All Questions
2,633 questions
8
votes
2
answers
436
views
Can we count the number of simple modules for a reduced enveloping algebra?
Let $G$ be a reductive algebraic group over a field of positive characteristic $p$, which I'll assume to be very good for $G$. Then the Lie algebra $\mathfrak{g}$ is restricted and each simple $\...
- 599
8
votes
1
answer
1k
views
Maximal Submodule of a Verma Module
Let $\mathfrak{h}$ be a Cartan subalgebra of a $\mathbb{C}$-semi simple Lie algebra $\mathfrak{g}$. Given $\lambda \in \mathfrak{h}^*$, $M(\lambda)$ the Verma module of highest weight $\lambda$ and $N(...
- 83
8
votes
4
answers
2k
views
list of Hall basis
Anyone know a place where the standard Hall basis is listed up to at least
5-fold brackets?
And for graded Lie algebras?
The rules are clear but I'd rather not turn the crank myself.
Google search ...
- 3,880
8
votes
2
answers
808
views
Lie algebras and non-smoothness of centralisers in bad characteristic
Let $G$ be a simple algebraic group over an algebraically closed
field $k$ of characteristic $p>0$. For $x\in G$, let $C_{G}(x)$
denote the centraliser, considered as a group scheme over $k$. If
$p$...
- 3,823
8
votes
1
answer
1k
views
Commutativity and Kostant sections
Let $(e, h, f)$ be an $\operatorname{SL}_2$-triple in $\mathfrak g$. My understanding is that $e + C_{\mathfrak g}(f)$ is called a Kostant section only in case $e$ is regular; but I don't impose this ...
- 12.9k
8
votes
1
answer
943
views
Automorphisms over finite field that do not lift to an automorphism in characteristic zero
My main question is the following: is there an automorphism of the affine space $\mathbb{A}^n$ (automorphism of an algebraic variety) defined over a finite field which does not lift to an automorphism ...
- 7,722
8
votes
1
answer
429
views
Koszulness of some DG-algebras and a paper by Kohno and Oda
This is a follow-up of my previous question Formality of the 2nd ordered configuration space of a closed Riemann surface.
At page 131 of [B], R. Bezrukavnikov states Proposition 4.1, in which he ...
- 66.3k
8
votes
2
answers
1k
views
$p$-adic exponentials for $p$-adic Lie groups
Let $G$ be a $p$-adic Lie group, $\text{Lie}(G)$ its Lie algebra.
Is there any reasonable notion of exponential map $\text{exp} : \text{Lie}(G)\to G$?
user113393
8
votes
2
answers
767
views
What is Jantzen's formula for the determinant of the Shapovalov form in the case of generalized Verma modules?
The best reference I found is
[Kac, Kazhdan '79]
which extends the results of Shapovalov and Jantzen to the case of infinite dimensional Lie algebras.
Theorem 1 of this paper gives the Shapovalov ...
8
votes
1
answer
330
views
Is the Chevalley-Eilenberg cohomology the only interesting cohomology for Lie algebra?
When talking about the cohomology space of a Lie algebras, it comes naturally to refer to the Chevalley-Eilenberg cohomology, is there other interesting type of cohomology for Lie algebra?
- 175
8
votes
1
answer
774
views
A variant on characteristic $p$ de Rham cohomology
I was thinking about de Rham cohomology in characteristic $p$, and in particular the recent question about Poincare residues, and I came up with the following construction.
Let $k$ be a perfect field ...
- 156k
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 $r = s$, ...
- 643
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 ...
- 449
8
votes
1
answer
167
views
Symmetries of the flag variety
Let $\mathfrak g$ be a finite dimensional simple Lie algebra over $\mathbb C$, and let $\mathcal B=G/B$ be the associated Flag variety.
Is it true that the obvious map
$$
\mathfrak g\to \Gamma (T\...
- 43.2k
8
votes
1
answer
490
views
ad (A^n) is a polynomial in ad A ?
Let $k$ be a field and $n$ a nonnegative integer. For any matrix $U\in\mathrm{M}_n\left(k\right)$, let $\mathrm{ad} U$ denote the map $\mathrm{M}_n\left(k\right)\to \mathrm{M}_n\left(k\right),\ V\...
- 33.8k
8
votes
1
answer
331
views
If C is a fusion category over a field of nonzero characteristic and dim C = 0, is Z(C) ever fusion?
If $C$ is a fusion category and $\dim(C) \neq 0$ (the latter is automatic in characteristic zero, but not in nonzero characteristic), then the Drinfel'd center $Z(C)$ is fusion. More generally, if $C$...
- 28.1k
8
votes
2
answers
2k
views
BGG resolution and representations of parabolic subalgebras
Everything here is over $\mathbb{C}$.
Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra and let $\mathfrak{p}$ be a parabolic subalgebra (relative to some fixed Borel subalgebra that is ...
- 8,559
8
votes
2
answers
1k
views
Quasi-Lie algebras in nature?
