All Questions
2,633 questions
2
votes
0
answers
29
views
Ordering of norms and the Shapovalov form on highest weight modules
Let $\mathfrak{g}$ be a complex semisimple Lie algebra, and let $\mathfrak{U}(\mathfrak{g})$ be its universal enveloping algebra. Fix a Cartan subalgebra $\mathfrak{h} \subset \mathfrak{g}$, and ...
- 31
6
votes
1
answer
154
views
Derivations and central extensions of some infinite dimensional simple Lie algebras in characteristic zero
Let $F$ be a field of characteristic zero and consider the polynomial algebra $A=F[x_1,x_2,\ldots,x_n]$ in $n$ indeterminates over $F$. I recall that the derivations of $A$ form a simple Lie algebra $...
- 99
1
vote
1
answer
102
views
Multiplicities and double and triple tensor products of simple $\frak{g}$-modules
Given a complex simple Lie algebra $\frak{g}$ and a simple module $V_{\lambda}$ for some dominant weight $\lambda$. Consider the tensor product decomposition
$$
V_{\lambda} \otimes V_{\lambda} \simeq ...
- 467
16
votes
0
answers
188
views
Representation theory of Pin groups
I am (still) thinking about branching rules from $\mathfrak{so}(n+m)$ to $\mathfrak{so}(n) \oplus \mathfrak{so}(m)$, using Proctor's paper as the starting point.
Proctor describes this rule for $m = 2$...
- 1,574
2
votes
0
answers
132
views
A question about q-binomials at roots of unity
I have a question about a lemma $9.3.6$ in the book A Guide to Quantum Groups written by Vyjayanthi Chari and Andrew Pressley. This question comes from page 301, "The restricted specialization&...
- 137
2
votes
1
answer
144
views
Paper request: Graev's classification of SU(2,2) irreducible unitary representations
I am interested in Graev's paper in "M. L. Graev:Dokl. Akad. Nauk SSSR,98, 517 (1954); Amer. Math. Soc. Transl.,66, 1 (1968)." in which the irreducible unitary representations of SU(2,2) are ...
- 679
0
votes
0
answers
72
views
Question about action of exponential of Lie algebras (Faraut and Koranyi's book)
I'm trying to understand a statement on page 49 in the book "Analysis on Symmetric Cones" by Faraut and Koranyi.
The situation is as follows. Let $\Omega$ be a symmetric cone in $V$ and ...
- 633
9
votes
2
answers
378
views
Reference for an old result of P. M. Cohn
As it was shown by Malcev, unlike the commutative case, in which every domain can be embedded in a field, there are noncommutative domains that can't be embedded in a division ring.
For noncommutative ...
- 3,318
4
votes
1
answer
114
views
Sum of two positive roots which is not a root: uniqueness of heights of the summands
Consider a (finite reduced irreducible crystallographic) root system $\Phi$ and four positive roots $\alpha,\beta,\gamma,\delta$ such that $\{\alpha,\beta\} \neq \{\gamma,\delta\}$ and $\alpha+\beta=\...
- 3,186
3
votes
1
answer
472
views
Tips for how I can proceed with my Lie theoretical problem?
$\DeclareMathOperator\SL{SL}$I am looking at a map from a Lie group into a Lie algebra $\phi$:
$$\phi: \SL(n)\rightarrow \mathfrak{sl}_n$$
$$ P \rightarrow U_1^\dagger P U_1 + U_2^\dagger P U_2.$$
$P$ ...
0
votes
0
answers
124
views
Do the following two notions of quantum groups sometimes coincide?
On the one hand there is the notion of quantum groups due to Drinfeld and Jimbo. In there notion a quantum group is defined as a deformation of the universal enveloping algebra of a semisimple Lie ...
- 163
2
votes
0
answers
34
views
Analytic continuation of a bi-holomorphic automorphism on an irreducible bounded symmetric domain
I need your help.
Let $\Omega \subseteq \mathbb{C}^n$ be a type $IV_n$ Cartan domain, i.e;
$\Omega$ =$\{ z \in \mathbb{C}^n$: $1-2Q(z,\bar z)+|Q(z, z)|^2>0,\qquad Q(z, \bar z)<1 \}$
where $Q(z,...
7
votes
0
answers
121
views
Cohomology of current algebras in higher dimensions
I am interested in the cohomology of current algebras in $n$ dimensions with coefficients in non-trivial modules. An $n$-dimensional current algebra is a Lie algebra of smooth functions on ${\mathbb R}...
- 1,579
6
votes
1
answer
317
views
Which Lie groups are covers of matrix groups?
I would like to ask a variation on a question (not yet answered) I previously asked on math.SE, namely:
Which Lie groups are covers of matrix Lie groups? That is, which Lie groups $G$ admit discrete ...
- 3,115
5
votes
0
answers
148
views
Tensor product of a Verma module of the highest weight and a Verma module of the lowest weight, $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$
$\DeclareMathOperator\sl{\mathfrak{sl}}\newcommand\hw{\mathrm{hw}}\newcommand\lw{\mathrm{lw}}$Consider $\mathfrak{g}=\sl_2(\mathbb{C})$. Fix $\lambda,\mu\in\mathbb{C}\setminus \frac{1}{2}\mathbb{Z}_{\...
