Two-step solvable Lie algebras and weight graphs

By using the concept of weight graph associated to nonsplit complex nilpotent Lie algebras \mathfrakg\mathfrak{g}, we find necessary and sufficient conditions for a semidirect product \mathfrakg?^? T_i\mathfrak{g}\overrightarrow{\oplus } T_{i} to be two-step solvable, where $T_{i}TT over \mathfrakg\mathfrak{g} which induces a decomposition of \mathfrakg\mathfrak{g} into one-dimensional weight spaces without zero weights. In particular we show that the semidirect product of such a Lie algebra with a maximal torus of derivations cannot be itself two-step solvable. We also obtain some applications to rigid Lie algebras, as a geometrical proof of the nonexistence of two-step nonsplit solvable rigid Lie algebras in dimensions n\geqslant 3n\geqslant 3.     

A Twining Character Formula For Demazure Modules

We prove a formula for the twining characters of certain Demazure modules, over a Borel subalgebra \mathfrakb\mathfrak{b} of a finite dimensional complex semisimple Lie algebra \mathfrakg\mathfrak{g}. This formula describes the twining character of the Demazure module by the w\omega-Demazure operator associated to an element of the Weyl group that is fixed by the Dynkin diagram automorphism w\omega of \mathfrakg\mathfrak{g}. Our result is a refinement of the twining character formula for the irreducible highest weight \mathfrakg\mathfrak{g}-modules of symmetric dominant integral highest weights, and also of the ordinary Demazure character formula.     

Bounded reductive subalgebras of \mathfrak{s}{\mathfrak{l}_n}

Let \mathfrakg \mathfrak{g} be a reductive Lie algebra and \mathfrakk ì \mathfrakg \mathfrak{k} \subset \mathfrak{g} be a reductive in \mathfrakg \mathfrak{g} subalgebra. A ( \mathfrakg,\mathfrakk \mathfrak{g},\mathfrak{k} )-module M is a \mathfrakg \mathfrak{g} -module for which any element m ∈ M is contained in a finite-dimensional \mathfrakk \mathfrak{k} -submodule of M. We say that a ( \mathfrakg,\mathfrakk \mathfrak{g},\mathfrak{k} )-module M is bounded if there exists a constant C _M such that the Jordan-H?lder multiplicities of any simple finite-dimensional \mathfrakk \mathfrak{k} -module in every finite-dimensional \mathfrakk \mathfrak{k} -submodule of M are bounded by C _M . In the present paper we describe explicitly all reductive in \mathfraks\mathfrakl_n \mathfrak{s}{\mathfrak{l}_n} subalgebras \mathfrakk \mathfrak{k} which admit a bounded simple infinite-dimensional ( \mathfraks\mathfrakl_n,\mathfrakk \mathfrak{s}{\mathfrak{l}_n},\mathfrak{k} )-module. Our technique is based on symplectic geometry and the notion of spherical variety. We also characterize the irreducible components of the associated varieties of simple bounded ( \mathfrakg,\mathfrakk \mathfrak{g},\mathfrak{k} )-modules.     

Cohomology of solvable lie algebras and solvmanifolds

The cohomology H _{\mathfrakg\mathfrak{g} ) of the tangent Lie algebra \mathfrakg\mathfrak{g} of the group G with coefficients in the one-dimensional representation \mathfrakg\mathfrak{g} \mathbbK\mathbb{K} defined by [(W)\tilde] \mathfrakg \tilde \Omega _\mathfrak{g} of H ¹(G/ \mathfrakg\mathfrak{g} .}     

