This survey paper is an exposition of old and recent results of Kostant and al. on the relationships between the exterior algebra of a simple Lie algebra and the action of the Casimir operator on it. Our exposition relies on u-cohomology and it is basically self-contained.
For every simple Lie algebra $mathfrak{g}$ we consider the associated Takiff algebra $mathfrak{g}^{}_{ell}$ defined as the truncated polynomial current Lie algebra with coefficients in $mathfrak{g}$. We use a matrix presentation of $mathfrak{g}^{}_{e
ll}$ to give a uniform construction of algebraically independent generators of the center of the universal enveloping algebra ${rm U}(mathfrak{g}^{}_{ell})$. A similar matrix presentation for the affine Kac--Moody algebra $widehat{mathfrak{g}}^{}_{ell}$ is then used to prove an analogue of the Feigin--Frenkel theorem describing the center of the corresponding affine vertex algebra at the critical level. The proof relies on an explicit construction of a complete set of Segal--Sugawara vectors for the Lie algebra $mathfrak{g}^{}_{ell}$.
Let $SL_2$ be the rank one simple algebraic group defined over an algebraically closed field $k$ of characteristic $p>0$. The paper presents a new method for computing the dimension of the cohomology spaces $text{H}^n(SL_2,V(m))$ for Weyl $SL_2$-modu
les $V(m)$. We provide a closed formula for $text{dim}text{H}^n(SL_2,V(m))$ when $nle 2p-3$ and show that this dimension is bounded by the $(n+1)$-th Fibonacci number. This formula is then used to compute $text{dim}text{H}^n(SL_2, V(m))$ for $n=1, 2,$ or $3$. For $n>2p-3$, an exponential bound, only depending on $n$, is obtained for $text{dim}text{H}^n(SL_2,V(m))$. Analogous results are also established for the extension spaces $text{Ext}^n_{SL_2}(V(m_2),V(m_1))$ between Weyl modules $V(m_1)$ and $V(m_2)$. In particular, we determine the degree three extensions for all Weyl modules of $SL_2$. As a byproduct, our results and techniques give explicit upper bounds for the dimensions of the cohomology of the Specht modules of symmetric groups, the cohomology of simple modules of $SL_2$, and the finite group of Lie type $SL_2(p^s)$.
For the abelian Chern-Simons field theory, we consider the quantum functional integration over the Deligne-Beilinson cohomology classes and we derive the main properties of the observables in a generic closed orientable 3-manifold. We present an expl
icit path-integral non-perturbative computation of the Chern-Simons links invariants in the case of the torsion-free 3-manifolds $S^3$, $S^1 times S^2$ and $S^1 times Sigma_g$.
For symplectic Lie algebras $mathfrak{sp}(2n,mathbb{C})$, denote by $mathfrak{b}$ and $mathfrak{n}$ its Borel subalgebra and maximal nilpotent subalgebra, respectively. We construct a relationship between the abelian ideals of $mathfrak{b}$ and the c
ohomology of $mathfrak{n}$ with trivial coefficients. By this relationship, we can enumerate the number of abelian ideals of $mathfrak{b}$ with certain dimension via the Poincare polynomials of Weyl groups of type $A_{n-1}$ and $C_n$.
We give a complete study of the Clifford-Weyl algebra ${mathcal C}(n,2k)$ from Bose-Fermi statistics, including Hochschild cohomology (with coefficients in itself). We show that ${mathcal C}(n,2k)$ is rigid when $n$ is even or when $k eq 1$. We find
all non-trivial deformations of ${mathcal C}(2n+1,2)$ and study their representations.