**1**

vote

**0**answers

85 views

### When is the Ad (Adjoint Representation) Morphism a Closed Map

Given a Lie group $\mathfrak{G}$ with finite centre and with Lie algebra $\mathfrak{g}$, I am looking at a simple proof that negative definite Killing form implies compactness. This proof is given ...

**0**

votes

**0**answers

32 views

### Fixed Point Algebras of Adjoint Actions of Banach Lie groups

I have the following question:
Let a be an element in a connected Banach Lie group G (over K, where K is the reals or the complex numbers).
We assume that G is not trivial, that has more than one ...

**1**

vote

**1**answer

153 views

### Basics on lattice in classical groups

as a beginner,I am not sure whether this question is too basic to post here./-\。
Many textbook will talk about the prototypical example SL(n,Z)\SL(n,R), which can be identified with the space of ...

**1**

vote

**0**answers

50 views

### Examples of Lie subalgebras of universal enveloping algebras

I'm looking for non-trivial examples of triples $(\mathfrak g, L, \psi)$, where $\mathfrak g$ and $L$ are finite-dimensional non-abelian Lie algebras over field $\Bbbk$ and $\psi\colon U_{\mathfrak ...

**8**

votes

**1**answer

163 views

### Variety of nilpotent Lie algebras or $p$-groups

Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either:
1) Let $\mathcal{L}$ ...

**-2**

votes

**0**answers

36 views

### How to show a matrix can't be written as exponential? [migrated]

How can I show the matrix
$$A = \left(
\begin{array}{c c}
-2 & 0 \\
0 & -1 \\
\end{array}
\right)$$
can't be written as $A = exp(a)$?
I've tried to write A like
$$A = \left(
...

**3**

votes

**3**answers

152 views

### Nilradical of a Lie algebra associated to a associative algebra

Let $A$ be a finite dimensional (unital) $K$-Algebra. By $A^{\circ}$ we denote the associated $K$-Lie-algebra of $A$ with respect to the product $a\circ b:=ab-ba$. In addition, we denote by ...

**0**

votes

**0**answers

43 views

### Basis for Witt algebra in general format

The Witt algebra $W(n,m)$ is defined as the set of element ${∑f_jD_j such that f_j∈A(n,m)}$ with usual Lie bracket. I am a bit confused about basis for $W(n,m)$? What is the meaning of $"f_jD_j"$ ? I ...

**5**

votes

**1**answer

215 views

### Harish-Chandra isomorphism for compact symmetric spaces

I would be interested to have an explicit description of the algebra of invariant differential operators on functions on a compact symmetric space $G/K$. A reference would be especially useful.
...

**0**

votes

**1**answer

72 views

### A problem on about a matrix norm on $\mathfrak{su}(4)$

Given a fixed $B \in \mathfrak{su}(4)$ is it possible to solve for $F$:
$\sigma^{\text{max}}\left(\frac{A}{F(A)} + B \right) = 1$, $\forall A \in \mathfrak{su}(4)$. A theorem in ...

**1**

vote

**0**answers

64 views

### A criteria for a subalgebra of M(n,C) being M(n,C) [migrated]

Suppose $S$ is a subalgebra of the matrix algebra $M_n(\mathbb{C})$. If for any vector $v$ and $w$ in $\mathbb{C}$, there always exists a matrix $A$ in $S$, depending on $v$ and $w$ of course, which ...

**5**

votes

**0**answers

187 views

### Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course.
Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...

**2**

votes

**0**answers

59 views

### First Variation of Dyson Series/Magnus Expansion

Given the matrix differential equation $\frac{dU_t}{dt}=A_t U_t$ there are at least two ways to write a formal solution. Both the Dyson series: $U_t = \mathcal{T} e^{\int_{0}^{t} A_t dt}$ and the ...

**8**

votes

**2**answers

274 views

### Sums of degrees of irreducible complex characters

The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant to determine the dimension of a maximal torus of the Lie algebra associated to ...

**2**

votes

**1**answer

124 views

### Weight multiplicities for some particular representations of SO(2m).

I am looking for explicit formulas for the weight multiplicities of some particular irreducible representations of $SO(2m)$.
It is possible that they have been already computed; in this case I will ...

**-1**

votes

**0**answers

54 views

### Complex conjugation of positive roots [migrated]

I have a simple question about root systems. Suppose that $G$ is a connected reductive group over the reals $\mathbb{R}$, and $T\subset G$ is a maximal torus (by this I mean that $T_{\mathbb{C}}$ is a ...

**4**

votes

**2**answers

120 views

### Invariant planes of a nilpotent matrix with two Jordan blocks of size two

Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map
$$
N = \begin{pmatrix}
0 & 1 & & \\
0 & 0 & & \\
& & 0 ...

