# Tagged Questions

**5**

votes

**1**answer

118 views

### Motivating the existence of Canonical Bases for Representations

In Representation Theory, the theme of the existence of a canonical basis has been explored quite a lot. I will limit myself in this question to the kind of canonical bases that arise from the ...

**14**

votes

**5**answers

2k views

### Matrix representation for $F_4$

Has anyone ever bothered to write down the 26-dimensional fundamental representation of $F_4$? I wouldn't mind looking at it. Is it in $\mathfrak{so}(26)$?
I'm familiar with the construction of the ...

**-3**

votes

**0**answers

58 views

### Prove that two Lie groups have homeomorphic universal covers if and only if their corresponding Lie algebras are isomorphic as Lie groups [on hold]

I asked this on MSE but am not really convinced by the answer. Can anyone determine whether the statement is true to begin with before I start attempting a proof?
...

**8**

votes

**0**answers

389 views

+100

### Local proof of Grothendieck-Riemann-Roch theorem

There is a theorem by Feigin and Tsygan(Theorem 1.3.3 here) which they call "Riemann-Roch" theorem.
Given a smooth morphism $f:S\to N$ of relative dimension $n$ and a vector bundle $E/S$ of rank $k$ ...

**6**

votes

**2**answers

323 views

### Socle of tilting modules in the BGG category $\mathcal{O}$ over a semisimple Lie algebra

Suppose that $\mathfrak{g}$ is a finite dimensional, complex, semisimple Lie algebra. Let $\mathcal{O}$ be the BGG category over $\mathfrak{g}$.
Tilting module theory play an important role in the ...

**15**

votes

**1**answer

996 views

### Modern reference for maximal connected subgroups of compact Lie groups

What's the nicest place to see a list of the maximal connected subgroups of compact Lie groups? Is there anything on-line?
I looked at Tits' Bourbaki talk on Dynkin's and others' work, but he admits ...

**4**

votes

**2**answers

211 views

### A finiteness property for semi-simple algebraic groups

Let $G$ be a semi-simple algebraic group over a field $K$, I am considering a question about whether there exists a finite set of semi-simple $K$-subgroups, say $H_1,...,H_r$, such that for any ...

**4**

votes

**0**answers

169 views

### A soft question on Gauge Equivalence in Integrable Systems

I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations ...

**3**

votes

**1**answer

70 views

### (Co-) Homology of free nilpotent Lie Algebras

What is actually known about the (co-) homology of free nilpotent Lie Algebras over $\mathbb{C}$ and coefficients also in $\mathbb{C}$?
I.e. the nilpotent Lie Algebras that one gets by a quotient of a ...

**3**

votes

**0**answers

69 views

### Isomorphisms between extension group and $\mathfrak{u}$-cohomology

Let $\mathcal{O}^{\mathfrak{p}}$ be a parabolic subcategory (see, Humphrey's BGG's Ch.9) of the BGG category $\mathcal{O}$ (w.r.t the Cartan $\mathfrak{h}$ and Borel $\mathfrak{b}$) over a ...

**6**

votes

**1**answer

677 views

### Nilpotent matrices related to Lie algebras of special orthogonal groups in characteristic 0

In terms of matrix theory, the question I'm led to is the following: Start with an $n$-dimensional vector space over an algebraically closed field of characteristic 0 such as $\mathbb{C}$, which has ...

**2**

votes

**0**answers

100 views

### Are singular critical points isolated for control systems on compact semisimple Lie groups

Given a control system on $\mathrm{SU}(n)$ (or any other compact, semi-simple Lie group I suspect) of the form:
$\frac{d U_t}{dt} = (A + w(t)B)U_t$
where $A,B \in \mathfrak{su}(n)$ generate the ...

**5**

votes

**1**answer

317 views

### A criterion for orbits of complex reductive group to be closed

I am having some trouble understanding Nakajima's proof of the Kempf-Ness theorem in [1]. At the end (proof of Proposition 3.9(6)), his argument is basically the following:
Let $G=K_{\Bbb C}$ be a ...

