All Questions
2,633 questions
2
votes
0
answers
105
views
Embeddings of symplectic group into the orthogonal group
Let $\mathfrak{sp}$ denote the complex symplectic Lie algebra and $\mathfrak{so}$ the complex orthogonal one. Do we have an embedding
$$
\mathfrak{sp}_{2n-2} \hookrightarrow \mathfrak{so}_{2n}?
$$
In ...
- 2,751
3
votes
0
answers
112
views
What are the possibilities of the general fibres in an Iitaka fibration?
This question is motivated by complex algebraic geometry.
If $X$ is a complex algebraic variety with Kodaira dimension in $[1,\dim X-1]$, then the Iitaka fibration (the rational map induced by the ...
- 9,501
4
votes
0
answers
120
views
Why are all "non-swinging" representations self-dual?
Let $\mathfrak{g}$ be a semisimple (say complex) Lie algebra, and $V$ an irreducible finite-dimensional representation of $\mathfrak{g}$. Denote by $w_0$ the longest element of the Weyl group, i.e. ...
- 1,574
1
vote
0
answers
214
views
Lie algebra cohomology: Künneth formula for semidirect product?
In trying to understand an example of Lie algebra for which every one-dimensional extension splits (see this question), but not every extension splits, I found it necessary to use the following ...
- 1,225
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 $\DeclareMathOperator{\ad}{ad}$
$$ \...
- 217
3
votes
0
answers
111
views
Modules over the unipotent subalgebra as direct summands of modules over a semisimple Lie algebra
Let $\mathfrak g$ be a semisimple finite-dimensional Lie algebra over the field of complex numbers $\mathbb C$. Let $\mathfrak n\subset\mathfrak g$ be the maximal unipotent subalgebra of $\mathfrak g$...
- 15.6k
22
votes
6
answers
2k
views
About the definition of E8, and Rosenfeld's "Geometry of Lie groups"
I've been searching the literature for a direct definition of the group $E_8$ (over a general field, but even a definition of just one incarnation would be great). I knew (from talking to people) that ...
- 2,287
3
votes
0
answers
151
views
Is there a classical version of Yetter-Drinfeld modules?
One motivation for the notion of the Drinfeld double $D(H)$ of an Hopf algebra $H$ is that it is defined exactly so that modules over $D(H)$ correspond to Yetter-Drinfeld modules over $H$.
If we think ...
- 3,446
3
votes
0
answers
122
views
Torsion of Fermat hypersurfaces
An interesting invariant of a rationally chain-connected variety $X/k$ is the exponent of the group,
$$ \ker{(\mathrm{CH}_0(X_K) \xrightarrow{\deg} \mathbb{Z})} $$
where $K = k(X)$ is the function ...
- 3,730
0
votes
1
answer
130
views
Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules
Let $\lambda \in \mathcal{P}^+$ be a dominant weight for $\frak{sl}(n,\mathbb{C})$. When does it hold that the zero weight space, of the associated finite-dimensional $L(\lambda)$, is non-trivial?
...
- 327
3
votes
2
answers
651
views
Representations of $\mathrm{SO}_n$ versus representations of $\mathrm{Spin}_n$ [duplicate]
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$Since $\Spin_n$ is a compact simply connected simple Lie group, its irreducible representations are equivalent to the irreducible ...
- 327
2
votes
0
answers
80
views
The dual of elements $E$, $F$, and $H$ of $U_h(\mathfrak{sl}_2)$ corresponds to which element of $F_h(\mathrm{SL}_2)$ by isomorphism?
$\newcommand{\sl}{\mathfrak{sl}}\DeclareMathOperator\SL{SL}$Let $U_h(\sl_2)$ be the quantized universal enveloping algebra of $\sl_2(\mathbb{C})$ and $F_h(\SL_2)$ be the quantized function algebra of $...
- 255
7
votes
2
answers
777
views
Why is the generalized flag variety a “variety”?
In several places (for example, Chriss & Ginzburg’s book “Representation Theory and Complex Geometry”), the author says that the set $X$ of Borel subalgebras of a semi-simple Lie algebra $\...
- 159
1
vote
0
answers
75
views
Carleman approximation for functions from $\mathbb R$ to (closed convex subset of) a Lie algebra
I am looking for an approximation result dealing with continuous functions of a real parameter with values in (some subset of) the unitary algebra. However, I wouldn't be surprised if the following ...
