All Questions
2,633 questions
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
2
votes
1
answer
358
views
Tensor product of fundamental representations
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $V_1,\cdots, V_n$ be the fundamental representations (the irreducible ones with fundamental weights $\omega_1,\cdots,\omega_n$). Take a $k$-...
- 391
4
votes
1
answer
284
views
Double centralizer theorem for ($\ell$-adic) Lie algebras
$\DeclareMathOperator\GL{GL}$I'm reading the proof of Serre's open image theorem from his book "Abelian $\ell$-adic representations and elliptic curves". This is chapter IV Section 2.2 of ...
- 63.9k
0
votes
0
answers
73
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Problem in understanding the proof of cocycle condition for cocommutator
Let $G$ be a Poisson–Lie group with Poisson bivector field $\pi$. Let $\pi^{R} \colon G \longrightarrow \bigwedge^2 \mathfrak{g}$ be defined by $$\pi^R (x) = (d_x R_{x^{-1}} \otimes d_x R_{x^{-1}}) \...
- 199
1
vote
0
answers
101
views
Singular vectors/null states in algebra $\mathfrak{su}(2)_{-4/3}$
I encounter recently admissible affine Lie algebras when visiting some physics problems. I am reading Adamovic's A construction of admissible $A_1^{(1)}$-modules of level $−4/3$.
In section 3, it is ...
- 12.9k
6
votes
1
answer
160
views
Lie coalgebra with no finite-dimensional subcoalgebras
In Walter Michaelis' paper Lie Coalgebras, he gives on page 9 an explicit example of a Lie coalgebra which is not the union of its finite-dimensional Lie subcoalgebras. In fact, Michaelis' example has ...
- 12.9k
15
votes
1
answer
597
views
Proof for which primes H*G has torsion
In 1960 Borel proved a beautiful result:
**Theorem**. Let G be a simple, simply connected Lie group. Suppose that *p* is a prime that does not divide any of the coefficients of the highest root (...
- 2,821
5
votes
2
answers
735
views
Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?
Let $(M,g)$ be a (connected, paracompact, $C^{\infty}$-smooth) Riemannian manifold with Riemannian metric $g$. The exponential map is defined for each point $p \in M$ to be the map $\exp_p : T_p M \to ...
CommunityBot
- 1
1
vote
0
answers
179
views
Relationship between singular values, traces and Hermitian conjugate
I am working on a following problem in my free time (which is a simplified version of a problem described here - arxiv.org/abs/0711.2613):
Let $A$, $B$ be zero-trace $4 \times 4$ matrices that meet ...
2
votes
0
answers
90
views
Suzuki-Ree Lie algebras
Do the Suzuki and Ree groups of Lie type have associated Lie algebras over finite fields in the same way that the other groups of Lie type do? These algebras would be 5-dimensional over $\mathbb{F}_{2^...
- 2,773
2
votes
0
answers
145
views
How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?
It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
- 63.9k
3
votes
0
answers
127
views
Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$
Let $K$ be quadratic imaginary field. Let $E$ be an elliptic curve which has CM over $R_K$
($R_K$ is ring of integers of $K$).
According to SIlverman's ''ADvanced topics in the arithmetic of elliptic ...
- 1,541
10
votes
1
answer
399
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Basic algebra of $\mathcal{O}_0(\mathfrak{sl}_n(\mathbb{C}))$ — Reference request
It is well known that the principal block $\mathcal{O}_0$ of the BGG category $\mathcal{O}$ of a semisimple Lie algebra is equivalent to the category of finitely generated modules over a certain ...
- 26.5k
2
votes
0
answers
47
views
Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
- 63.9k
3
votes
1
answer
1k
views
Highest weight orbit characterization (reformulated and extended)
Edit 1: I think that the question was not stated clearly enough so modified it a little.
Edit 2:
I thought over the physics that lies behind this question which led me to reformulation of the original ...
- 12.9k
11
votes
1
answer
1k
views
Baker–Campbell–Hausdorff formula: prime divisors of denominators
Consider the Baker–Campbell–Hausdorff formula (Wikipedia page):
$$Z(X,Y) := X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] + \dotsb$$
Many sources, including the Wikipedia ...
- 4,713
5
votes
3
answers
2k
views
Complete classification of six dimensional non-semi simple Lie algebra
I would aim to know the complete classification of 6 dimensional non-semi simple Lie algebra (here the dimension stands for the generators; or the dimension $\leq 6$).
In this paper, in page 7, it ...
- 12.1k
3
votes
1
answer
203
views
Free $S^1$-action on compact homogeneous spaces
Let $M = G/K$ be a compact homogeneous space, i.e. $G$ a connected compact Lie group, and $K$ a closed subgroup inside $G$ that contains no nontrivial normal subgroup of $G$.