**2**

votes

**1**answer

228 views

### The Quasimirs and Casimirs of a Lie algebra

Casimirs are not completely trivial to compute, and somewhat ill defined (if $C_4$ is a quartic casimir and $C_2$ a quadratic, $C_4'=C_4+p*C_2^2+q*C_2$ is another quartic casimir). With generalized ...

**3**

votes

**1**answer

91 views

### Vanishing of power of nilpotent operator $\mathrm{ad} \, \;e$ in different characteritics

This question needs some background:
(1) In his influential 1959 paper here, Kostant studied the adjoint representation of a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (which can be ...

**5**

votes

**1**answer

316 views

### A Hom-Tensor identity - $\text{Hom}_{R}(P,B)\otimes _SC \cong \text{Hom}_{R}(P,B \otimes_S C) $

let $R,S$ be associative algebras over $\mathbb{C}$. Let $\mathcal{C} \subseteq$ $R$-Mod be a full abelain subcategory of $R$-Mod which is the category of $R$-modules. Let $B$ and $C$ be, a ...

**10**

votes

**2**answers

408 views

### From Weyl groups to Weyl groupoids?

I'm trying to find a framework where the choices in the classical construction of a root system of a semi-simple lie algebra are not needed.
Let $\mathfrak{g}$ be a semisimple lie algebra.
...

**4**

votes

**1**answer

105 views

### Cartan subspaces for general algebraic representations

So I feel like asking the following likely open-ended question: What good generalizations of the notion of Cartan subspace do we have?
To be precise, let $G\curvearrowright V$ be an algebraic ...

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vote

**0**answers

41 views

### Integrable modules and comodules

Let $G$ be a semisimple Lie group and $\mathfrak{g}$ its Lie algebra. Do we have the following result: $V$ is an integrable $U(\mathfrak{g})$-module if and only if $V$ is a $\mathbb{C}[G]$-comodule? ...

**5**

votes

**1**answer

492 views

### Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$

Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...

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votes

**0**answers

49 views

### Are there nilpotent Manin Triples?

Let $\mathfrak{g}$ be a Lie bialgebra and denote by $\mathfrak{d}$ the double of $\mathfrak{g}$, i.e. $\mathfrak{d}$ is a Manin triple. Are there known examples or conditions on $\mathfrak{g}$ for ...

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**0**answers

54 views

### Try to understand the relation between Poisson Lie groups and Lie bialgebras

I am trying to understand the relation between Poisson Lie groups and Lie bialgebras. In the book Lectures on quantum groups, Page 20, Theorem 22 says that given a Poisson Lie group, we can construct ...

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votes

**2**answers

167 views

### Recover Poisson bracket on $C[G]$ using the Lie cobracket $\delta: g \to \Lambda^2 g$

By a theorem of Drinfeld, there is a one to one correspondence between Lie bialgebras and Poisson Lie groups. Therefore given a Lie cobracket $\delta: g \to \Lambda^2 g$, there is a Poisson bracket on ...

**2**

votes

**1**answer

77 views

### Projections of orbifolds

A while back I came across orbifolds, in particular the quotients $SU(2)/U(1)\cong S^2$, $SU(3)/(SU(2)\times U(1)\cong \mathbb{C}P^2$ and $SU(3)/(U(1)\times U(1))$. The way I needed them, was as an ...

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votes

**1**answer

853 views

### Roadmap to Geometric Representation Theory (leading to Langlands)?

I believe there has been at least one question similar to this one and yet I still think this particular question deserves to have a thread of its own.
I'm becoming increasingly fascinated by stuff ...

