All Questions
298 questions
5
votes
1
answer
256
views
Definition of a Dirac operator
So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as:
$D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...
- 63.9k
4
votes
2
answers
422
views
GAP versus SageMath for branching to Lie subgroups
Which computer package is better, GAP or SageMath, for
decomposing an irreducible representation of a (simple) Lie group
$G$ into representations of a Lie subgroup. I am most interested when
...
- 2,449
4
votes
1
answer
202
views
Branching to Levi subgroups in SAGE and the circle action
In the SAGE computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup:
http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/...
- 8,597
8
votes
2
answers
2k
views
Does $SU(N)$ have pseudo-real representation?
For $N\ge 2$, does $SU(N)$ have a non-real pseudo-real irreducible representation? (The adjoint representation of $SU(N)$ is real).
A (complex, finite-dimensional) representation $R:SU(N)\to GL_n(\...
- 14.2k
9
votes
3
answers
576
views
Reference Request: Structure constants for G2
Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, ...
- 6,180
2
votes
1
answer
93
views
Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight
Show that if $\Theta$ is an infinitesimal weight of a real $T$-module $W$ ($T$ is a torus) then $-\Theta$ is also a weight.
It is an exercise of Bröcker's book on Representations of Compact Lie ...
- 26.3k
2
votes
2
answers
1k
views
Reductive Lie algebra of a Lie group
In the answer of my question:
On the full reducibility of representations of reductive Lie algebras
James E. Humphreys replied to me saying that:"the notion of "reductive" for a Lie algebra in ...
- 44.7k
9
votes
0
answers
161
views
Can semisimple orbits be written $\exp(\mathfrak{g})\cdot x$?
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $G$ be its adjoint group. If $x\in\mathfrak{g}^{rs}$ is a regular semisimple element, is its orbit
$$G\cdot x=\{\mathrm{Ad}_gx:g\in G\}$$
...
- 1,383
8
votes
2
answers
436
views
Infinite Krull-Schmidt categories?
In a Krull--Schmidt category, if
$$
X_{1}\oplus X_{2}\oplus \cdots \oplus X_{r}\cong Y_{1}\oplus Y_{2}\oplus \cdots \oplus Y_{s},
$$
where the $X_{i}$ and $Y_j$ are all indecomposable, then $r = s$, ...
- 35.2k
9
votes
1
answer
543
views
Polynomial invariants for simple algebraic groups
Let $G$ be a simple complex algebraic group. Let $V$ be a finite-dimensional algebraic representation of $G$. Thus, we can write $V=V_1\oplus \cdots \oplus V_n$ where $V_i$'s are irreducible ...
- 12.9k
8
votes
0
answers
382
views
Significance of half sum of non-simple positive roots
In representation theory, there are plenty of places that a $\rho$-shift makes an appearance, where $\rho$ is the half sum of positive roots. See, for instance, this post for some discussions of the ...
- 12.9k
9
votes
2
answers
2k
views
Fundamental representations and weight space dimension
For the Lie algebra $\frak{sl}_n$, its fundamental representations can be realised as the exterior powers of the first fundamental representation. From this we can see that their weight spaces are all ...
- 8,597
2
votes
0
answers
59
views
Finite-dimensional graded Lie algebras with $2$ generators
Does anyone know of a classification of those (complex) Lie algebras which are:
generated by two elements
$\mathbb{Z}$-graded Lie algebras
finite dimensional
- 3,574
8
votes
0
answers
411
views
Which representations of the Lie algebra of a Lie group come from representations of the group itself?
I think this is very classic mathematics, but I can't find a complete answer in the literature.
Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \...
- 1,586
2
votes
1
answer
271
views
Discrete decomposability of unitary representation
[INTRODUCTION]
Let $G$ be a non-compact simple Lie group, and $G'$ a reductive subgroup of $G$. Suppose that $\pi$ is a non-trivial (hence, infinite dimensional) irreducible unitary representation of ...
- 63.9k
5
votes
0
answers
137
views
Differential operators on $G/K$
Let $G$ be a connected Lie group and $K$ a compact subgroug of $G$. The question is about the algebra of the differential operators $Diff(G/K)$ on $G/K.$ Let $U(\mathfrak g)$ denote the universal ...
- 574
60
votes
8
answers
13k
views
Why the Killing form?
I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...
- 23.3k
4
votes
1
answer
288
views
The embedding of $\mathfrak{g}_2$ into $\mathfrak{b}_3$ ($\mathfrak{so}_7$) on Chevalley generators
Let $\mathfrak{g}_2$ / $\mathfrak{b}_3$ be the simple complex Lie algebra of type $\mathsf{G}_2$ / $\mathsf{B}_3$ (the latter is also known as $\mathfrak{so}_7$).
How is the embedding $\mathfrak{g}...