If $r(G) > r(K)$ (...
- 9,580
4
votes
0
answers
111
views
How many diagrams interlace a given Young diagram?
For a fixed partition $\lambda=(\lambda_1\geq\dots\geq \lambda_n)$ we say $\mu=(\mu_1\geq \dots \geq \mu_{n-1})$ $\textit{interlaces}$ $\lambda$ iff
$$\lambda_1\geq \mu_1\geq \dots \geq \mu_{n-1}\geq \...
1
vote
0
answers
71
views
Coxeter-Dynkin diagrams resulting in Lie-ing algebras
We know that certain structures in a Coxeter-Dynkin diagram will guarantee that it never encodes a (semi)simple Lie algebra, for example a quadruple node or the diagram $E_n,n>8$, which is "...
- 4,829
4
votes
0
answers
64
views
An analog of a BGG resolution in subregular case in positive characteristic
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$. For every regular weight $\lambda$ of $\mathfrak{sl}_n$, we have the ...
1
vote
1
answer
136
views
Two (or less) filtrations on coenveloping coalgebra
Conilpotent coenveloping coalgebra UC(T) of a conilpotent Lie coalgebra T is defined by an universal property, similar to usual enveloping algebra: it's a coassocative, conilpotent coalgebra UC(T) ...
- 1,790
12
votes
4
answers
1k
views
Real and quaternionic representations according to weights
According to this question, it is easy to know whether a (complex, finite-dimensional) representation is self-dual or not: just check if the weight distribution in space is symmetric about the origin.
...
- 630
1
vote
0
answers
64
views
Is the Lie bracket on $\mathfrak g^{\ast}$ induced from a cocommutator defined on $\mathfrak g\ $?
Let $G$ be a Poisson-Lie group. Let $\mathfrak g = \text {Lie} (G) = T_1 G$ be the corresponding Lie algebra. Then the Poisson structure on $G$ gives rise to a Lie bracket $[\cdot, \cdot]$ on $\...
- 199
3
votes
0
answers
120
views
Resolving the "wild" singularities of $\mathbb A^n/C_n$
Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
- 7,746
4
votes
0
answers
72
views
Real forms of complex Lie algebras with specified semisimple part
Let $\mathfrak g$ be a complex Lie algebra. If $\mathfrak g$ is semisimple then its real forms are completely understood. In particular there is a compact real form of $\mathfrak g$, which is unique ...
- 4,729
4
votes
0
answers
161
views
Differential invariants and Lie symmetries
I have the following question:
Does differential invariants have the same Lie symmetries?
I want to know about the relation of differential invariants of an action and their symmetry properties. ...
- 49
1
vote
1
answer
565
views
Kronecker product: Is it possible to simplify this product $e^{-A} \otimes e^{A}$ where $A$ is an invertible and symmetric matrix [closed]
Let $A$ be an invertible, symmetric and tridiagonal matrix of size $n \times n$. Assume that $A_{i,i}=a \neq 0$ for $i=1\dotsc n$ and all the elements in the sub- and super-diagonal of $A$ are $b \neq ...
- 20.2k
1
vote
1
answer
149
views
When is $R$ a direct summand of Frobenius pushforwards?
Let $(R,\mathfrak m)$ be a reduced Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e_* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module ...
- 20.5k
2
votes
0
answers
177
views
How do characters of representations in cohomology depend on the (positive-characteristic) field?
The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
- 12.9k
3
votes
3
answers
1k
views
Lie algebra bundle associated to a Lie group bundle
I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.
I am not comfortable with these notions and google ...
- 35.5k
5
votes
1
answer
398
views
Asymptotically nilpotent matrices
A matrix $A\in\textbf{Mat}_n(\mathbb{R})$ is called asymptotically nilpotent if for each vector $v$, ${\lim}_{k\to\infty}A^k(v) = 0$. Assume that $\mathcal{A}, \mathcal{B}$ be maximal (under ...
- 291
1
vote
1
answer
525
views
Centralizer of an element in a matrix Lie group whose Jordan form is given
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}$Let $G\subset \GL_n(\mathbb{C})$ be a complex matrix Lie group, ...
CommunityBot
- 1
5
votes
1
answer
311
views
Non-standard partial orders on root systems
Let $\frak{g}$ be a semisimple complex Lie algebra and let $\Delta$ be its associated root system with $\{\alpha_1, \dotsc, \alpha_l\}$ a choice of positive roots. As we all know - $\Delta$ admits a ...
- 156k
2
votes
0
answers
50
views
A construction of Weyl-equivariant maps from the space of regular Cartan triples to the space of tuples of complex polynomials (up to scalar factors)
Let $G$ be a compact semisimple Lie group and let $T$ be a maximal torus in $G$. On the Lie algebra level, we have a real Lie algebra $\mathfrak{g}$ and a (particular) real slice, say $\mathfrak{t}$, ...
