All Questions
2,633 questions
0
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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
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
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^{...
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
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
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
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
$$
V_{\lambda} \otimes V_{\lambda} \simeq ...
- 467
2
votes
0
answers
49
views
Action of Lusztig braid group operators on locally finite part
Let $\mathbf{U}_q(\mathfrak{g})$
be a Drinfel'd-Jimbo quantum group. The quantum group $\mathbf{U}_q(\mathfrak{g})$ acts on itself by the left adjoint action $ad(u)(x)=u_{(1)}u S(u_{(2)})$, where we ...
- 141
3
votes
0
answers
50
views
Root systems of maximally noncomact Cartan subalgebras
Let $G$ be a real reductive Lie group, and $K$ a maximal compact subgroup in $G$. Write $\mathfrak{g}$ for the Lie algebra of $G$, and $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the Cartan ...
- 951
6
votes
2
answers
794
views
Tensor algebra and universal enveloping algebra
Let $\mathfrak g$ be a Lie algebra which is not reductive. Let $T(\mathfrak g)$ and $U(\mathfrak g)$ be the tensor algebra and universal enveloping algebra of $\mathfrak g$ respectively. We have a ...
- 673
1
vote
0
answers
281
views
A question about decomposition of irreducible root system
Fix an irreducible root system $\Phi$ with rank $r$ and a root base $\Delta$ (we only care type ADE). Call a disjoint union $\Phi = \Phi_{1}\sqcup\dotsb\sqcup\Phi_{s}$ a decomposition of $\Phi$ if ...
- 93
7
votes
1
answer
262
views
A name for the Weyl group of $\frak{so_{2n}}$
For the $A$-series the Weyl group is the symmetric group $S_n$. For the $B$ and $C$ series the Weyl group is the hyperoctahedral group $\mathbb Z_2 \wr S_n$.
A) Does the $D$-series Weyl group $S_n \...
- 467
15
votes
6
answers
671
views
Why, conceptually, does the torus normalizer in $G_2$ split?
Background: Let $G$ be a complex reductive group, $T$ a maximal torus, $N$ the normalizer of $T$ in $G$, and $W = N/T$ the Weyl group. All in all, we have a group extension
$$ 1 \to T \to N \to W \to ...
- 835
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
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
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
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
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
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
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
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
0
votes
0
answers
42
views
Which finite group schemes over positive characteristic have projectivity detected by a set of (generalized) cyclic subgroups?
A finite algebra $B$ over a field $K$ has projectivity detected by a set $S = \{f_i : A \to B \mid i \in I\}$ of maps into $B$ from a fixed algebra $A$ if for any finite module $M$, $M$ is projective ...
- 542
0
votes
0
answers
69
views
A weakening of the definition of positive roots for a root system
Given a root system $\Delta$ a choice of positive/negative roots is a decomposition of the elements of $\Delta$ into two subsets $\Delta^+$ and $\Delta^-$ satisfying
$$\Delta^+ = - \Delta^-\tag{$*$}\...
- 467
4
votes
0
answers
79
views
Closed character formula for the module $L(a\omega_i)$
Let $\mathfrak{g}$ be a complex finite-dimensional simple Lie algebra with a fixed Cartan subalgebra $\mathfrak{h}$. Assume that $\omega_1, \omega_2, \dots, \omega_n\in\mathfrak{h}^{*}$ is the ...
- 41
2
votes
0
answers
34
views
Bounding norms of symplectic matrix factorisations and non-separable Hamiltonian flows
Problem setup: Let $e^{hJM}$ be the time-$h$ flow corresponding to the ODE $\dot{x} = JMx$, with $M = \left(\begin{array}{cc}
A & C\\
C^T & B\\
\end{array}\right)$ ...
- 21
0
votes
0
answers
61
views
Representation and Laplacian on the Heisenberg group
Let $\pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H=\mathbb C\times \mathbb R$. For $\varphi\in L^2(\mathbb R)$, we have
$$\pi_{\lambda} (z,t)\varphi(\xi)=e^{i\...
- 650
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
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 $\mathfrak{r}$, ...
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
2
votes
0
answers
48
views
Proof of redundancy for defining relation in current algebra $J$ presentation
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$. The $\mathfrak{g}$-current (Lie) algebra is $\mathfrak{g} \otimes \mathbb{C}[t]$, with Lie bracket given by $[a \otimes t^m, b \otimes t^n]...