- 1,066
3
votes
0
answers
106
views
Induced $(\mathfrak{g},K)$-modules
Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$. Fix a maximal compact subgroup $K$ of $G$ such that the intersection $K'=K\cap G'$ is a maximal compact ...
- 951
3
votes
1
answer
211
views
Coinduced modules in the BGG category $\mathcal O$ over complex semisimple Lie algebras
For a given finite-dimensional complex semisimple Lie algebera $\mathfrak g$, we fix Cartan $\mathfrak h$ and Borel subalgebras $\mathfrak b$, then we have the BGG category $\mathcal O$. As usual, we ...
- 52.9k
6
votes
1
answer
169
views
Existence of a real eigenvalue is a necessary condition for the density of all the orbits of a Lie subgroup of $GL(\mathbb{R},d)$
Good morning,
I would like to pose the following (maybe naive) question. Let $\mathfrak{a}\subset \mathfrak{gl}(\mathbb{R},d)$ be any lie subalgebra, and $A$ be the connected, simply connected ...
- 63.9k
38
votes
18
answers
24k
views
Learning about Lie groups
Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.
- 558
19
votes
5
answers
4k
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 ...
- 1,991
4
votes
2
answers
350
views
Generating Irreducible representations of a simple lie algebra with Schur functors
Let $\mathfrak{g}$ be a simple lie algebra over $\mathbb{C}$. Let $Rep(\mathfrak{g})$ denote the category of finite dimensional $\mathfrak{g}$-modules. For every $V \in Rep(\mathfrak{g})$ define $...
- 5,039
5
votes
0
answers
126
views
Modular $S$-matrix for an extended affine Lie algebra
This is a refinement of this old question of mine. In order to find an answer, I've been working my way through q-alg/9511026, which contains all the information I need.
In this paper, the authors ...
4
votes
0
answers
91
views
Good range and fair range
Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable ...
- 951
9
votes
1
answer
444
views
Young tableaux for exceptional Lie algebras
Irreducible representations for the $A$-series Lie algebras are labelled Young diagrams, with a basis of each given by Young tableaux. Moreover, analogues exist for the $B,C$, and $D$ series.
Does ...
- 16.5k
1
vote
1
answer
159
views
Abstracting the properties of the category $\frak{g}$-modules
Given a semisimple Lie algebra $\frak{g}$ over $\mathbb{C}$, and a finite dimensional irreducible representation $V$, with dual representation $V^*$, we know that the decomposition of $V \otimes V^*$ ...
- 8,524
21
votes
2
answers
727
views
On a drawing in Dixmier's Enveloping Algebras
This image
comes from Dixmier's book, 'Enveloping Algebras' ('Algèbres enveloppantes').
Dixmier writes that
The curves shown on p. XIV have their origin in the study of U(sl(3)).
They are due ...
- 410
1
vote
1
answer
128
views
Diagonalisation of invariant hermitian forms and irreducible representations of tori actions
here is my question:
Suppose that the torus $T^n = (S^1)^n$ acts on $\mathbb{C}^n$ by linear transformations $$(e^{i \theta_1},...,e^{i \theta_n}).(z_1,...,z_n) = (e^{i \theta_1}.z_1,...,e^{i \...
- 14.2k
2
votes
1
answer
261
views
Representation of a Lie algebra from a representation of Lie group
Let $\pi$ be a unitary representation of a Lie group $G$ on a Hilbert space $H_{\pi}$. Thus $\pi:G \to B(H_{\pi})$. One can extend $\pi$ to a representation of $L^1(G)$ via the formula
$\pi(f)=\int_{...
- 23k
4
votes
1
answer
319
views
How to write down the connection morphism in the long exact sequence in Čech cohomology explicitly in this specific case?
Fix an integer $k$. Let $X=G/P$ be a complex rational homogeneous variety. I assume here $G$ is a simply connected semi simple complex Lie group and $P=P_k$ is a maximal parabolic subgroup defined by ...
- 1,810
2
votes
1
answer
358
views
Is the Adjoint Action self dual over finite fields?
Given a finite group $G$, and representation $\rho: G \to H(\mathbb{F}_q)$ where $H$ is some classical algebraic group ($Gl$, $Sl$, $O$, $SO$, $SP$, $GSP$, $U$, etc), is the induced Adjoint ...
- 52.9k
7
votes
2
answers
668
views
Branching laws for $SO(n)$
The branching laws for the $SO(n-1)$ as a subgroup of $SO(n)$ are well known and easy to find. See for example the Wikipedia article:
https://en.wikipedia.org/wiki/Restricted_representation#...
- 574
6
votes
1
answer
376
views
Does $SU(2)\cong Sp(1)\subset SO(5)$ factor through $Spin(5)\cong Sp(2)$ as the standard embedding $Sp(1) \to Sp(2)$?
