### lie algebras - How to find Casimir operators?

Given a general Lie algebra, is there a general procedure to find all its Casimir operator?...Read more

Given a general Lie algebra, is there a general procedure to find all its Casimir operator?...Read more

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

Schur multipliers for group extensions and for Lie groups alsoWhere are they written for Lie algebras?...Read more

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

Let $\mathfrak{g}$ be positively graded Lie algebra over $\mathbb{Q}$, concentrated in even degrees. Question: If $\mathfrak{g}$ is not free, must there exist linearly independent elements $a,b\in\mathfrak{g}$ such that $[a,b]=0$?...Read more

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

Let $D$ be a division algebra and $n\in \mathbb{N}$. If $D$ is a field, then it is well-known that the diagonal-matrices form a Cartan subalgebra of $gl(n,D)$. Is there a complete description of all Cartan subalgebras?...Read more

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

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

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

$SU(2)$ 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

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

Let L be a Lie algebra with a one-dimensional maximal subalgebra. Is the following true?Over a perfect field of characteristic 0 or p > 3, every such finite-dimensional Lie algebra is either 2-dimensional, or is a form of sl(2). General structure theory seems to indicate this....Read more

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

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