All Questions
43 questions
27
votes
3
answers
3k
views
Is there a 'nice' interpretation of virtual representations?
Let $G$ be a compact group and let $R(G)$ be the representation ring of $G$. Additively, $R(G)$ is generated by the irreducible representations of $G$. Usually one only deals with those ...
- 5,191
12
votes
1
answer
2k
views
Relationship between the Witt algebra and vector fields on the circle
I have seen some (apparently contradictory) claims by mathematicians and physicists regarding the existence of an (infinite dimensional) Lie group whose Lie algebra is the Virasoro algebra.
The ...
- 1,329
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
254
views
A weight generalization of root systems?
For any simple complex Lie algebra $\frak{g}$, with a given choice of Cartan subalgebra $\frak{h}$, we have an associated root system $R \subseteq \frak{h}^*$. The properties of $R$ can be formalized ...
- 123
6
votes
2
answers
358
views
Duals of the spinor representations of $\frak{so}_{2n}$
For the $D_n$-series simple Lie algebra $\frak{so}_{2n}$
a curious phenomenon occurs for the fundamental representations corresponding to the spinor nodes of the Dynkin diagram, which is to say the ...
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 ...
- 327
5
votes
1
answer
320
views
Convention on Clifford Product [duplicate]
When studying the Clifford Algebra associated to some $(V,Q)$, one finds two basic identities differing by a sign:
$vv=Q(v)$ (see, for instance, Wikipedia)
$vv=-Q(v)$ (see, for instance, MathWorld or ...
- 2,091
5
votes
1
answer
560
views
What is the Schouten bracket for the Chevalley-Eilenberg complex with coefficients in a nontrivial module?
Let $\mathfrak g$ be a Lie algebra. The Chevalley-Eilenberg complex is defined to be $\wedge^* \mathfrak g$ with differential $d\colon \wedge^* \mathfrak g\to \wedge^{*-1}\mathfrak g$ defined by $$d(...
- 4,898
5
votes
0
answers
428
views
Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book
This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course.
Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
- 181
5
votes
0
answers
518
views
Generalized Haar Measures and Semiring-Valued Integrals on Lie Groups
In an applied research problem I am currently working on, I am using non-commutative semiring convolution to formulate some interesting types of calculations on images and solid objects. For discrete ...
- 2,392
4
votes
0
answers
534
views
Why are Lie Algebras/ Lie Groups so much like crossed modules, and not?
A crossed module consists of a pair of groups $G$ and $H$ with a group homomorphism, $t:H \rightarrow G$, and $\alpha: G \times H \rightarrow H$ that defines an action of $G$ on $H$, $\tilde{\alpha}$: ...
- 5,264
3
votes
2
answers
2k
views
Is there an almost-direct product decomposition for disconnected reductive algebraic groups?
$\textbf{Some definitions:}$
Let $G$ be an algebraic group (for me that is the complex points of an affine algebraic group). We say $G$ is reductive if its unipotent radical (maximal connected normal ...
- 397
3
votes
1
answer
355
views
The normalizer of SU(n) in U(m)?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$.
Below I specify a specfic way to embed $...
- 10.5k
3
votes
1
answer
472
views
Tips for how I can proceed with my Lie theoretical problem?
$\DeclareMathOperator\SL{SL}$I am looking at a map from a Lie group into a Lie algebra $\phi$:
$$\phi: \SL(n)\rightarrow \mathfrak{sl}_n$$
$$ P \rightarrow U_1^\dagger P U_1 + U_2^\dagger P U_2.$$
$P$ ...
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 ...
- 55
3
votes
1
answer
190
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 finite-...
- 159
3
votes
0
answers
143
views
Summing over roots of a simple Lie algebra and Deligne series
For a simple Lie algebra $\mathfrak{g}$ we can define a Killing form $K(X,Y) \equiv \frac{1}{2 h^\vee}\operatorname{tr}(\mathfrak{ad}_X \mathfrak{ad}_Y)$, where $\mathfrak{ad}_X Y \equiv [X, Y]$ as ...