**6**

votes

**2**answers

187 views

### “Diagonalizing” Littlewood-Richardson coefficients

Let's consider the Littlewood-Richardson coefficients $c^{\lambda}_{\mu \nu}$ so that
\begin{equation}
V_\mu \otimes V_\nu = \bigoplus_\lambda V_\lambda^{\oplus c^{\lambda}_{\mu \nu}}
\end{equation}
...

**4**

votes

**0**answers

56 views

### Harmonicity on semisimple groups

I asked this on Math.SE and got no answer, so I'll try my luck here.
Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in ...

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vote

**0**answers

55 views

### Why the sub-Laplacian $\Delta_{sub}$ on the Heisenberg group $H^3 $ is sub-elliptic but not elliptic?

I want to know why the sub-Laplacian $\Delta_{sub}= X^2 + Y^2$ on the Heisenberg group $H^3 = \mathbb C \times \mathbb R$ is sub-elliptic but not elliptic, where $X$ and $Y$ are the left-invariant ...

**2**

votes

**1**answer

82 views

### multiplicity-free action on $SO(n+1)/SO(n-1)$

I'm trying to show that the Lie group $G=SO(n+1) \times SO(2)$ acts multiplicity-free on the cotangentbundle $T^* (SO(n+1)/SO(n-1))$.
That means:
1)
There exists an ...

**6**

votes

**1**answer

400 views

### What is this Lie algebra?

Consider two matrices $A,B \in \mathfrak{su}(N)$ which are both diagonal in the standard basis and non-zero.
If we consider the new matrix $\tilde{B} := FBF^{\dagger}$ where $F$ is the `quantum' ...

**3**

votes

**1**answer

122 views

### The Jacobi identity of a Lie algebra?

Let $g$ be a finite dimensional real Lie algebra and $(,)$ be a nondegenerate invariant symmetric bilinear form on $g$. Let $r\in g\bigotimes g$ be a skew-symmetric solution of the MCYBE. We may ...

**17**

votes

**6**answers

639 views

### Automorphism group of real orthogonal Lie groups

I would like to know what is the "outer-automorphism group" $Out$ of $SO(p,q)$ and $O(p,q)$, where $p+q >0$. My working definition of $Out$ is as follows:
Let us denote by $Aut(G)$ the ...

**4**

votes

**1**answer

129 views

### When this Ad-invariant function on a Lie algebra is zero?

Let $G$ be a compact Lie group with Haar measure $dg$ and (finite-dimensional real) Lie algebra $\frak g$. Endow $\frak g$ with an $\hbox{Ad}$-invariant norm $\|\cdot\|_{\frak g}$ so that $\frak g$ ...

**3**

votes

**1**answer

91 views

### Unclear asymmetry in Lie-algebra module structure on space of linear transformations Hom(V,W)

Let $L$ be a (finite dimensional) Lie-algebra. Let $V, W$ be finite-dimensional vector spaces. If $V,\; W$ are in addition $L$-modules (see, e.g., 6.1 in Humphreys Introduction to Lie Algebras), then ...

**6**

votes

**2**answers

140 views

### Can we count the number of simple modules for a reduced enveloping algebra?

Let $G$ be a reductive algebraic group over a field of positive characteristic $p$, which I'll assume to be very good for $G$. Then the Lie algebra $\mathfrak{g}$ is restricted and each simple ...

**4**

votes

**0**answers

106 views

### Lie algebras whose derivation algebra is nilpotent

Let $L$ be a Lie algebra and denote by $\mathrm{Der}(L)$ the derivation algebra of $L$. If $L$ is finite-dimensional, then a theorem of Leger and Togo (Duke
Math. J. $\bf{26}$ (1959), 623 – 628, DOI: ...

**3**

votes

**0**answers

43 views

### filteration for Leibniz algebras [closed]

Let $L$ be restricted Lie algebra, we define p-filteration $$ L=L_{1} \supseteq L_{1} ... \supseteq L_{n}... $$, satisfying the following conditions:
$ [L_{i},L_{j}] \subseteq L_{i+j}$ and ...