Slices for biparabolics of index 1

Let \mathfraka \mathfrak{a} be an algebraic Lie subalgebra of a simple Lie algebra \mathfrakg \mathfrak{g} with index \mathfraka \mathfrak{a}  ≤ rank \mathfrakg \mathfrak{g} . Let Y( \mathfraka ) Y\left( \mathfrak{a} \right) denote the algebra of \mathfraka \mathfrak{a} invariant polynomial functions on \mathfraka^* {\mathfrak{a}^*} . An algebraic slice for \mathfraka \mathfrak{a} is an affine subspace η + V with h ? \mathfraka^* \eta \in {\mathfrak{a}^*} and V ì \mathfraka^* V \subset {\mathfrak{a}^*} subspace of dimension index \mathfraka \mathfrak{a} such that restriction of function induces an isomorphism of Y( \mathfraka ) Y\left( \mathfrak{a} \right) onto the algebra R[η + V] of regular functions on η + V. Slices have been obtained in a number of cases through the construction of an adapted pair (h, η) in which h ? \mathfraka h \in \mathfrak{a} is ad-semisimple, η is a regular element of \mathfraka^* {\mathfrak{a}^*} which is an eigenvector for h of eigenvalue minus one and V is an h stable complement to ( \textad  \mathfraka )h \left( {{\text{ad}}\;\mathfrak{a}} \right)\eta in \mathfraka^* {\mathfrak{a}^*} . The classical case is for \mathfrakg \mathfrak{g} semisimple [16], [17]. Yet rather recently many other cases have been provided; for example, if \mathfrakg \mathfrak{g} is of type A and \mathfraka \mathfrak{a} is a “truncated biparabolic” [12] or a centralizer [13]. In some of these cases (in particular when the biparabolic is a Borel subalgebra) it was found [13], [14], that η could be taken to be the restriction of a regular nilpotent element in \mathfrakg \mathfrak{g} . Moreover, this calculation suggested [13] how to construct slices outside type A when no adapted pair exists. This article makes a first step in taking these ideas further. Specifically, let \mathfraka \mathfrak{a} be a truncated biparabolic of index one. (This only arises if \mathfrakg \mathfrak{g} is of type A and \mathfraka \mathfrak{a} is the derived algebra of a parabolic subalgebra whose Levi factor has just two blocks whose sizes are coprime.) In this case it is shown that the second member of an adapted pair (h, η) for \mathfraka \mathfrak{a} is the restriction of a particularly carefully chosen regular nilpotent element of \mathfrakg \mathfrak{g} . A by-product of our analysis is the construction of a map from the set of pairs of coprime integers to the set of all finite ordered sequences of ±1.     

On divisible weighted Dynkin diagrams and reachable elements

Let e be a nilpotent element of a complex simple Lie algebra $ \mathfrak{g} Let e be a nilpotent element of a complex simple Lie algebra \mathfrakg \mathfrak{g} . The weighted Dynkin diagram of e, D(e) \mathcal{D}(e) , is said to be divisible if D(e)

Lifting automorphisms of generalized adjoint quotients

Let G be a reductive algebraic group over an algebraically closed field K of characteristic zero. Let p:\mathfrakg^r ? X = \mathfrakg^r//G \pi :{\mathfrak{g}^r} \to X = {\mathfrak{g}^r}//G be the categorical quotient where \mathfrakg \mathfrak{g} is the adjoint representation of G and r is a suitably large integer (in general r ≥ 5, but for many cases r ≥ 3 or even r ≥ 2 suffices). We show that every automorphism φ of X lifts to a map F:\mathfrakg^r ? \mathfrakg^r \Phi :{\mathfrak{g}^r} \to {\mathfrak{g}^r} commuting with π. As an application we consider the action of φ on the Luna stratification of X.     

On quasiorder lattices and topology lattices of algebras

In this paper, it is shown that the dual [(\textQord)\tilde]\mathfrakA \widetilde{\text{Qord}}\mathfrak{A} of the quasiorder lattice of any algebra \mathfrakA \mathfrak{A} is isomorphic to a sublattice of the topology lattice á( \mathfrakA ) \Im \left( \mathfrak{A} \right) . Further, if \mathfrakA \mathfrak{A} is a finite algebra, then [(\textQord)\tilde]\mathfrakA @ á( \mathfrakA ) \widetilde{\text{Qord}}\mathfrak{A} \cong \Im \left( \mathfrak{A} \right) . We give a sufficient condition for the lattices [(\textCon)\tilde]\mathfrakA\text, [(\textQord)\tilde]\mathfrakA \widetilde{\text{Con}}\mathfrak{A}{\text{,}} \widetilde{\text{Qord}}\mathfrak{A} , and á( \mathfrakA ) \Im \left( \mathfrak{A} \right) . to be pairwise isomorphic. These results are applied to investigate topology lattices and quasiorder lattices of unary algebras.     

Solvable Lie algebras are not that hypo

We study a type of left-invariant structure on Lie groups or, equivalently, on Lie algebras. We introduce obstructions to the existence of a hypo structure, namely the five-dimensional geometry of hypersurfaces in manifolds with holonomy SU(3). The choice of a splitting \mathfrakg^* = V₁ ?V₂ {\mathfrak{g}^*} = {V_1} \oplus {V_2} , and the vanishing of certain associated cohomology groups, determine a first obstruction. We also construct necessary conditions for the existence of a hypo structure with a fixed almost-contact form. For nonunimodular Lie algebras, we derive an obstruction to the existence of a hypo structure, with no choice involved. We apply these methods to classify solvable Lie algebras that admit a hypo structure.     