- 857
3
votes
0
answers
244
views
On the Gelfand-Kirillov Conjecture
The base field $k$ is of zero characteristic.
Notation: $A_{n,s}(k):= A_n(k(x_1,\ldots,x_s))$, the Weyl agebra over a purely transcedental extension of the base field; $F_{n,s}(k)$, the Weyl field, is ...
- 3,318
3
votes
0
answers
177
views
Lie algebra cohomology of loop algebra
Let $G$ be a simple algebraic group over the complex numbers. Is it true that the Lie algebra cohomology $H^*(L\mathfrak{g}, \mathbb{C})$ of the loop Lie algebra $L\mathfrak{g}=\mathfrak{g} \otimes \...
- 3,019
3
votes
0
answers
73
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 [0,...
- 2,099
3
votes
0
answers
126
views
Irreducible representations in BGG category $\mathcal{O}$ over (finitely) direct sum of general linear Lie superalgebra
Let $\mathfrak{g} = \oplus_i^k\mathfrak{gl}(m_i|n_i)$ be a direct sum of general linear Lie superalgebras $\mathfrak{gl}(m_i|n_i)$'s with the Cartan subalgebra $\mathfrak{h} = \oplus_i^k \mathfrak{h}...
- 159
3
votes
0
answers
501
views
Some counter examples in group theory
In this question, which we flag it as a community wiki question, we search for a big list of groups $G$ which can not be isomorphic to a structure mentioned in $i.$ for some $i \in \{1,2,\ldots,...
3
votes
0
answers
307
views
Construction of an algebra with prescribed representation of the automorphism group.
For this discussion, $G$ is a compact semisimple Lie Group.
For many of the common representations of compact groups, there is a realization of the representation as the automorphisms of some ...
- 5,191
2
votes
3
answers
318
views
Realization of irreducible $\mathfrak{S}_d$-modules and the representation theory of Lie algebra
Let $n$ be a positive integer. It is well-known that a method to realize irreducible $\mathfrak{S}_d$-modules is to construct the so-called Specht modules $S^{\mu}$ which are submodules in the so-...
- 159
2
votes
2
answers
755
views
Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?
Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that $\mathrm{Aut}(\mathbb{Z}^n)\...
- 33
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
2
votes
1
answer
196
views
Weight space dimension of the fundamental representation $\pi_n$ for type $C_n$
Will the fundamental representation $\pi_n$ of type $C_n$, for $n > 3$, have weight spaces of dimension greater than $1$? Is there some online resource where weight space multiplicities can be ...
2
votes
1
answer
123
views
Polar decomposition with respect to the nonstandard involution of quaternionic matrices?
The quaternions admit infinitely many involutions. But up to isomorphism, there are only two: The standard one $t+xi+yj+zk\mapsto t-xi-yj-zk$ and the nonstandard one $\phi:t+xi+yj+zk\mapsto t-xi+yj+zk$...
- 4,943
2
votes
0
answers
90
views
Lie Groups and Lie algebras related to Jordan algebras
Let $J$ be a Jordan algebra. I knew three relative Lie groups/Lie algebras to $J$.
In the paper "The Capelli Identity, Tube Domains, and the Generalized Laplace Transform"
Jacobson [J] has ...
- 719
2
votes
0
answers
81
views
Jordan decomposition for simple Lie algebra
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let us fix a Cartan, a Borel, and generators $x_\alpha$ of negative simple roots. Then $N:=\sum x_\alpha$
is a principal (=regular) nilpotent ...
- 2,751
2
votes
0
answers
81
views
The centralizer and normalizer of products of (SU(n) $\times$ SU(p) $\times$ …) in U(m)
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}
$Consider the special unitary group $\SU(n)$ and the unitary group $\U(m)$.
Below I specify a specfic way to embed $...