$SU(2)$ can be seen as a subgroup of $SO(5)$ through the following chain of subgroups
$$
SU(2) \subset SO(4) \subset SO(5).
$$
If we identify $SU(2)\cong Sp(1)$, does the inclusion $Sp(1) \to SO(5)$ ...
- 44.7k
2
votes
1
answer
320
views
Gelfand pairs and (self)-dual representations
For a Lie group $G$ with compact Lie subgroup $K$, we say that $(G,K)$ is a pair of Gelfand type if the representation $L^2(G/K)$ of $G$ is multiplicity free, that is, if it is a direct integral of ...
- 9,486
1
vote
1
answer
93
views
How to show that the structure constant on $\mathcal{G}^*$ is $C_{c}^{ab} = f_{cd}^b r^{ad} + f_{cd}^a r^{db}$?
Let $(\mathcal{G}, \mathcal{G}^*, \delta)$ be a Lie bialgebra. Suppose that the structure constant on $\mathcal{G}^*$ and $\mathcal{G}$ are
\begin{align}
& [t^a, t^b]_* = C_c^{ab} t_c, \\
& [...
- 422
1
vote
1
answer
162
views
How to compute the index of a given weight?
I am learning the Borel–Weil–Bott theorem in rational homogeneous varieties. Working on them, I need to know two questions:
How to judge if a weight is singular?
How to compute the index of a given ...
CommunityBot
- 1
10
votes
4
answers
2k
views
Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$
One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,
$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$
the ring of ...
9
votes
0
answers
470
views
Branching rules for compact Lie groups
Let $G$ be a compact connected Lie group, and let $H\subset G$ be a closed subgroup. For an irreducible representation $\pi:G\to\mathrm{End}_\mathbb{C}(V)$ of $G$ ($\dim\pi<\infty$) I want to know ...
- 1,959
3
votes
0
answers
111
views
Simple $\mathfrak{g}$-modules preserved by twisting
Let $G$ be a semi-simple lie group (simply connected for simplicity), $\mathfrak{g}$ its lie algebra. Write $\overline{G}=Inn(\mathfrak{g})$ for the adjoint form of $G$ which we identify here with ...
- 1,077
53
votes
5
answers
8k
views
Beautiful descriptions of exceptional groups
I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
- 12.9k
8
votes
1
answer
3k
views
The Quotients $SO(n)/SO(n-1)$, $O(n)/O(n-1)$ and $SO(n)/O(n−1)$
The branching laws for the restricted representation of $SO(n)$ with respect to the subgroup $SO(n-1)$ are discussed in this Wikipedia article. Am I correct in reading from this that any given ...
- 6,349
7
votes
0
answers
166
views
"Non standard" formulas for eigenspaces in $V_\rho$
In the context of the Simple Lie Algebras Representations, let $\rho$ be half-the-sum of the positive roots and let $V_\rho$ be the irreducible representation of highest weight $\rho$.
Let$\mu$ be a ...
- 233
2
votes
1
answer
90
views
Plancherel expansion for Spin(n-1,1)
I am interested in the principal series (unitary irreducible) representations of $Spin(n-1,1)$, and in the generalized Pancherel's formula for the delta function on the group in terms of a sum (and an ...
- 31.5k
15
votes
2
answers
416
views
Constructing $E_8$ from its branching to $A_8$
Background/motivation: One of the usual constructions of [the adjoint representation of] the $E_8$ exceptional Lie group (found, e.g., in J. F. Adams's, "Lectures on Exceptional Lie Groups", esp. chap....
- 108k
11
votes
0
answers
432
views
Connection between Gelfand-Tsetlin basis and SSYT basis in Schur module
Consider an $n$-dimensional complex vector space $V$ with a chosen basis $e_1,\ldots,e_n$. This basis defines a Cartan decompostion of $GL(V)\cong GL_n$ and for an (integral dominant) highest weight $\...
- 3,513
3
votes
1
answer
190
views
Critical points of characters on semisimple groups
Let $G$ be a semisimple connected complex Lie group, compact real Lie group or linear algebraic group. Let $\chi$ be the character of a finite dimensional irreducible representation of $G$ (I am ...
CommunityBot
- 1
2
votes
1
answer
143
views
Trivial representation of $SL_n$ in $V(\alpha_1)\otimes \cdots \otimes V(\alpha_m)$
Suppose $\alpha_1,\cdots, \alpha_m$ are partitions of lenght at most $n$ with $c = \frac{1}{n}\sum_s\sum_i \alpha_s(i)$ an integer. If the representation of $SL_n$ $V(\alpha_1)\otimes \cdots \otimes V(...
- 291
2
votes
0
answers
573
views
Clebsch–Gordan(CG) coefficients for SO(N) and Sp(N) group
I know how to calculate the CG coefficients for $SU(N)$, but there are other simple Lie group like $SO(N)$ and $Sp(N)$. But up to now I can't find any textbook tells me how to calculate these and I ...
- 249