All Questions
38 questions
2
votes
0
answers
126
views
A property of matrices formed by pairing of roots and coroots
Let $A$ be an $n\times n$ integral matrix, define its level $l(A)$ as
$$l(A) := \begin{cases}0 &\det A = 0 \\ \text{smallest integer } N \text{ such that } NA^{-1} \text{ is integral} &\det A \...
- 528
3
votes
1
answer
108
views
Roots of polynomial $\sum_{\sigma \in W} x^{l(\sigma)}$
Let $W$ be Weyl group of a root system $\Phi$ (of finite dimensional simple Lie algebra). For $\sigma\in W$, $l(\sigma)$ be the its length. Consider the following polynomial
$$P_\Phi(x) = \sum_{\sigma ...
- 528
4
votes
0
answers
79
views
Closed character formula for the module $L(a\omega_i)$
Let $\mathfrak{g}$ be a complex finite-dimensional simple Lie algebra with a fixed Cartan subalgebra $\mathfrak{h}$. Assume that $\omega_1, \omega_2, \dots, \omega_n\in\mathfrak{h}^{*}$ is the ...
- 41
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
11
votes
0
answers
436
views
A rather strange algebra
Let $k$ be an algebraic closed field of zero characteristic and $X$ an affine smooth variety, with $A=\mathcal{O}(X)$ the algebra of regular functions and $\mathcal{V}$ the Lie algebra of vector ...
- 3,318
3
votes
0
answers
222
views
What is a Gelfand-Tsetlin subalgebra?
For context on general Gelfand-Tsetlin theory, see for instance this MO post.
Let's work over $\mathbb{C}$. Fix $n>0$. There is a natural chain of embeddings of the general linear Lie algebras $\...
- 3,318
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
6
votes
1
answer
255
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
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 ...
2
votes
0
answers
74
views
Explicit example of prime ideal not an intersection of maximal ideals, in universal enveloping algebra
Let $A$ be a $\mathbb k$-algebra. If $A$ is affine commutative, by Nullstellensatz, then every prime ideal of $A$ is an intersection of maximal ideals.
To justify the notion of being primitive in ...
- 485
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
4
votes
1
answer
228
views
When is the enveloping algebra finitely generated over its center as a module?
Let $g$ be a Lie algebra with enveloping algebra $U(g)$ and $Z(g)$ the center of $U(g)$.
Question 1: When is $Z(g)$ noetherian?
Question 2: When is $U(g)$ a finitely generated $Z(g)$-module? Is this ...
- 26.5k
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 ...
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 ...
4
votes
2
answers
360
views
Double centralizer in special linear algebra
It is well known that for a matrix $A$ in $\mathfrak{sl}_n(\mathbb{C})$, we have the following equivalence:
$$\dim Z(A) \text{ is minimal} \leftrightarrow A \text{ is cyclic}$$
where $Z(A)$ is the ...
- 617
6
votes
2
answers
501
views
Complete reducibility and field extension
Let $\pi$ be a representation of a Lie algebra $L$ in a finite-dimensional linear space $V$ over the field $F$. Let $K$ be a field extension of $F$. Let $\pi_K=\pi\otimes K$ be the corresponding ...
- 71
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
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
9
votes
0
answers
244
views
Homology of an interesting Lie algebra
Let $U$ and $V$ be finite dimensional complex vector spaces (or perhaps graded vector spaces).
Let $E(U)$ be the "square zero extension" $\mathbb C \oplus U$, made into a commutative ring in such a ...
- 40.2k
0
votes
0
answers
101
views
Spherical Rings
My question is concerned with filtered rings. It is a classical result that if $R$ is a finitely generated commutative ring graded by a semigroup $S$ then $S$ is also finitely generated.
The reverse ...
- 501
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
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
6
votes
0
answers
164
views
Systematic treatment of folding and valued graphs
I'm going to say beforehand that this question has something of a "am I missing something?" flavor. I'm in that odd position mathematicians often find themselves, where a topic has been ...
- 44.7k
4
votes
2
answers
186
views
Covering derivations of a quotient algebra
Let $(\mathcal{A},+,·)$ an algebra and $\mathcal{I}$ an ideal of $\mathcal{A}$.