**0**

votes

**0**answers

174 views

### PBW proof proposal

One version of the PBW theorem states:
$\omega $:$\mathfrak {S} \mapsto \mathfrak {E} $ is an isomorphism of algebras.
I am curious if this is a possible proof for the PBW theorem, part is taken ...

**0**

votes

**1**answer

67 views

### equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution

I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...

**11**

votes

**2**answers

372 views

### Who originated the standard symbols for Lie groups GL, SL, SU, etc.?

Who was first to use symbols GL, SL, O, SO, U, SU, Sp and their projective versions, and how did this notation become standard?
The notation appears in fairly modern form in Weyl's "The Classical ...

**5**

votes

**3**answers

376 views

### Characterising the adjoint representation of SU(N)

One can show that the adjoint representation of $\mathrm{SU}(n)$, the image of the map $\mathrm{Ad}:\mathrm{SU}(n) \rightarrow \mathrm{Aut}(\mathrm{su(n)})\subset \mathrm{GL}(\mathrm{su}(n))$, is an ...

**1**

vote

**1**answer

46 views

### Bi-invariant one forms on compact Lie groups

I'm hoping to learn of a criterion for the existence of a bi-invariant one form on a Lie group $G$. I'm looking for a reference that there are no such one forms son $SU(n)$ (as long as this is in fact ...

**2**

votes

**0**answers

119 views

### Can the commuting condition in Jordan-Chevalley decomposition be replaced with this global criterion?

Let $G$ be a reductive linear algebraic group defined over an algebraically closed field $k$ of arbitrary characteristic, and write $\mathfrak{g}$ for its Lie algebra.
The Jordan-Chevalley ...

**0**

votes

**0**answers

57 views

### deformation of Lie algebra

How can we deform a given Lie algebra? In particular, in the attachment file how can we arrive at the commutation relations (20) by starting from the commutation relation (19)?
...

**8**

votes

**1**answer

153 views

### Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids

When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie ...

**0**

votes

**0**answers

92 views

### A special Lie subalgebra

Motivated by comments of the following post A question on involutions on the Lie algebra of vector fields we ask the following question:
Let $L$ be a Lie algebra. We consider the Lie subalgebra ...

**6**

votes

**1**answer

163 views

### Laplace-Beltrami operator on a Lie group

For an arbitrary Lie group, is it always possible to chose a left-invariant Riemannian metric such that the Laplace-Beltrami operator $\Delta$ is given by
$$\Delta f = \delta^{i j} X_i X_j f$$
for ...

**2**

votes

**1**answer

265 views

### Is every closed subgroup of $\text{GL}_n(K[[x]])$ finitely generated?

Let $n \in \mathbb{N}$, $K$ a finite field. Denote by $K[[x]]$ the (profinite) ring of formal power series over $K$. Note that $\text{GL}_n(K[[x]])$ is a profinite group.
Is every closed subgroup of ...

**3**

votes

**0**answers

59 views

### Explicit description of graded (counital) cofree cocommutative coalgebras

Let $k$ be a field of characteristic $p \neq 2$, and $V = \oplus V_{n}$ be a graded vector space over $k$.
Then, can one compute the graded (counital) cofree cocommutative coalgebra $C(V)$ ...

**1**

vote

**1**answer

131 views

### Euler-Poincare equations with constraints

It is well known that the stationary curves (say $\Xi(t)$) of a regular Lagrangian $\mathcal{L}$ on a compact, semi-simple Lie group $G$ have the property that $\xi(t) = \frac{d \Xi(t)}{dt} ...

**2**

votes

**0**answers

85 views

### (Co)Homology of groups vs. Lie algebras: polynomial rings

For Lie groups (or algebraic groups over fields) there is a strong relation between the cohomology of the group and the cohomology of its Lie algebra. Some MO-question where this is discussed can be ...

**10**

votes

**2**answers

435 views

### Geodesics on $SU(4)$

Are the geodesics of the following metrics on $SU(4)$ known or easy (in a way not known to me!) to find?
In the adjoint representation, one can express the Killing form as a matrix and consider it as ...

**18**

votes

**1**answer

294 views

### Why should affine lie algebras and quantum groups have equivalent representation theories?

Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$ and let $\hat{\mathfrak{g}}$ be the Kac-Moody algebra obtained as the canonical central extension of the algebraic loop algebra ...

**0**

votes

**0**answers

48 views

### Involution on $\chi^{\infty}(\mathbb{R}^{n})$

Is there a Lie algebraic involution $\theta$ on $\chi^{\infty}(\mathbb{R}^{n})$ which restriction to linear vector fields is the involution
$\theta(A)=-A^{tr}$ for $A\in M_{n}(\mathbb{R})$?
...

**0**

votes

**0**answers

97 views

### A linear/Lie algebra problem

Let $\mathfrak{g}$ be a complex linear Lie algebra of dimension $n$. If there exists a basis $\{e_1,\dots,e_n\}$ of $\mathfrak{g}$ such that ...

