0
votes
0answers
92 views

A special Lie subalgebra

Motivated by comments of the following post A question on involutions on the Lie algebra of vector fields we ask the following question: Let $L$ be a Lie algebra. We consider the Lie subalgebra ...
3
votes
1answer
186 views

What is twisted in a twisted Poisson structure?

The usual definition of a twisted Poisson algebra involving a 3-form does not seem to refer to a Poisson algebra that is then twisted, i.e. twisting either the commutative product or the Lie bracket. ...
4
votes
1answer
292 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 ...
14
votes
3answers
570 views

Can one show the equivalence of the abstract and classical Jordan decompositions for simple Lie algebras without complete reducibility?

The following fact is basic in the theory of complex Lie algebras: Theorem. Let ${\mathfrak g} \subset {\mathfrak gl}_n({\bf C})$ be a simple Lie algebra, and let $x \in {\mathfrak g}$. Let $x = ...
1
vote
0answers
167 views

universal enveloping algebras and commutator subalgebras

Let $A$ and $B$ are Lie subalgebras of a Lie algebra $L$. $U(A)$, $U(B)$ and $U(L)$ are the universal enveloping algebras of $A$, $B$ and $L$, respectively. Let $[A, B]$ be the Lie subalgebras ...
2
votes
1answer
263 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 : ...
4
votes
2answers
272 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 ...
5
votes
2answers
223 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 ...
1
vote
1answer
193 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 ...
10
votes
1answer
348 views

What are the simple Lie superalgebras of type E?

Background Simple finite dimensional Lie superalgebras over $\Bbb C$ have been classified. There are the Cartan type superalgebras (algebras of purely odd vector fields), two strange families P(n) ...
7
votes
3answers
439 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 ...
9
votes
2answers
446 views

Does any identity holding in all finite-dimensional Lie algebras hold in all Lie algebras?

Equivalently, is the free Lie algebra on finitely many generators over a fixed field $k$ (say of characteristic not equal to $2$) residually finite-dimensional in the sense that any nonzero element ...
3
votes
2answers
436 views

German term for “restricted Lie algebra” ?

Can anyone tell me the German term for "restricted Lie algebra" ? Many thanks in advance ! Kind regards, Stephan Kroneck.
3
votes
1answer
468 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 ...
5
votes
0answers
315 views

Lie locally nilpotent associative algebras

Let $A$ be an associative algebra over a field. Then $A$ can be regarded as a Lie algebra via the Lie bracket defined by $[a,b]=ab-ba$ for every $a,b\in A$. The algebra $A$ is called Lie locally ...
5
votes
1answer
530 views

PBW theorem over a Q-algebra, without freeness or flatness

Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-Lie algebra, which is not necessarily free as a $k$-module. Let $S\left(L\right)$ denote the symmetric algebra of $L$ (over $k$), constructed ...
4
votes
1answer
375 views

Commutator formulas in a universal enveloping algebra

I am interested in finding formulas for commutators of symmetrized monomials in a universal enveloping algebra. Let $C(x_1,\ldots, x_n)= (1/n!)\sum x_{\sigma(1)}\cdots x_{\sigma(n)}$ where the sum ...
0
votes
0answers
189 views

Good and/or standard notation for the abelianization of a Lie algebra

I'd like to solicit good notations for the abelianization of a Lie algebra $\mathfrak g$. One could write $\mathfrak g/[\mathfrak g,\mathfrak g]$, or even $H_1(\mathfrak g)$ but I'd like something ...
6
votes
2answers
566 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 ...
7
votes
4answers
904 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 ...
7
votes
4answers
1k 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 ...
2
votes
0answers
391 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 $H$, $\tilde{\alpha}$: ...
14
votes
5answers
3k views

Introduction to W-Algebras/Why W-algebras?

Hi, 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 ...
9
votes
2answers
1k 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 ...