4**

votes

**0**answers

106 views

### When does finite presentability of the associated graded Lie algebra of a group imply the group is finitely presented?

Let $G$ be a finitely generated group; let $L(G)$ denote the graded Lie algebra (over $\mathbb{Q}$) associated to the lower central series of $G$. I would like to know conditions for when the finite ...

**3**

votes

**1**answer

108 views

### necessary and sufficient conditions for littlewood richardson coefficients to be non zero

Is there any necessary and sufficient conditions for $V(\tau)$ to be an irreducible component of the tensor product of two irreducible representations $V(\lambda)$ and $V(\mu)$ of a simple lie algebra ...

**1**

vote

**1**answer

90 views

### tensor product of two irreducibles having same maximal weight

Is there any explicit decomposition of tensor product of two finite dimensional irreducible modules of simple lie algebras whose highest weights are same?

**2**

votes

**0**answers

83 views

### Automorphisms of Nilmanifolds

Let $\mathfrak{g}$ be an n-dimensional, rational, nilpotent Lie algebra with simply connected that lie group $G$. It is stated in some papers that if $A$ is an automorphism of $\mathfrak{g}$ which is ...

**2**

votes

**1**answer

174 views

### “Quantum Littlewood-Richardson” Rule?

Let $\frak{g}$ be a complex semi-simple Lie algebra, and $\lambda,\mu \in P^+$ two positive dominant weights with corresponding irreducible representations $V(\lambda)$ and $V(\mu)$. The tensor ...

**4**

votes

**2**answers

267 views

### Lie's Theorem in characteristic $p$

Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ ...

**2**

votes

**2**answers

224 views

### Lie group about the quantum harmonic oscillator [closed]

We konw that in quantum harmonic oscillator $H=a^\dagger a$, $a^\dagger$, $a$, $1$ will span a Lie algebra, where $a, a^\dagger$ is annihilation and creation operator, $H$ is the Hamiltonian operator. ...

**0**

votes

**0**answers

78 views

### Explicit calculation of module of derivations on noncommutative polynomial ring

Let $R$ be a commutative unital associative ring and set $R<x,y>$ to be the $R$-algebra of non-commuting polynomials in two variables over $R$.
Explicitly how would one go about computing ...

**1**

vote

**0**answers

111 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 ...

**-2**

votes

**1**answer

109 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?

**1**

vote

**2**answers

205 views

### Are two distinct Weyl chambers always disjoint?

Let $G$ be a real semisimple Lie group; we suppose $G$ is connected and centerless. Let $\mathfrak{g}$ be its Lie algebra, $\mathfrak{a}$ a Cartan subspace of $\mathfrak{g}$ (i. e. a maximal abelian ...

**10**

votes

**1**answer

276 views

### Cohomological Proof of Serre Relations for a Symmetrizable Kac-Moody Algebra

In 'Infinite Dimensional Lie Algebras, 3rd edition', Kac mentions at the top of p. 170 that 'a simple cohomological proof of Theorem 9.11 was found by O. Mathieu (unpublished)'.
Does anyone know how ...

**1**

vote

**1**answer

215 views

### homogeneous algebras

Let $A$ be a finite dimensional algebra over finite field (not necessarily associative). Then $A$ is said to be homogeneous if $Aut(A)$ acts transitively on the one-dimensional subspace of A. If A is ...

**0**

votes

**1**answer

123 views

### Comparison of two infinite dimensional Lie Algebras

Is there an example of a real analytic (compact) manifold $M$ such that the following two lie algebras are isomorphic Lie algebras:
$\chi^{\infty}(M)$, the Lie algebra of all smooth vector ...

**1**

vote

**0**answers

50 views

### A canonical map Aut$_{\mathsf{Lie}_R}(\mathfrak{n} \rtimes_\pi \mathfrak{g}) \to$ Aut$_{\mathsf{Lie}_R}(\mathfrak{n})$

Let $\mathfrak{n}$, $\mathfrak{g} \in \mathsf{Lie}_R$ be two Lie algebras over a commutative ring $R$, s.t. $\mathfrak{g}$ acts on $\mathfrak{n}$ as a derivation: $\pi:\mathfrak{g} \to ...