- 51
6
votes
2
answers
240
views
What is the highest weight of the representation of special orthogonal group $SO(n)$ on the space of harmonic polynomials $\mathcal H_m(\mathbb R^n)$?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\diag{diag}$
Let
$$
\mathcal{H}_m(\mathbb{R}^n)=\left\{P\in \mathbb{C}[x_1,\dotsc ,x_n]\left| \begin{align}
P\text{ is homogeneous of degree }m\text{ ...
- 605
4
votes
1
answer
201
views
Lie algebra cohomology and Lie groups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}\DeclareMathOperator\Sp{Sp}$Let $G$ be a complex Lie group and $\mathfrak{g}$ its Lie algebra (which is over $\mathbb{C}$). My first question is:
(...
- 985
2
votes
0
answers
80
views
Does restricting the eigenvalues of Hermitian matrices to the interval $[0,2\pi)$ make the exponential map to the unitary group bijective?
Let $U \in U(n)$ be a generic unitary matrix. Since the unitary group $U(n)$ is compact and connected, I know that the exponential map is surjective, i.e. that every $U \in U(n)$ has the form $U = e^{...
7
votes
3
answers
303
views
Enveloping algebra of affine Lie algebra is (not) noetherian
I work over an algebraically closed field of characteristic $0$. Let $\mathfrak{g}$ be a semisimple Lie algebra, $\hat{\mathfrak{g}}=\mathfrak{g}[t,t^{-1}]\oplus\mathbb{C}K\oplus\mathbb{C}D$ the ...
- 1,391
2
votes
1
answer
244
views
Decomposition of an $\text{SL}_n(\mathbb{C})$ representation
Let $W = V \oplus V^*$, where $V$ is the standard $\text{SL}_n(\mathbb{C})$ rep and $V^*$ is its dual. I'm ultimately trying to decompose the space $(W \otimes \bigwedge^2 W) / {\bigwedge^3 W}$.
This ...
- 181
5
votes
1
answer
304
views
Iwasawa decomposition of a non-compact semisimple Lie group?
A (non-compact) symmetric space is of the form $G/K$, where $G$ is a (non-compact) semisimple Lie group defined over $\mathbb{R}$, and $K$ is a maximal compact subgroup of $G$.
Let $M = G/K$ be a rank-...
- 650
3
votes
0
answers
192
views
How can I prove this stronger version of Fedder's Criterion?
I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...
- 317
3
votes
0
answers
50
views
Root systems of maximally noncomact Cartan subalgebras
Let $G$ be a real reductive Lie group, and $K$ a maximal compact subgroup in $G$. Write $\mathfrak{g}$ for the Lie algebra of $G$, and $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the Cartan ...
- 951
0
votes
1
answer
85
views
Centralisers of elements in a Lie algebra
I am considering the centraliser subgroup of some specific elements in a Lie algebra.
Let $\mathfrak{g}$ be a Lie algebra over characteristic $0$. Let us consider some elements $x:=\{x_1,x_2,\ldots , ...
- 147
2
votes
0
answers
49
views
Action of Lusztig braid group operators on locally finite part
Let $\mathbf{U}_q(\mathfrak{g})$
be a Drinfel'd-Jimbo quantum group. The quantum group $\mathbf{U}_q(\mathfrak{g})$ acts on itself by the left adjoint action $ad(u)(x)=u_{(1)}u S(u_{(2)})$, where we ...
- 141
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 53
3
votes
1
answer
85
views
Isogeny of compact Lie group with central circle
Suppose $G$ is a connected, compact Lie group and $S^1 \subset G$ is a central subgroup. Can I write $G$ as a quotient of a product group $$G=(S^1 \times H)/Z$$
where the $S^1$ factor maps onto the ...
- 959
3
votes
0
answers
128
views
Spectra of Coxeter diagrams and representations of Coxeter groups
Let $S$ be a Coxeter diagram, i.e. an unoriented graph, edges labelled by weights $3,4,5,...$ . This includes (affine) Dynkin diagrams,
Then
Theorem 3.1.3 of Brouwer and Haemers' Spectral Graph ...
- 5,711
1
vote
0
answers
58
views
Linear algebraic group, absolute root system, computing roots
Let $G(F)$ be a reductive linear algebraic group, where $F$ is a local field. Let $T(F)$ be a maximal anisotropic torus of $G$ that splits over a quadratic extension of $F$. Is there an efficient ...
- 11
2
votes
0
answers
40
views
Reconstruction of a Poisson-Lie group structure from a Lie bialgebra $\mathfrak{g}$
Let $(\mathfrak{g}, [,], \delta)$ be a Lie bialgebra where $\delta$ is the cobracket. It is well-known that there exists a simply connected Poisson-Lie group $G$ such that $\mathfrak{g} = \mathrm{Lie}(...
- 255
1
vote
0
answers
148
views
Which elements lie in a Cartan subalgebra?