- 10.5k
2
votes
0
answers
111
views
The centralizer and normalizer of products of (Spin(n) $\times \dots$) in U(m)
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$
Consider the spin group $\Spin(n)$ and the unitary group $\U(16)$.
Below I specify a specfic way to embed $(\Spin(...
- 10.5k
2
votes
0
answers
61
views
Generalizing polynomial identities for rings
For a ring $R$, a polynomial identity of $R$ is a polynomial (in non-commuting variables) $f(x_1,\ldots,x_n)\in \mathbb{Z}[x_1,\ldots, x_n]$ such that for any choice of $a_i\in R$, $f(a_1,\ldots, a_n)=...
- 21
2
votes
0
answers
127
views
Multiplicative subgroups of $GL(V)$ which are almost additively closed
Edit:
According to comments of YCor and Vincent, I revise the question.I appreciate their comments:
Let $G$ be a group.
We say that $G$ is a special group if it is isomorphic to a subgroup $H$ ...
- 356
2
votes
0
answers
207
views
Does the "diagonal map" of the universal algebra of a Lie algebra depend on its association with the Lie algebra?
This is another naive question that popped up as I'm learning Lie theory.
Given a lie algebra $L$ over a field $k$ of characteristic 0, we can define its universal algebra $UL$ as a suitable quotient ...
- 10.7k
2
votes
0
answers
111
views
A question about the associative classes of parabolic subgroups
Let $\mathbb{A}$ denote the adele group of $\mathbb{Q}$. Suppose that $P_1$ and $P_2$ are parabolic subgroups of a reductive algebraic group $G$, and consider their Langlands decomposition
$$
P(\...
- 21
1
vote
1
answer
275
views
The normalizer of $\operatorname{Spin}(2N)$ in $\operatorname{U}(2^{N-1})$?
$\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$ I can show that
$$
\U(2^{N-1})\supset \Spin(2N)
$$
when $2N > 4$ or a positive integer $N > 2$, so $\Spin(2N)$ can be embedded in $\U(2^...
- 10.5k
1
vote
1
answer
187
views
Is a finite dimensional graded algebra isomorphic to the equivariant de Rham complex of a Lie group?
Edit: According to essential comment of YCore I revise the question.
Let $A$ be a finite dimensional graded algebra which is a unital, super commutative and associative algebra. Is there a Lie group $...
- 356
1
vote
1
answer
314
views
A nice/simple relationship between the Chevalley generators of $\mathfrak{sp}_n$ and the Chevally generators of $\mathfrak{sl}_n$?
The Lie algebra $\mathfrak{sl}_n$ is defined to be the trace free matrices in $M_n(\mathbb{C})$. The Lie algebra $\mathfrak{so}_n$ is defined to be the matrices $A$ in $M_n(\mathbb{R})$ satisfying $A +...
- 1,144
1
vote
1
answer
106
views
Cosemisimple pointed Hopf algebras
Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Every cosemisimple pointed Hopf $\mathbb{K}$-algebra $A$ is easily seen to be cocommutative. Does this imply that $A$ is the ...
- 1,479
1
vote
0
answers
92
views
The $U({\frak g})v$-module generated by a single element of a $U({\frak g})v$-module
Let $\frak{g}$ be a finite dimensional complex semisimple Lie algebra and let $U(\frak{g})$ be its universal enveloping algebra. Take $V$ an infinite dimensional module over $U(\frak{g})$. Let $v \in ...
1
vote
0
answers
102
views
On uniqueness of mapping preserving triality
I'm reading Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society 39 (2): 145–205 regarding $\mathrm{Spin}(8)$ and trialities. My question is about this sentence:
The ...
- 11
0
votes
0
answers
228
views
Coordinate ring of a flag variety
Edited:
[If G here is a simply connected semismple complex algebraic group.
A partial flag variety $G/P$ can be naturally embedded as a closed subset of $\prod_j \mathbb{P} (L(\omega_j)^*)$.
The ...
- 201