Pontrjagin–Thom maps and the homology of the moduli stack of stable curves

We study the singular homology (with field coefficients) of the moduli stack [`(\mathfrakM)]<sub>g, n</sub>{\overline{\mathfrak{M}}_{g, n}} of stable n-pointed complex curves of genus g. Each irreducible boundary component of [`(\mathfrakM)]_{g, n}{\overline{\mathfrak{M}}_{g, n}} determines via the Pontrjagin–Thom construction a map from [`(\mathfrakM)]<sub>g, n</sub>{\overline{\mathfrak{M}}_{g, n}} to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This detects many new torsion classes in the homology of [`(\mathfrakM)]_{g, n}{\overline{\mathfrak{M}}_{g, n}}.     

Enveloping algebras of Slodowy slices and Goldie rank

Let U( \mathfrakg,e ) U\left( {\mathfrak{g},e} \right) be the finite W-algebra associated with a nilpotent element e in a complex simple Lie algebra \mathfrakg = \textLie(G) \mathfrak{g} = {\text{Lie}}(G) and let I be a primitive ideal of the enveloping algebra U( \mathfrakg ) U\left( \mathfrak{g} \right) whose associated variety equals the Zariski closure of the nilpotent orbit (Ad G) e. Then it is known that I = \textAn\textn_{U( \mathfrakg )}( Q_e ?_{U( \mathfrakg,e )}V ) I = {\text{An}}{{\text{n}}_{U\left( \mathfrak{g} \right)}}\left( {{Q_e}{ \otimes_{U\left( {\mathfrak{g},e} \right)}}V} \right) for some finite dimensional irreducible U( \mathfrakg,e ) U\left( {\mathfrak{g},e} \right) -module V, where Q _e stands for the generalised Gelfand–Graev \mathfrakg \mathfrak{g} -module associated with e. The main goal of this paper is to prove that the Goldie rank of the primitive quotient U( \mathfrakg )

Modular Lie algebras and the Gelfand–Kirillov conjecture

Let ${\mathfrak{g}}Let \mathfrakg{\mathfrak{g}} be a finite dimensional simple Lie algebra over an algebraically closed field \mathbbK\mathbb{K} of characteristic 0. Let \mathfrakg_\mathbbZ{\mathfrak{g}}_{{\mathbb{Z}}} be a Chevalley ℤ-form of \mathfrakg{\mathfrak{g}} and \mathfrakg_\Bbbk=\mathfrakg_\mathbbZ?_\mathbbZ\Bbbk{\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk, where \Bbbk\Bbbk is the algebraic closure of  \mathbbF_p{\mathbb{F}}_{p}. Let G_\BbbkG_{\Bbbk} be a simple, simply connected algebraic \Bbbk\Bbbk-group with \operatornameLie(G_\Bbbk)=\mathfrakg_\Bbbk\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U(\mathfrakg_\Bbbk)U({\mathfrak{g}}_{\Bbbk}) to show that if the Gelfand–Kirillov conjecture (from 1966) holds for \mathfrakg{\mathfrak{g}}, then for all p≫0 the field of rational functions \Bbbk (\mathfrakg_\Bbbk)\Bbbk ({\mathfrak{g}}_{\Bbbk}) is purely transcendental over its subfield \Bbbk(\mathfrakg_\Bbbk)^G_\Bbbk\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions \mathbbK(\mathfrakg){\mathbb{K}}({\mathfrak{g}}) is not purely transcendental over its subfield \mathbbK(\mathfrakg)^\mathfrakg{\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}} if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄, then the Lie field of \mathfrakg{\mathfrak{g}} is more complicated than expected.     

Algebras of twisted chiral differential operators and affine localization of {\mathfrak {g}} -modules

We propose a notion of algebra of twisted chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of “smallest” such modules are irreducible [^(\mathfrakg)]{{\hat{{\mathfrak{g}}}}} -modules, and all irreducible \mathfrakg{{\mathfrak{g}}} -integrable [^(\mathfrakg)]{{\hat{{\mathfrak{g}}}}} -modules at the critical level arise in this way.     

Determination of 7-Dimensional Indecomposable Nilpotent Complex Lie Algebras by Adjoining a Derivation to 6-Dimensional Lie Algebras

For any complex 6-dimensional nilpotent Lie algebra \mathfrakg,\mathfrak{g}, we compute the strain of all indecomposable 7-dimensional nilpotent Lie algebras which contain \mathfrakg\mathfrak{g} by the adjoining a derivation method. We get a new determination of all 7-dimensional complex nilpotent Lie algebras, allowing to check earlier results (some contain errors), along with a cross table intertwining nilpotent 6- and 7-dimensional Lie algebras.     