Let $\mathfrak{g}$ be a Lie subalgebra of the real upper triangular matrices (i.e. $\mathfrak{g}$ is a completely solvable real Lie algebra). $\mathfrak{g}$ has an exponential radical $\mathfrak{r}$, ...
17
votes
2
answers
2k
views
How to think of algebraic geometry in characteristic p?
How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's ...
5
votes
1
answer
189
views
Why do most eigenspaces of a Lie algebra automorphism have finitely many orbits?
I'm interested in understanding the following lemma, which Vogan states (Lemma 4.8) in his paper The Local Langlands Conjecture (omitting the "well-known" proof).
Suppose $G$ is a complex ...
- 835
4
votes
0
answers
79
views
Closed character formula for the module $L(a\omega_i)$
Let $\mathfrak{g}$ be a complex finite-dimensional simple Lie algebra with a fixed Cartan subalgebra $\mathfrak{h}$. Assume that $\omega_1, \omega_2, \dots, \omega_n\in\mathfrak{h}^{*}$ is the ...
- 41
2
votes
1
answer
88
views
Number of real forms of a (not semisimple, solvable) Lie algebra
Consider the Lie algebra $\mathfrak{r}_2=\mathfrak{aff}(\mathbb{C})$ of the group of affine maps of $\mathbb{C}$, and let $\mathfrak{g}=\mathfrak{r}_2 \oplus \mathfrak{r}_2$.
I am interested in ...
- 23
3
votes
1
answer
85
views
Restriction of scalar commutes with taking maximal subtorus for semisimple group G
I was wondering such a question: for a semisimple complex Lie group $G$, whether it is true that the maximal subtorus of $\mathrm{Res}_{\mathbb{C}/\mathbb{R}}(G)$ is $\mathrm{Res}_{\mathbb{C}/\mathbb{...
- 455
7
votes
1
answer
2k
views
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
6
votes
1
answer
284
views
Multiplication factors in folding root systems and Lie algebras by automorphisms
When Stembridge, in the paper Folding by automorphisms, considers folding by automorphism $\sigma$ he considers the root system generated by for each orbit $J$.
$$\sum_{i \in J} \alpha_i .$$
Whereas ...
- 83
4
votes
1
answer
91
views
Lie algebra with finitely generated envelope
If a Lie algebra $\mathfrak g$ is finitely generated, its enveloing algebra $U\mathfrak g$ is finitely generated as an associative algebra. In fact, taking the enveloping algebra of the surjection $\...
- 985
1
vote
0
answers
79
views
The partial orders on the elements of a root system coming from the positive spans of the weights and the roots
Let $(\Delta,V)$ be a root system with a choice of positive roots $\Delta^+$. Denote the $\mathbb{N}_0$-span of the positive roots by $\mathcal{O}^+$, and the $\mathbb{N}_0$-span of the associated ...
1
vote
0
answers
139
views
Integral convex polytopes formed from the weight diagrams of representations of $\mathfrak {sl}_4$($\mathbb{C}$)
I'm a student studying undergraduate abstract algebra and doing a summer research project in the mathematics department at my school. I'm barely familiar with the rudiments of representation theory; I ...
- 111
1
vote
2
answers
197
views
What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
- 9,501
4
votes
3
answers
275
views
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
1
vote
1
answer
114
views
Source for highest weight vectors for $\text{SL}_n(\mathbb{C})$ representations
I have a list of a lot of irreducible $\text{SL}_n(\mathbb{C})$ representations and need to know what all of their highest weight vectors are. For example, using the notation from Fulton and Harris, I ...
- 181
7
votes
1
answer
336
views
Nilpotent orbits of a parabolic subgroup
Suppose $G$ is a reductive group over an algebraically closed field of characteristic $0$ with parabolic $P$, Levi quotient $M$, and unipotent radical $U$. We denote the nilpotent elements of $\mathrm{...
- 953
11
votes
0
answers
437
views
A rather strange algebra
Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector ...
- 3,318
1
vote
0
answers
58
views
Can a maximal rank subgroup of a simply connected Lie group have simply connected factors?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\rank{rank}$Take a simply connected Lie group $G$ such as $\SU(N)$ and a maximal rank subgroup $H$, i.e. $\rank(G) = \rank(H)$. Assume that $H$ takes ...
1
vote
1
answer
181
views
Relation between the root lattice of $\mathrm{SO}(7)$ and the root lattice of $G_2$
The root lattice of $\mathfrak{so}(7)$ is given by the following 18 roots:
$$
\left(\begin{array}{c}0\\0\\1\end{array}\right)
,
\left(\begin{array}{c}0\\0\\-1\end{array}\right)
,
\left(...
- 369
3
votes
0
answers
196
views
A property of an irreducible root system
Let $\Phi$ be an irreducible root system. Let $\alpha_k$ be a simple root. I recently observed that the number of positive roots which are bigger than $\alpha_k$ and of height $m$ is same as the ...
- 673
1
vote
0
answers
65
views
Some details about relationship between central charges and second cohomology group of the Lie algebra
S. Weinberg in his book "The quantum theory of fields" talks about central charge that appear in Lie algebra of a given Lie group. To be more precise, on page 83 in the book, he computes the ...
- 287