**3**

votes

**1**answer

87 views

### Vector fields, diffeomorphism subgroups and lie group actions

Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization:
Let $\{X_j\} \in Vect(M)$ be a ...

**2**

votes

**0**answers

63 views

### Action of orthogonal group on the free Lie algebra

This question is somewhat related and inspired by this post of professor Montgomery.
The free Lie algebra $L(V)$ generated by an $r$-dimensional vector space $V$ is, in the language of ...

**3**

votes

**0**answers

68 views

### False optima for control on Lie groups?

Consider the equation
$\frac{d Y_t}{dt} = (A + w(t)B) Y_t$
evolving on a compact semi-simple Lie group $G$ where $\frak{g}$ is the Lie algebra of $G$ and $A,B \in \frak{g}$ and:
$J:G \rightarrow ...

**7**

votes

**1**answer

103 views

### Symmetries of the flag variety

Let $\mathfrak g$ be a finite dimensional simple Lie algebra over $\mathbb C$, and let $\mathcal B=G/B$ be the associated Flag variety.
Is it true that the obvious map
$$
\mathfrak g\to \Gamma ...

**4**

votes

**0**answers

139 views

### Affine Steinberg groups vs Steinberg groups over Laurent polynomials

Let $R$ be a commutative ring and $\Phi$ be a finite (also called spherical) reduced irreducible root system of rank $\geq 2$. I will denote by $\mathrm{St}(\Phi,R)$ the Steinberg group of type $\Phi$ ...

**8**

votes

**1**answer

412 views

### Kazhdan-Lusztig graph for the Springer fiber of the minimal special unipotent class?

This graph was determined in the case of simply-laced root systems by Igor Dolgachev and Norman Goldstein here. For other root systems the original question should be modified, leading to a precise ...

**3**

votes

**0**answers

74 views

### Automorphism group of Lie algebra of bounded operators

What is the automorphism group of the complex Lie algebra of bounded operators on a complex Hilbert space, with the commutator as Lie bracket? What for the real Lie algebra of bounded antihermitian ...

**2**

votes

**1**answer

112 views

### Commuting nilpotent matrix collection

For every large enough $m\in\Bbb N$ are there $c=\alpha m$ (for some fixed $\alpha>0$) square matrices $A_1,\dots,A_c$ that commute with each other with nonzero product ($\forall ...

**3**

votes

**0**answers

73 views

### Poincaré-Birkhoff-Witt theorem for Leibniz algebras

Leibniz algebras can be seen as a non-skew-symmetric generalization of Lie algebras. I have already taken a look at some papers related to Leibniz algebras and extending main results of Lie algebras ...

**3**

votes

**0**answers

79 views

### How to write down solutions of Yang-Baxter equations for $sl_3$ explicitly?

In the paper, Stolin classifies all quasi-Frobenius subalgebras of $sl_3$. How to write down solutions of Yang-Baxter equations for $sl_3$ explicitly using these quasi-Frobenius subalgebras? Thank you ...

**2**

votes

**0**answers

79 views

### Lie algebra (co)homology of the Lie algebra of differential operators

Let $X$ be an algebraic variety (or a complex manifold) over $\mathbb C$. Let $D(X)$ be its algebra of differential operators. Mostly I am interested in algebraic differential operators, but the case ...

**5**

votes

**0**answers

81 views

### Analytic continuation of $\mathfrak{so}(n)$ algebras to real $n$?

In a 1988 paper "The Lie algebras $\mathfrak{gl}(\lambda)$ and cohomologies of Lie algebras of differential operators", Feigin defined the analytic continuation of $\mathfrak{sl}(n)$ algebras (over ...

**11**

votes

**6**answers

915 views

### Representation Theory of Lie Groups: Reference Request

I am looking for a reference that describes the correspondence between the (finite-dimensional) representations of (real) Lie groups and the representations of their Lie algebras. More precisely, ...