Is easy to check that if $D\in Der(\mathcal{A})$ with $D(\mathcal{I})\subseteq I$ then $D$ induces a derivation $D_I$ ...
- 43
1
vote
0
answers
187
views
Examples of Lie subalgebras of universal enveloping algebras
I'm looking for non-trivial examples of triples $(\mathfrak g, L, \psi)$, where $\mathfrak g$ and $L$ are finite-dimensional non-abelian Lie algebras over field $\Bbbk$ and $\psi\colon U_{\mathfrak g}\...
- 1,422
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
4
votes
1
answer
755
views
Commutators for quantum Lie algebras
Can the usual definition of a Lie algebra via commutators be simply adapted
to quantum Lie algebras? Graphically you have the IHX scheme, with the X
being a virtual crossing (so to say). Does it ...
- 4,829
5
votes
1
answer
427
views
Does $\text{Mat}_n(k)$ have some universal properties similar to its universal enveloping algebra?
Let $k$ be a field and $\text{Mat}_n(k)$ be $n \times n$ matrices over $k$. Let's consider $\text{Mat}_n(k)$ as an associative algebra and denote $gl_n(k)$ be the same $k$-linear space as $\text{Mat}...
- 9,019
2
votes
1
answer
492
views
finite dimensional irreducible representation of finite dimensional nilpotent Lie algebra
Let $k$ be a field, $L$ be a finite dimensional nilpotent Lie
algebra over $k$ and $M$ be a finite dimensional irreducible
representation of $L$. Assume that there is a linear function $\rho
: L\...
- 491
5
votes
2
answers
584
views
BGG-like resolutions and translations
This question originated from my confusion after I read the following paragraph (page 31, section 4.8) in Resolutions and Hilbert series of the unitary highest weight modules of the exceptional ...
- 8,597
5
votes
2
answers
487
views
Lie algebra embeddings and the center of their enveloping algrabras
Let $\mathfrak{g}_1\subset \mathfrak{g}_2$ be a Lie algebra embedding. Assume both are semisimple. For instance take the standard diagonal embedding $\mathfrak{sl}(2, \mathbb{C})\subset \mathfrak{sl}(...
- 501
1
vote
1
answer
322
views
translation functors in parabolic category $\mathcal{O}$
I'm looking for a reference for a treatment of translation functors (as defined e.g. in this [question][1]) in parabolic versions of BGG category $\mathcal{O}$.
I am mainly interested in the ...
- 8,597
7
votes
3
answers
617
views
Why/when classification of simple objects is "simple" ? E.g. (unknown) classification of simple Lie algebras in char =2,3...
Classification of simple finite-dim Lie algebras for char >=5 has been accomplished not so long time ago, and char p=2,3 is open problem.
I wonder what is known/expected for char p=2,3 ?
More vague ...
- 24.9k
3
votes
1
answer
694
views
Reference Request: Basis in terms of ring of symmetric polynomials
As part of the result of solving the problem I am working on, my advisor and I translated the task of finding a basis for $R(T_{sl_{\mathbb{C}}(n)})$ in terms of $R(sl_{\mathbb{C}}(n))$ into the ...
- 799
13
votes
4
answers
2k
views
Geometric interpretation of Universal enveloping algebras
Given a complex Lie algebra $\mathfrak g$, we can form its universal enveloping algebra and interpret it as a noncommutative space.
Is this perspective useful? What does this space "look like"?
How ...
- 13.2k
13
votes
4
answers
3k
views
What is a "block" in an abelian category?
In the literature and in some posts here, there has been variation in the undefined use of the term "block" for a category of modules over a ring, or more abstractly an abelian category (all of which ...
- 52.9k
24
votes
6
answers
7k
views
Introduction to W-Algebras/Why W-algebras?
Does anyone know of an introduction and motivation for W-algebras?
Edit: Okay, sorry I try to add some more background. W algebras occur, for example when you study nilpotent orbits: Take a nice ...
- 13.2k
15
votes
3
answers
3k
views
Which is the correct universal enveloping algebra in positive characteristic?
This is an extension of this question about symmetric algebras in positive characteristic. The title is also a bit tongue-in-cheek, as I am sure that there are multiple "correct" answers.
Let $\...
- 54.6k