Question on standard filtration Ex 3.7(a) James Humphreys "Representations of Semisimple Lie Algebras in the BGG Category O"

Exercise 3.7(a) James Humphreys's "Representations of Semisimple Lie Algebras in the BGG Category O"Let $V \in \mathcal{O}$ be a module which admits a standard filtration. Suppose that there is a surjective homomorphism $$\phi: V \rightarrow M(\mu)$$ where $M(\mu)$ is a Verma module. Show that ker$\phi$ also admits a standard filtration.My attempt: Suppose that $\lambda$ is maximal among the weights of $V$ and let $M(\lambda)$ be the corresponding submodule. If $\mu \neq \lambda$, then the composition of the embedding $ M(\lambda) \rightarrow ...Read more

ds.dynamical systems - Lie algebra admitting some hyperbolic automorphism is nilpotent

Let $\mathfrak{g}$ be a finite dimensional Lie algebra over $\mathbb{R}$ and $\phi:\mathfrak{g}\to\mathfrak{g}$ be a Lie algebra automorphism. Viewing $\mathfrak{g}$ as a linear space and $\phi$ a linear automorphism, we can say $\phi$ is hyperbolic if the eigenvalues of $\phi$ are disjoint from $\lbrace z\in\mathbb{C}:|z|=1\rbrace$.Then Proposition 3.6 in Smale's paper (here) says that:Suppose that $\phi:\mathfrak{g}\to\mathfrak{g}$ is a Lie algebra automorphism which is hyperbolic as a linear map. Then $\mathfrak{g}$ must be nilpotent.He also...Read more

lie algebras - I don't get a part of Bernstein's / Deligne-Morgan's proof of Poincaré-Birkhoff-Witt

Question: I am talking about the proof given on pages 50-52 of Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, and Edward Witten (editors), Quantum Fields and Strings: A Course for Mathematicians, Volume 1, AMS 1999 (on google books and in the usual internet sources).The problem is easily described: In the middle of page 52, the authors say "and (1.3.7.7) gives that [...]". But I don't see how (1.3.7.7) gives the equation that follows.Sidenotes: The proof was rather readable and...Read more

A possible mistake in Kac's "Infinite Dimensional Lie Algebras"

I have a paperback 3rd edition and on page 65 you can find Proposition 5.8. My question is about part (c): If $A$ is of indefinite type, then $$ \overline{X} = \{ h \in \{ \frak h_{\mathbb{R}} \}| \left \langle \alpha, h \right \rangle \geq 0 \text{ for all } \alpha \in \Delta_+^{im} \}, $$ where $\overline{X}$ denotes the closure of $X$ in the metric topology of $\frak h_{\mathbb{R}}$.Some context: $A$ is a generalized Cartan matrix, $\frak h_{\mathbb{R}}$ is a real form of the Cartan subalgebra in the Kac-Moody algebra associated to $A$ an...Read more

lie algebras - Constructing Affine Kac-Moody Groups

Does anyone know a simple construction for Affine Kac-Moody groups? There is a book by Kumar ("Kac-Moody groups, their flag varieties, and representation theory") that does the construction for the general Kac-Moody case, but I find the presentation dense. There is also a section that constructs a one-dimensional extension of the loop group by loop rotation, which is a fairly transparent definition. However, I don't know how to add on the final central extension.Even if the answer to my question is "There is no simpler construction," could some...Read more

On Engel-anticommutative algebras

Let $\mu:\mathbb{R}^n \times \mathbb{R}^n \longrightarrow \mathbb{R}^n$ be a alternating bilinear map, i.e. $\mu(X,Y)=-\mu(Y,X)$ (anticommutativity) and so, let $\mathfrak{a}=(\mathbb{R}^n,\mu)$ be a skew-symmetric algebra (this one is not necessarily a Lie algebra).Questions:Is there a "famous" example of a skew-symmetric algebra that is not a Lie algebra?We assume that for any $X \in \mathfrak{a}$, $\mu(X,\cdot):\mathbb{R}^n \longrightarrow \mathbb{R}^n$ is a nilpotent linear transformation. Is it true that $\mathfrak{a}$ is a nilpotent algeb...Read more

lie algebras - 2-cocycle on LSU(2)

$SU(2)$ is a Lie group, with a Lie algebra $\mathfrak{su}(2)$. I now consider the loop group $ LSU(2) = \{ \gamma: S^1 \to SU(2); \gamma \mathrm{ smooth} \} $. It is well-known that there exists non-trivial 2-cocycles on it, obtained from the invariant bilinear form on $\mathfrak{su}(2)$ (see Pressley and Segal for example).The 2-cocycle on the Lie-algebra is very simple to write. However, the 2-cocycles on the group are rarely written explicitly. I know that it may be quite difficult for general groups but what about a group as "simple" as SU(...Read more

lie algebras - Unique dimension of Cartan subalgebras?

If $L$ is a Lie algebra over an algebraic closed field $K$ of characteristic zero, then all Cartan subalgebras are conjugated. Hence, they have all the same dimension. If $K$ is not algebraic closed but still of characteristic zero, then let $F$ the algebraic closure of $K$ and $H$ a Cartan subalgebra of $L$. Then $H\otimes F$ is a Cartan subalgebra of $L\otimes F$ with $dim_K(H)=dim_F(H\otimes F)$. Thus, the dimension of all Cartan subalgebras in characteristic zero is identical.In the modular case this statement is wrong. There are several ex...Read more

lie algebras - Does the vanishing of the Poisson bracket on $S(\mathfrak{g})^{\mathfrak{g}}$ inspire the disover of Duflo's isomorphism theorem?

For any finite dimensional Lie algebra $\mathfrak{g}$, we know that the universal enveloping algebra $U(\mathfrak{g})$ is a deformation of the symmetric algebra $S(\mathfrak{g})$. In fact let's define$$U_t(\mathfrak{g}):=\text{T}(\mathfrak{g})/(X\otimes Y-Y\otimes X-t[X,Y]).$$Then $S(\mathfrak{g})=U_0(\mathfrak{g})$ and $U(\mathfrak{g})=U_1(\mathfrak{g})$. Moreover we have the symmetrization map$$I_{PBW}:S(\mathfrak{g})\longrightarrow U_t(\mathfrak{g})$$which pulls back the product on $U_t(\mathfrak{g})$ to a product on $S(\mathfrak{g})$. We ca...Read more

lie algebras - 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 following problem:Finding a basis for $\mathbb{Z}[x_{1},x_{2},...x_{n+1}], \text{with condition} \prod x_{i}=1$ in terms of $$\mathbb{Z}[x_{1}+x_{2}..+x_{n+1}, \sum_{i\le j} x_{i}x_{j}, \sum_{i\le j\le k} x_{i}x_{j}x_{k},...]$$The later ring is obviously a ring of elementary symmetric polynomials. We are wondering if anyone from ring theory has alread...Read more