A Lie algebra over $\mathbb Z$ is defined to be an abelian group with a bracketing operation $[\cdot,\cdot]$, satisfying the Jacobi identity and the relation $[x,x]=0$ for every $x$. On the other hand,...
- 4,898
8
votes
2
answers
619
views
Relationship between $q$-Weyl dimension formula and $q$-analog of weight multiplicity?
$\DeclareMathOperator\dim{dim}$For a dominant (integral) weight $\lambda$ and any (integral) weight $\mu$ of a simple Lie algebra $\mathfrak{g}$, Lusztig's $q$-analog of weight multiplicty $K_{\lambda,...
- 24.2k
8
votes
1
answer
863
views
Modern sources on Lie algebraic methods in combinatorics
I am looking for modern textbooks, expository papers and journal articles regarding the use of Lie algebras in combinatorics. In particular, I am interested to know the extent to which people still ...
- 376
8
votes
2
answers
462
views
The action of $GL_{\infty}$ on the infinite wedge space
This is a question from the book "Highest weight representations of infinite dimensional Lie algebras, 2nd ed" by V. G. Kac, A. K. Raina, and N. Rozhkovskaya.
Consider the following objects:
the ...
- 671
8
votes
1
answer
808
views
Automorphisms of curves in positive characteristic
It is well known that over an algebraically closed field of characteristic zero a general curve (for an open subset of $M_g$) of genus $g\geq 3$ is automorphism-free.
Is this result still true over ...
user68440
8
votes
3
answers
502
views
Polarizations generate the ring of invariants?
The symmetric group $S_n$ acts on $\mathbb R^n$ by permuting the coordinates and the ring of polynomial invariants is generated by the elementary symmetric polynomials. If we restrict the action to ...
- 187
8
votes
1
answer
1k
views
PBW basis and canonical basis
Consider the example of $\mathfrak{g} = sl_3$. Then
$$
\mathfrak{g} = \mathfrak{n} \oplus \mathfrak{h} \oplus \mathfrak{n}^{-},
$$
where $\mathfrak{n}$ is generated by $E_{12}, E_{13}, E_{23}$, $\...
- 6,211
8
votes
1
answer
374
views
Chevalley restriction theorem for exterior algebras
Suppose $G$ is semisimple Lie group, $\mathfrak{g}$ is its Lie algebra, $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, and $W$ is the correspondent Weyl group.
Chevalley restriction theorem ...
- 1,276
8
votes
1
answer
375
views
Motivation behind Panyushev's "constant-averages-along-orbits" conjecture
In his article "On orbits of antichains of positive roots" (European Journal of Combinatorics 30 (2009) 586–594, Dmitri Panyushev discusses an interesting self-map on the set of antichains of a finite ...
- 19.7k
8
votes
3
answers
570
views
Variations on a theme of O'Bryant, Cooper and Eichhorn concerning power series over $\mathbb Z/2\mathbb Z$
Define 2 power series over the field $\mathbb Z/2\mathbb Z$ by $f=1+x+x^3+x^6+\dots$, the exponents being the triangular numbers, and $g=1+x+x^4+x^9+\dots$, the exponents being the squares. Write $f/g$...
- 5,422
8
votes
2
answers
378
views
Relation between Lie Algebra Cohomology and Number of Relations of a Cyclic Module?
Let $\mathfrak{g}$ be a finite dimensional Lie algebra over $k$, let $U$ be its enveloping algebra, and let $M$ be a $\mathfrak{g}$-module (not necessarily finite dimensional). Call the invariant ...
- 13k
8
votes
1
answer
390
views
What is the ring structure on Lusztig's integral form of quantum $\mathfrak{sl}(2)$?
Consider the quantum group $U_q(\mathfrak{sl}_2)$, with generators $E,F,K$ such that $[E,F]=\frac{K-K^{-1}}{q-q^{-1}}$. Write $[n]=\frac{q^n-q^{-n}}{q-q^{-1}}$, and $[n]!=[n][n-1]\dotsm[1]$.
In ...
- 492
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
8
votes
1
answer
591
views
History of the study of Verma modules in terms of Kazhdan Lusztig Theory
Let $\mathfrak{g}$ be a complex finite dimensional semisimple Lie algbera, $W$ be the Weyl group, $\rho$ be the half sum of positive roots, $M(\eta)$ be the Verma module of weight $\eta$ and $L(\eta)$ ...
- 1,875
8
votes
1
answer
452
views
What is the relation between Coxeter transformations of generalized Cartan matrices and Coxeter transformations of finite-dimensional algebras?
Note: This question now has a sister :-)
The Coxeter transformation of a generalized Cartan matrix:
In the paper
The spectral radius of the Coxeter transformations for a generalized Cartan matrix, ...
- 253
8
votes
1
answer
308
views
One identity in Lie algebras
Let $L$ be a (non-restricted) Lie algebra over a field of prime characteristic $p,$ $UL$ be its universal enveloping algebra and $a_1,\dots, a_p \in L$ (the number of elements is equal to the ...
