Related papers: Factorisation of Lie Resolvents

Let G be a reductive complex algebraic group and V a finite-dimensional G-module. From elements of the invariant algebra C[V]^G we obtain by polarization elements of C[kV]^G, where k\geq 1 and kV denotes the direct sum of k copies of V. For…

The generalized p-characters for the restricted Lie color are defined. The simplicity of the induced modules for the FP triples are determined. As applications, we have obtained an analogue of the Kac-Weisfiler theorem for the algebraic Lie…

The Waring Problem over polynomial rings asks how to decompose a homogeneous polynomial $p$ of degree $d$ as a finite sum of $d$-{th} powers of linear forms. In this work we give an algorithm to obtain a real Waring decomposition of any…

Given a prime $p$, and $v_p(a)$ stand for the $p$-adic valuation of the element $a$ in a finite extension $K$ of $\mathbf{Q}_p$, or more generally the field $\mathbf{C}_p$ which is the complete field of the algebraic closure $\mathbf{Q}_p$…

This survey article is an introduction to some of Lusztig's work on the character theory of a finite group of Lie type $G(F_q)$, where $q$ is a power of a prime~$p$. It is partly based on two series of lectures given at the Centre Bernoulli…

We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma$ be a finite index subgroup of $\mathrm{SL}(2,\mathbb{Z})$ with an action on a complex simple Lie algebra $\mathfrak…

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A layer sum is introduced as the sum of formal exponentials of the distinct weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that the character of…

Given a faithful finite-dimensional representation $V$ of a finite group $G$ over any field $\mathbb{F}$, we show that any irreducible ${\mathbb{F}}G$-module $W$ appears, as a submodule or a quotient, in $\mathrm{Sym}^m(V)$ for some integer…

We provide a series of examples of finite groups G and mod p representations V of G whose stable endomorphisms are all given by scalars such that V has a universal deformation ring R(G,V) which is large in the sense that R(G,V)/pR(G,V) is…

Decomposition classes provide a way of partitioning the Lie algebras of an algebraic group into equivalence classes based on the Jordan decomposition. In this paper, we investigate the decomposition classes of the Lie algebras of connected…

We construct a family of exact functors from the BGG category of representations of the Lie algebra sl to the category of finite-dimensional representations of the degenerate (or graded) affine Hecke algebra H of GL. These functors…

Let $Tr$ denote the trace $\mathbb{Z}$-module homomorphism defined on the ring $\mathcal{O}_{L} $ of the integers of a number field $L.$ We show that $Tr(\mathcal{O}_{L})\varsubsetneq \mathbb{Z}$ if and only if there is a prime factor $p$…

Let $F$ be an algebraically closed field of characteristic zero, and let $p$ be an odd prime. We show that the center of the generic division algebra of degree $p$ is stably rational over $F$. Equivalently, if we let $V=M_p(F) \oplus…

Let F be a non-Archimedean locally compact field of residue characteristic p, let G be an inner form of GL(n,F) with n>0, and let l be a prime number different from p. We describe the block decomposition of the category of finite length…

Let $S=K[x_1,...,x_n]$ or $S=K[[x_1,...,x_n]]$ be either a polynomial or a formal power series ring in a finite number of variables over a field $K$ of characteristic $p > 0$ with $[K:K^p] < \infty$. Let $R$ be the hypersurface $S/fS$ where…

We develop the theory of p-Lie algebras of finite Morley rank. In particular, we obtain a quite complete characterization in the soluble case

The present paper studies Hecke rings derived by the automorphism groups of certain algebras $L_p$ over the ring of $p$-adic integers. Our previous work considered the case where $L_p$ is the Heisenberg Lie algebra (of dimension 3) over the…

Let $K$ be an arbitrary field of characteristic zero and $A$ a commutative associative $ K$-algebra which is an integral domain. Denote by $R$ the fraction field of $A$ and by $W(A)=RDer_{\mathbb K}A,$ the Lie algebra of $\mathbb…

Let $p>0$ be a prime, $G$ be a finite $p$-group and $\Bbbk$ be an algebraically closed field of characteristic $p$. Dave Benson has conjectured that if $p=2$ and $V$ is an odd-dimensional indecomposable representation of $G$ then all…

Given a prime $p \ge 5$ and an abstract odd representation $\rho_n$ with coefficients modulo $p^n$ (for some $n \ge 1$) and big image, we prove the existence of a lift of $\rho_n$ to characteristic $0$ whenever local lifts exist (under some…