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 ...
- 309
1
vote
0
answers
137
views
A question about Araki's 1962 paper on classification of irreducible symmetric spaces
I am looking at Sôhô Araki's 1962 paper for the classification of real semisimple lie algebras. Here's the link to the paper: On root systems and an infinitesimal classification of irreducible ...
- 251
9
votes
2
answers
481
views
Is there a nice q-analogue of the Jacobi identity in a quantized enveloping algebra?
In a Lie algebra $\mathfrak{g}$ the Jacobi identity $\newcommand{\bracket}[2]{\left[#1\,#2\right]} \bracket{x}{\bracket{y}{z}} + \bracket{z}{\bracket{x}{y}} + \bracket{y}{\bracket{z}{x}} = 0$ holds. ...
- 1,161
3
votes
0
answers
142
views
Solvability of a matrix exponential equation - generalized matrix logarithm
For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation
$$
G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) .
$$
Basic ...
- 1,081
1
vote
0
answers
67
views
Poisson bracket on $T^*T\mathrm{SU}(1,1)$
Consider the cotangent bundle of the tangent bundle $T^*TG$ of a Lie group $G$. Denote its the Lie algebra by $\mathfrak{g}$. By left translations, we have the trivialization $T^*G \cong G \times \...
- 829
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 ...
- 1,422
1
vote
0
answers
102
views
How do you find the generators of $E_{10}$?
$E_{10}$ is a hyperbolic Lie group, i.e. one that not only has infinite generators, but one where the number of generators grows exponentially when you commutate more elements of the algebra. So at ...
- 255
14
votes
3
answers
1k
views
About enveloping algebras of direct sums
This question is imported from MSE. It is linked to this one in the case of semi-direct products.
My question Let us consider a Lie $R$-algebra ($R$ is a commutative ring) written as a (module) ...
- 3,738
1
vote
1
answer
105
views
Problem in understanding a fact about Belavin-Drinfeld triple
A Belavin-Drinfeld triple associated to a simple Lie algebra $L$ is a triple $(\Gamma_1, \Gamma_2, \tau)$ where $\Gamma_1, \Gamma_2 \subseteq \Gamma$ ($\Gamma$ is a set of simple roots or fundamental ...
- 199
2
votes
0
answers
119
views
Resolution of singularities of the resultant locus
We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $...
- 7,746
1
vote
1
answer
88
views
Generic finite subgroups, associated to small finite fields, of reductive algebraic groups
Theorem 1 of [LS] Liebeck and Seitz - On the subgroup structure of exceptional groups says:
Let $X = X(q)$ be a quasisimple group of Lie type in characteristic $p$, and suppose that $X < G$, where ...
- 12.9k
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 ...
9
votes
2
answers
789
views
Bender-Knuth involutions for symplectic (King) tableaux
First let me recall the combinatorial theory of the characters of $\mathfrak{gl}_m$, a.k.a., Schur polynomials. For a partition $\lambda$, a semistandard Young tableaux of shape $\lambda$ is a filling ...
- 24.2k
2
votes
1
answer
107
views
Proof of restrictableness of Lie algebra without basis
$\DeclareMathOperator\ad{ad}$According to the Strade and Farnsteiners' textbook, a Lie algebra $L$ is restrictable if and only if $\ad(L)$ is $p$-subalgebra of $\operatorname{Der}(L)$. Also, the ...
- 145
8
votes
2
answers
684
views
Slodowy slice intersecting a given orbit "minimally"?
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Is it true that for any $X\in\mathfrak{g}$, there exists an $\mathfrak{sl}_2$-triple $(e,h,f)$ in $\mathfrak{g}$ such that
We have $X\in e+Z_{\...
- 1,856
1
vote
1
answer
197
views
Semisimple Lie algebra and convexity
There is any relation between semisimple lie algebras and symmetric cones? I'm saying this because the classification of the euclidean Jordan algebras, dual by Koecher-Vinberg theorem to homogeneous ...
3
votes
1
answer
316
views
Complete $2$-step solvable Lie algebras
A Lie algebra is complete if its center is zero and all its derivations are inner. I would like to study a class of Lie algebras, in particular
Let $C$ be the class of finite dimensional $2$-step ...