**4**

votes

**2**answers

282 views

### Dimension of the nilpotent centralizer of a nilpotent matrix

Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}_n(k)$. For any partition of $n$, $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $A_{\lambda}$ be the ...

**-3**

votes

**1**answer

288 views

### Lie algebraic Grassmannian

Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given.
We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$.
For ...

**1**

vote

**0**answers

89 views

### Subgroups of $GL(n,\mathbb{R})$ which are $Aut(L)$ for some Lie structure [closed]

What is a sufficient condition for a lie subgroup $G$ of $GL(n,\mathbb{R})$ to be the automorphism group of a Lie structure on $\mathbb{R}^{n}$. In particular does $O(n)$ satisfies this property?

**3**

votes

**1**answer

153 views

### A Manifold for which $\chi^{\infty}(M)$ is rich

Is there a manifold $M$ for which $\chi^{\infty} (M)$, the lie algebra of smooth vector fields on $M$ contains all finite dimensional Lie algebras(Up to isomorphism)?
A weaker question:
Is there a ...

**1**

vote

**1**answer

158 views

### The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$

Let $G$ be a compact connected Lie group and $O_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^*$ and $\mathfrak{g}^*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let ...

**10**

votes

**2**answers

358 views

### Which nilpotent Lie algebras appear as nilradicals of parabolic subalgabras?

I am interested to identify (ideally classify) nilpotent Lie algebras that occur as nilradicals of parabolic subalgebras in (say) reductive Lie algebras.
For example, all Heisenberg Lie algebras ...

**5**

votes

**1**answer

352 views

### Root space decomposition

What is the root system for the special unitary lie algebra $\mathfrak{su}(p, q)$. Remind that these are matrices of the form
$\left(
\begin{array}{cc}
X & Y \\
\overline{Y}^t & Z ...

**3**

votes

**1**answer

113 views

### Reference to definition of matrix log with domain SO(3) which is Borel measurable

I was trying to set up an inverse to matrix exponential $\exp:\mathrm{Skew}(3\times 3)\to SO(3)$ that "covers" the biggest possible domain and is Borel measurable. I was wondering if there is a ...

**2**

votes

**0**answers

104 views

### (Graded) Lie algebras with “nice” irreps

"(The only Lie algebras for which) all finite-dimensional representations are completely reducible (are the semisimple Lie algebras)" (Chapman, here on MO). Evidently this can't no longer hold when ...

**2**

votes

**0**answers

106 views

### On Eigenvalues of the symmetric linear transformation related to a lie algebra's representation?

Let $\mathfrak{g}$ be a quadratic (finite dimensional) lie algebra and $\rho:\mathfrak{g}\rightarrow \mathfrak{gl}(W)$ be an anti-symmetric representation of $\mathfrak{g}$ on a finite dimensional ...

**1**

vote

**0**answers

148 views

### Replacing the Lie commutator with something else [closed]

Take a vector space $V={A,B,C,...}$ (of matrices), and the commutator $[A,B]=AB-BA$, then a Lie algebra of $V$ is characterized by $[V,V]$ staying in $V$. (Loosely speaking.)
What happens ...

**1**

vote

**1**answer

229 views

### labeling state vectors in representation space of a simple lie algebra

Given a simple lie algebra (over ${\mathbb C}$ or ${\mathbb R}$). What is the number
of operators such that their eigenvalues sufficiently label all state vectors in the algebra's representation ...

**2**

votes

**0**answers

458 views

### Mathematica package for Lie algebra computations?

I am interested in performing Lie algebra computations in Mathematica. I did a bit of searching and found several packages (LieART, KILLING, SuperLie, maybe more), and wondered if anyone would ...