- 1,484
8
votes
1
answer
618
views
Closure order on nilpotent orbits in exceptional Lie algebras
Let $G$ be a simple algebraic group over the algebraically closed field $k$ of positive characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent $G$...
- 1,336
8
votes
1
answer
439
views
One more question about PBW
Let $k$ be a commutative ring with unit and $L$ be a Lie $k$-algebra.
Let $U(L)$ be the universal enveloping $k$-algebra of $L$ (one can define it as a quotient of the tensor algebra, as it is ...
- 8,455
8
votes
1
answer
520
views
The exceptional Lie algebra $\mathfrak{g}_2$ and binary cubics
How is the exceptional 14-dimensional Lie algebra $\mathfrak{g}_2$ related to the covariant algebra for the binary cubic?
Here are some details on this question. This algebra is generated by 4 forms,...
- 1,277
8
votes
2
answers
1k
views
Killing form vs its counterpart in a given represenation
Let $\mathfrak{g}$ be a semi-simple Lie algebra and let $\phi:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ be its finite-dimensional complex irreducible representation. You can define two non-degenerate ...
- 965
8
votes
1
answer
540
views
Vanishing of H-cohomology
This looks elementary, but somehow I am stuck, please bear with me:
Let $H$ be a differential 3-form, nowhere vanishing, but not necessarily closed. What is a sufficient condition that the sequence ...
- 19.9k
8
votes
1
answer
705
views
Kazhdan-Lusztig theorem for composition factors of Verma modules
The Kazhdan-Lusztig Conjecture (which is actually a theorem) gives the character of some irreducible modules of a (say) simple complex Lie algebra $\mathfrak{g}$ in terms of characters of Verma ...
- 457
8
votes
1
answer
406
views
Characterizations of Jacobson-Morozov parabolics associated to a nilpotent
Let $x \in \mathfrak{g}$ (or $x \in G$) be a nilpotent (resp. unipotent) element of a simple Lie algebra (resp. linear algebraic group). One can associate to this data a Jacobson-Morozov parabolic ...
- 1,423
8
votes
1
answer
982
views
Is there a really big ring of differential operators in characteristic p?
$k$ is a field of characteristic $p$.
$k[t]$ has canonical first-order differential operator $\partial$
As an endomorphism of $k[t]$, $\partial^p=0$.
First way to fix it:
Use the divided power ...
8
votes
1
answer
800
views
higher Casimirs for $\mathfrak{sl}$
The Wikipedia universal enveloping algebra suggest a way to obtain higher Casimir operators (e.g. generators of the center of $\mathfrak{U(g)}$ for $\mathfrak{g}$ semisimple) by evaluating certain ...
- 8,597
8
votes
1
answer
151
views
Criterion for existence of a homogeneous space associated to a Lie pair
Recall that every finite-dimensional real Lie algebra $\mathfrak{g}$ is the Lie algebra of a simply-connected Lie group, which is unique up to isomorphism.
This statement generalises somewhat to ...
- 33.3k
8
votes
1
answer
236
views
Simple identity on Lie algebras in a note of Koszul
In a 1947 Comptes Rendus note (T224, p. 448), Koszul makes the following claim (paraphrased, hopefully correctly), which seems like it should have a simple proof I am missing.
Given a compact, ...
- 2,995
8
votes
1
answer
747
views
Deligne's exterior power
In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism
$$p : A^{\otimes n} \to A^{\otimes n}, ...
- 63.1k
8
votes
1
answer
295
views
What are twisted Verma modules? Basic properties?
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbf{C}$, $\lambda$ be a weight in the nonnegative Weyl chamber and $w_1,w_2\in W$. Then the twisted Verma modules are defined e.g. in Andersen and ...
- 5,711
8
votes
1
answer
443
views
When does the enveloping algebra functor lift to the category of bialgebras?
Let $\mathrm{Ass} $ denote the operad, whose algebras are associative unital algebras, considered as a dg-operad.
Denote $\mathrm{Ch} $ the category of chain complexes over a commutative ring $\...
- 1,361
8
votes
1
answer
650
views
Harish-Chandra isomorphism for compact symmetric spaces
I would be interested to have an explicit description of the algebra of invariant differential operators on functions on a compact symmetric space $G/K$. A reference would be especially useful.
Below ...
- 21.8k
8
votes
1
answer
775
views
Is an irreducible holomorphic symplectic manifold a simple Lie algebra?
The tangent bundle of a hyper-Kahler manifold gives a quadratic Lie algebra in the derived category. Can this be regarded as a simple Lie algebra according to Vogel's definition?
A point of view that ...
- 9,132
8
votes
1
answer
852
views
Symmetrization map for universal enveloppings. What are Harish-Chandra images of symmetrization (poisson-center of S(gl_n) ) ?
For any Lie algebra symmetrization maps Poisson center of S(g) to center of U(g).
Consider g=gl_n, the Poisson center of S(gl_n) is isomorphic to algebra of symmetric polynomials in eigenvalues.
...
- 24.9k