- 442
2
votes
0
answers
81
views
Complex semisimple Lie algebra modules with non-semisimple Cartan action
Let $\frak{g}$ be a complex semisimple Lie algebra. I would like to know about infinite-dimensional representations $M$ of $\frak{g}$ for which the Cartan $\frak{h} \subseteq \frak{g}$ does not act ...
- 257
77
votes
7
answers
21k
views
What is the symbol of a differential operator?
I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion....
- 54.6k
2
votes
0
answers
152
views
Line bundles on toric varieties associated to Weyl chamber
I am interested in studying toric varieties associated to the fan of Weyl chambers. General information would be best but I am also interested in the specific case of the Weyl chamber of $\mathfrak{sl}...
- 41
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 ...
- 413
2
votes
1
answer
71
views
How to define inverse of a non-degenerate triangular structure?
Let $\mathfrak {g}$ be a non-degenerate triangular Lie bialgebra with the non-degenerate triangular structure $r \in \bigwedge^2 \mathfrak {g}.$ Then how does it induce $r^{-1} \in \bigwedge^2 \...
- 199
10
votes
1
answer
434
views
Viewing exceptional Lie algebras via the classical ones
I've been trying to understand the exceptional Lie algebras through the classical ones that I am more familiar with. In particular I wanted to get a handle on the root spaces and most discussions that ...
- 954
3
votes
0
answers
86
views
Identifying the adjoint representation of an affine Kac-Moody algebra
We work over $\mathbb{C}$. Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra (the question is still relevant for the non-affine case, but I'm specifically interested in the affine case). Suppose ...
- 1,077
3
votes
1
answer
195
views
A representation of $\frak{sl}_n$ as partial derivatives on polynomials
As is known to all, the Lie algebra $\frak{sl}_2$ admits a very nice representation on
$$
\mathbb{K}[X,Y]
$$
the polynomials in two variables, given by
$$
E \mapsto X\frac{\partial }{\partial Y}, ~~ F ...
- 1,144
1
vote
1
answer
229
views
10-dimensional irreducible representations of $so(10,\mathbb{C})$
Are there 10-dimensional irreducible representations of the Lie algebra $so(10,\mathbb{C})$ which are not isomorphic to the standard representation?
- 21.8k
12
votes
2
answers
855
views
Groups associated with infinite dimensional Lie algebras
There is a classical correspondence between Lie algebras (over $\mathbb{R}$ or $\mathbb{C}$) and Lie groups in the finite dimensional case: to every Lie group $G$ there is an associated Lie algebra $\...
- 1,089
10
votes
1
answer
757
views
Can the numerator in Weyl's character formula be written as a determinant?
I paraphrase part of the wikipedia article on the Weyl character formula: Weyl character formula.
If $\pi$ is an irreducible finite-dimensional representation of a complex semisimple Lie algebra $\...
- 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
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 $\...
- 181
2
votes
0
answers
59
views
graded reps of Lie algebras literature
I am currently studying 'advanced' representation theory from a physicist's perspective, including topics like super-Lie algebras. I've come across various gradings (excluding the ℤ2
grading), such as ...
- 21
0
votes
0
answers
99
views
How to show that quantum $sl_2 (\mathbb C)$ is a Hopf algebra deformation of $U (sl_2 (\mathbb C))\ $?
The quantum $sl_2 (\mathbb C)$ is the non-commutative, non-cocommutative Hopf algebra $U_h (sl_2 (\mathbb C))$ over the ring $\mathbb C [[h]]$ generated by $E, F$ and $H$ with the relations $:$
$$[H, ...
- 41
0
votes
0
answers
97
views
Methods for calculating (one-parameter subgroup) actions
For $G$ a Lie group and $\mathfrak{g}$ its Lie algebra, I am interested in one-parameter subgroup actions on “functions” $f$ of the form
\begin{equation}
\mathrm{e}^{t L(z)} f(z)
\end{equation}
...
- 679
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
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 $r(G) > r(K)$ (...
- 143
10
votes
1
answer
636
views
Killing form: clean proof that two spaces are orthogonal?
Let $G$ be a linear algebraic group whose Lie algebra $\mathfrak{g}$ is semisimple. Let $x$ be a regular semisimple element of $G$. Write $\mathfrak{t}$ for the Lie algebra of the maximal torus $T = C(...
- 20.2k