**23**

votes

**2**answers

1k views

### Why do Lie algebras pop up, from a categorical point of view?

Groups pop up as automorphism groups in any category.
Rings pop up as endomorphism rings in any additive category.
Is there a similar way to attach a Lie algebra to an object in a category of a ...

**3**

votes

**2**answers

154 views

### A Lie algebra identity

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Fix a basis $e_1,\dots,e_n$ of $\mathfrak{g}$ and let $e^1,\dots,e^n$ be its dual basis. We also use $e^i$ to denote the left-invariant ...

**7**

votes

**3**answers

590 views

### nth term in the Baker-Campbell-Hausdorff formula

I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...

**1**

vote

**2**answers

198 views

### A question on involutions on the Lie algebra of vector fields

Edite According to the essential comment of Ian Agol I revise the question as follows
For a smooth manifold $M$, is there a non identity involution $\theta$ on the lie algebra $\chi^{\infty}(M)$ ...

**2**

votes

**0**answers

57 views

### Springer Isomorphisms for Adjoint Simple Exceptional Groups

I'm trying to understand explicitly a construction of Springer isomorphisms for adjoint exceptional groups given by Bardsley and Richardson. Their construction is as follows. Let $G$ be an adjoint ...

**5**

votes

**1**answer

296 views

### The surjectivity of the exponential map for the isometry group

Little is known on general conditions guaranteeing that the exponential map between a Lie algebra and an associated Lie group is surjective.
Let $M$ be a noncompact connected Riemann manifold, and ...

**2**

votes

**1**answer

40 views

### Termination conditions for matrix Lie alebra basis generation via “P. Hall algorithm”

Suppose $g_1,...,g_n\in\mathbb{M}_{d\times d}(\mathbb{C})$ are matrices and we are interested in finding the smallest matrix Lie algebra containing them, that is, the matrix Lie algebra generated by ...

**0**

votes

**0**answers

98 views

### Non-semisimple Lie algebra tensors

Let $\mathfrak{L}$ be a non-semisimple Lie algebra. Let $T_i$ be its generators. As usual, define the structure constants $C_{ij}^k$ by $[T_i,T_j]=C_{ij}^kT_k$ and the metric tensor $g_{jm}$ by ...

**7**

votes

**2**answers

295 views

### Quadratic Casimir of fundamental irreps of simply-laced Lie algebras

I have the following question, motivated by the expression for the character of level 1 highest weight integrable representations of simply-laced affine algebras (in terms of the string function). It ...

**5**

votes

**0**answers

121 views

### Generators for invariant tensors of lie algebras

EDITED FOR (hopeful) CLARITY:
For a simple Lie algebra $\mathfrak{g}$ (over $\mathbb{C}$) and its adjoint group G, the $G$-invariant polynomials on $\mathfrak{g}$ are linear combinations of products ...

**3**

votes

**1**answer

167 views

### Lie group GL(4) representation decomposition

Let $V$ be the defining representation of $GL(4,\mathbb C)\to GL(V)$ with $V=\mathbb{C}^4$.
Let $Ext\;V$ be the exterior square of $V$ which is a 6-dim repsentation.
My question: How does $$V\otimes ...

**4**

votes

**1**answer

112 views

### PBW basis and canonical basis

Consider the example of $\mathfrak{g} = sl_3$. Then
$$
\mathfrak{g} = \mathfrak{n} \oplus \mathfrak{h} \oplus \mathfrak{n}^{-},
$$
where $\mathfrak{n}$ is generated by $E_{12}, E_{13}, E_{23}$, ...

**4**

votes

**1**answer

145 views

### About the construction of the Universal Enveloping Lie Algebroid

Let $X$ be a reasonable smooth scheme over some base $S$. The tangent sheaf $T_X$ is a Lie algebroid, locally free as a $\mathcal{O}_X$ module, and its Universal Enveloping Lie Algebroid ...

**1**

vote

**0**answers

69 views

### Representations of $\mathfrak{so}(3)$ ($\mathfrak{so}(2,1)$) and $SO(3)$ ($SO(2,1)$)

(Apologies if this question is too basic!) I have explicit 5-dimensional real representations of $\mathfrak{so}(3)$ and $\mathfrak{so}(2,1)$, and I want to know whether it's necessarily true that the ...

**1**

vote

**1**answer

67 views

### Smallest subalgebra of $\mathfrak{su}(4)$ arrising from a control problem on $SU(4)$

What is the smallest subalgebra of $\mathfrak{su}(4)$ containing the span of the set $A = \{A, B_1, B_2\}$ where:
$A = i (J^x \sigma_x \otimes \sigma_x + J^y \sigma_y \otimes \sigma_y + J^z \sigma_z ...