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
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
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
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
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
$$
\mathcal{H}_m(\mathbb{R}^n)=\left\{P\in \mathbb{C}[x_1,\dotsc ,x_n]\left| \begin{align}
P\text{ is homogeneous of degree }m\text{ ...
- 605
3
votes
1
answer
85
views
Restriction of scalar commutes with taking maximal subtorus for semisimple group G
I was wondering such a question: for a semisimple complex Lie group $G$, whether it is true that the maximal subtorus of $\mathrm{Res}_{\mathbb{C}/\mathbb{R}}(G)$ is $\mathrm{Res}_{\mathbb{C}/\mathbb{...
- 455
5
votes
1
answer
304
views
Iwasawa decomposition of a non-compact semisimple Lie group?
A (non-compact) symmetric space is of the form $G/K$, where $G$ is a (non-compact) semisimple Lie group defined over $\mathbb{R}$, and $K$ is a maximal compact subgroup of $G$.
Let $M = G/K$ be a rank-...
- 650
3
votes
0
answers
196
views
A property of an irreducible root system
Let $\Phi$ be an irreducible root system. Let $\alpha_k$ be a simple root. I recently observed that the number of positive roots which are bigger than $\alpha_k$ and of height $m$ is same as the ...
- 673
5
votes
1
answer
344
views
Surjection onto endomorphisms of multiplicative group of a field
Let $k$ be an algebraically closed field of characteristic $p > 0$. Denote by $k^\times$ the multiplicative group of $k$. There is a ring homomorphism given by restriction to $k^\times$
$$
\mathbb{...
- 53
0
votes
0
answers
61
views
The theory of Groebner bases in Jordan case
There are many papers regarding spreading the theory of Groebner-Shirshov bases from Lie algebras to other nonassociative algebras. Also, it has been studied for associative algebras with operators ...
- 367
0
votes
0
answers
90
views
Some details about Kirillov-Kostant Poisson bracket
Let $G$ be a finite dimensional Lie group with Lie algebra $\mathfrak{g}$. The Kirillov-Kostant Poisson bracket on $\mathfrak{g}^*$ is defined as $$\{\cdot ,\cdot \} :C^{\infty}(\mathfrak{g}^*)\times ...
- 287
6
votes
1
answer
317
views
Which Lie groups are covers of matrix groups?
I would like to ask a variation on a question (not yet answered) I previously asked on math.SE, namely:
Which Lie groups are covers of matrix Lie groups? That is, which Lie groups $G$ admit discrete ...
- 3,115
-1
votes
1
answer
125
views
Killing form that is not diagonalizable?
An example Lie algebra $L$ with non-diagonalizable Killing form would have to be non-semisimple, and the Killing form complex. (Otherwise diagonalizability is obvious.) I tried with a few $L$, but (in ...
- 4,829
11
votes
0
answers
437
views
A rather strange algebra
Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector ...
- 3,318
6
votes
0
answers
96
views
Kac-Moody algebras of indefinite type
In his very influential book Infinite dimensional Lie algebras, which is still the main reference for Kac-Moody algebras, in section 0.4 of the introduction, Victor Kac discuss the problem of concrete ...
- 3,318
1
vote
1
answer
181
views
Relation between the root lattice of $\mathrm{SO}(7)$ and the root lattice of $G_2$
The root lattice of $\mathfrak{so}(7)$ is given by the following 18 roots:
$$
\left(\begin{array}{c}0\\0\\1\end{array}\right)
,
\left(\begin{array}{c}0\\0\\-1\end{array}\right)
,
\left(...
- 369
0
votes
0
answers
64
views
Proof of a folkloric result about PI-algebras [duplicate]
I am not not an specialist in PI-algebras, but I can say I have a rather good understanding on the subject.
It is, of course, interesting to discover if an algebra $A$ is a PI-algebra. But it is also ...
- 3,318
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
3
votes
0
answers
201
views
Theorem of highest weight of semisimple Lie algebras: what fails precisely for reductive case
Let $\mathfrak{g}$ a complex semisimple Lie algebra. It is well known that by the theorem of the highest weight, all finite-dimensional complex irreps of $\mathfrak{g}$ are (up to iso) classified by ...
- 5,986