Around splitting and reaping for partitions of ω

We investigate splitting number and reaping number for the structure (ω) ^ω of infinite partitions of ω. We prove that \mathfrakr_d ￡ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and \mathfraks_d 3 \mathfrakb{\mathfrak{s}_{d}\geq\mathfrak{b}} . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})} . To prove the consistency \mathfrakr_d < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and \mathfraks_d < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants \mathfrakr_pair{\mathfrak{r}_{pair}} and \mathfraks_pair{\mathfrak{s}_{pair}} . We also study the relation between \mathfrakr_pair, \mathfraks_pair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that cov(M),cov(N) ￡ \mathfrakr_pair ￡ \mathfraks_d,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and \mathfraks ￡ \mathfraks_pair ￡ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} .     

On algebraic integrability of Gelfand-Zeitlin fields

We generalize a result of Kostant and Wallach concerning the algebraic integrability of the Gelfand-Zeitlin vector fields to the full set of strongly regular elements in \mathfrakg\mathfrakl \mathfrak{g}\mathfrak{l} (n, ℂ). We use decomposition classes to stratify the strongly regular set by subvarieties X_D {X_\mathcal{D}} . We construct an étale cover [^(\mathfrakg)]_D {\hat{\mathfrak{g}}}_\mathcal{D} of X_D {X_\mathcal{D}} and show that X_D {X_\mathcal{D}} and [^(\mathfrakg)]_D {\hat{\mathfrak{g}}}_\mathcal{D} are smooth and irreducible. We then use Poisson geometry to lift the Gelfand-Zeitlin vector fields on X_D {X_\mathcal{D}} to Hamiltonian vector fields on [^(\mathfrakg)]_D {\hat{\mathfrak{g}}}_\mathcal{D} and integrate these vector fields to an action of a connected, commutative algebraic group.     

Polynomiality of invariants, unimodularity and adapted pairs

Let \mathfraka \mathfrak{a} be a finite-dimensional Lie algebra and Y( \mathfraka ) Y\left( \mathfrak{a} \right) the \mathfraka \mathfrak{a} invariant subalgebra of its symmetric algebra S( \mathfraka ) S\left( \mathfrak{a} \right) under adjoint action. Recently there has been considerable interest in studying situations when Y( \mathfraka ) Y\left( \mathfrak{a} \right) may be polynomial on index \mathfraka \mathfrak{a} generators, for example if \mathfraka \mathfrak{a} is a biparabolic or a centralizer \mathfrakg^x {\mathfrak{g}^x} in a semisimple Lie algebra \mathfrakg \mathfrak{g} .     

Dual partially harmonic tensors and Brauer-Schur-Weyl duality

Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let $ \mathfrak{B}_n^{(f)} Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let \mathfrakB_n^(f) \mathfrak{B}_n^{(f)} be the two-sided ideal of the Brauer algebra \mathfrakB_n( - 2m ) {\mathfrak{B}_n}\left( { - 2m} \right) over K generated by e ₁ e _3⋯ e _2f-1 where 0 ≤ f ≤ [n/2]. Let HT_f ^?n \mathcal{H}\mathcal{T}_f^{ \otimes n} be the subspace of partial-harmonic tensors of valence f in V ^⊗n . In this paper we prove that dimHT_f ^?n \mathcal{H}\mathcal{T}_f^{ \otimes n} and dim \textEn\textd_K\textSp(V)( V ^?n \mathord

Über die Bedingung Going up für {R \subset \hat{R}}

Let (R,\mathfrak m){(R,\mathfrak m)} be a noetherian, local ring with completion [^(R)]{\hat{R}} . We show that R ì [^(R)]{R \subset \hat{R}} satisfies the condition Going up if and only if there exists to every artinian R-module M with Ann_R(M) ì \mathfrakp{{\rm Ann}_R(M) \subset \mathfrak{p}} a submodule U ì M{U \subset M} with Ann_R(U)=\mathfrakp.{{\rm {Ann}}_R(U)=\mathfrak{p}.} This is further equivalent to R being formal catenary, to α(R) = 0 and to H^d_{\mathfrakq/\mathfrakp}(R/\mathfrakp)=0{H^d_{\mathfrak{q}/\mathfrak{p}}(R/\mathfrak{p})=0} for all prime ideals \mathfrakp ì \mathfrakq \subsetneq \mathfrakm{\mathfrak{p} \subset \mathfrak{q} \subsetneq \mathfrak{m}} where d = dim(R/\mathfrakp){d = {\rm {dim}}(R/\mathfrak{p})}.