- 5,215
4
votes
1
answer
349
views
Verma modules and Borel–Weil
Let $\mathfrak{g}$ be a semisimple Lie algebra and fix a root system. Let $\mathfrak{b}:=\mathfrak{h}\oplus\bigoplus_{\alpha\in R^+}\mathfrak{g}_\alpha$. The complex irreducible representation of $\...
- 27.9k
5
votes
2
answers
479
views
How to define cohomology of algebraic structures?
I learned that the Hochschild cohomology of an associative algebra $A$ with a bimodule $M$ is defined using the cochain
\begin{align*}
\cdots \rightarrow C^n(A,M) \stackrel{d^n}{\longrightarrow} C^{n+...
- 12.9k
2
votes
0
answers
89
views
Is there a character formula over Cartan triples?
Let $G$ be a compact semisimple Lie group, with $\mathfrak{g}$ its (real) Lie algebra. We also fix a choice of maximal torus $T$ in $G$, with corresponding Lie algebra $\mathfrak{t}$, itself a Lie ...
- 5,215
10
votes
0
answers
294
views
Each simple real Lie algebra as a representation of its maximal compact subalgebra
I am interested in a detailed description of the Cartan decomposition of each type of simple, real, finite-dimensional Lie algebra. (This is essentially a question about the classification of simple, ...
- 56.6k
54
votes
9
answers
9k
views
Nice proofs of the Poincaré–Birkhoff–Witt theorem
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $k$, with an ordered basis $x_1 < x_2 < ... < x_n$.
We define the universal enveloping algebra $U(\mathfrak{g})$ of $\...
- 30.3k
7
votes
0
answers
144
views
Learning roadmap for admissible representations of $\widehat{\mathfrak{g}}$ (affine Lie algebras)
Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over $\mathbf{C}$. A priori one might expect the representation theory of the affine Lie algebra $\widehat{\mathfrak{g}}$ (the Lie ...
- 5,711
2
votes
1
answer
98
views
3-dimensional principal subalgebra in the real case
I have a question on principal 3-dimensional subalgebras. I know such concept in the complex case, but I'm not sure of the definition in the real case. Also, if the definition of the principal 3-...
- 954
2
votes
2
answers
354
views
Particular reduced expression of the longest element of Weyl group
Let $I$ be the Dynkin diagram vertex set and $K$ be a proper nonempty subset of it. Let $w_0^K$ be the longest word of the Dynkin subdiagram $K$, which might be a disjoint union of connected Dynkin ...
- 12.9k
4
votes
1
answer
227
views
Compute de Rham-Witt sheaves
I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction.
It seems to be really difficult to compute anything with those beasts: like I cannot find any ...
0
votes
0
answers
49
views
Complemented subalgebra in a free Lie ring
A Lie ring is a triple $(G,+, [\ ,\ ]),$ where $(G,+)$ is an abelian group and $ [\ ,\ ]$ is a bilinear map satisfying
$[x,x]=0$
$[\ ,\ ]$ is bilinear
$[[x,y],z]+[[y,z],x]+[[z.x],y]=0,\ \forall\ x,...
- 101
0
votes
0
answers
174
views
Young tableaux — irreps correspondence for simple complex Lie algebras
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$I have learned that Young tableaux which were originally
introduced to study the irreducible representations of finite
symmetric groups $S_n$ ...
- 63.9k
2
votes
1
answer
220
views
Are finite-dimensional real representations of semisimple real Lie algebras completely reducible?
Suppose $\mathfrak{g}$ is a real form of a semisimple Lie algebra $\mathfrak{g}_\mathbb{C} = \mathfrak{g} \otimes_\mathbb{R} \mathbb{C}$. Then we have the following:
There is an equivalence of ...
- 618
1
vote
0
answers
75
views
Problem in understanding Theorem $6.2.9$ from Chari and Pressley
The theorem I am referring to here says that if we start with a Lie bialgebra $\mathfrak g$ determined by some skew-symmetric element $r \in \mathfrak g \otimes \mathfrak g$ satisfying classical Yang-...
- 199
3
votes
1
answer
403
views
A few reference questions about the Baker–Campbell–Hausdorff formula
I'm looking at the article Baker–Campbell–Hausdorff formula - Wikipedia and I have a few questions.
Under the "Special cases" section, there is a notation $\DeclareMathOperator{\ad}{ad}$
$$ \...
- 1,446
1
vote
0
answers
42
views
Do the Tits-Kantor-Koecher construction extend to non unital Jordan algebras?
I have an infinite dimensional Jordan algebra which is not-unital. I would like to study a Lie algebra related to it (if it exists). Natural choice would be to look at the KKT construction, but ...
- 295