**1**

vote

**0**answers

47 views

### Analytic vectors of elliptic elements of the enveloping algebra in a strongly continuous representation of a Lie group

Let $G$ be a semi-simple real Lie group, and $\rho$ a strongly continuous unitary representation of $G$ in a Hilbert space $H$. Let $\mathfrak{g}$ be its Lie algebra and $d\rho$ be the infinitesimal ...

**5**

votes

**2**answers

286 views

### Heisenberg subalgebras of affine Lie algebras

It seems to be "well-known" that (infinite-dimensional) Heisenberg subalgebras of an affine Lie algebra $\hat{\mathfrak{g}}$ corresponding to a finite-dimensional simple Lie algebra $\mathfrak{g}$ of ...

**4**

votes

**0**answers

88 views

### Adjoint orbit of two vectors

Let $G$ be a simple compact real Lie group and let $\mathfrak g$ be its Lie algebra. Let $u,v\in \mathfrak g$ be two distinct unit vectors and $H\subset \mathfrak g$ be a hyperplane with normal vector ...

**1**

vote

**1**answer

199 views

### Finding the 2nd homotopy group $\pi_2(G^\mathbb{C}/P)$

Let $G$ be a compact connected and simply connected Lie group and $G^\mathbb{C}$ be the complexification of Lie group (with is diffeomorphic with $G^\mathbb{C}\cong T^*G$) then I am looking for ...

**3**

votes

**0**answers

121 views

### Examples of divisible Lie algebras

We say that a nonzero Lie algebra $L$ is divisible, if for all elements $a$ and $b$ with $a\neq 0$, there exists $x\in L$ such that $[a, x]=b$. What are examples of divisible Lie algebras?

**3**

votes

**2**answers

243 views

### Generalizations of Lie algebras

I often stumble over the term "Lie superalgebra" (= "Lie algebra with a $\mathbb{Z}_2$ grading"). Obvious question: What about $\mathbb{Z}_3$ grading (and so on)? Is a Lie algebra with $\mathbb{Z}_n$ ...

**1**

vote

**1**answer

151 views

### Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure

I am looking for a proof or counterexample for following assertion
Each coadjoint orbit of a compact connected Lie group $G$ admits
a $G$-invariant generalized complex structure (In sense of ...

**3**

votes

**0**answers

120 views

### Physical relevance of either fundamental identity generalizing Jacobi [closed]

There are two fundamental identities for n-ary generalizations of the Jacobi identity.
One fundamental identity is right for Nambu mechanics and such, the other for L_\infty algebras as in CSFT.
Which ...

**6**

votes

**0**answers

284 views

### Injectivity of Lie group exponential function

If $G$ is a (finite-dimensional) Lie group, then the exponential function $\exp\colon\mathfrak{g}\to G$ is injective on some identity neighbourhood. If, moreover, $\mathfrak{g}$ is semi-simple and ...

**5**

votes

**2**answers

299 views

### Summary of Lie-Algebra integration tactics

If this is in the scope of MO, I would like to gather here the known tactics of
Lie algebra integration, since it appear surprisingly hard to find such a
compendium, library or any other kind of ...

**0**

votes

**2**answers

174 views

### A question on Lie algebras

To what extent, the following types of Lie algebras are classified :
Those Lie algebras $L$ such that every Lie Group $G$ with $Li(G)\sim L$, is necessarily compact.

**1**

vote

**1**answer

140 views

### adjoint action of a Levi subalgebra

We work over an algebraically closed field of characteristic 0.
Let $\mathfrak{g}$ be a reductive Lie algebra and let $\mathfrak{p}\supset\mathfrak{m}$ be a parabolic subalgebra, respectively a Levi ...

**6**

votes

**1**answer

406 views

### Origin of symbols used for half-sum of positive roots in Lie theory?

The Weyl character formula is a central result in the finite dimensional representation theory of semisimple Lie groups, algebraic groups, Lie algebras. Related questions on MO include these here ...