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)

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.     

An inverse spectral theorem for Kre?n strings with a negative eigenvalue

A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L²(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D_\mathfrakmD_x{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as

Discreteness and rationality of F-jumping numbers on singular varieties

We prove that the F-jumping numbers of the test ideal t(X; D, \mathfraka^t){\tau(X; \Delta, \mathfrak{a}^t)} are discrete and rational under the assumptions that X is a normal and F-finite scheme over a field of positive characteristic p, K _X  + Δ is \mathbb Q{\mathbb {Q}}-Cartier of index not divisible p, and either X is essentially of finite type over a field or the sheaf of ideals \mathfraka{\mathfrak{a}} is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero.     

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.     

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})}.     

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} .     

Automorphisms of a finitary factor power of an infinite symmetric group

We consider a semigroup FP_\textfin⁺ ( \mathfrakS_\textfin( \mathbbN ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) defined as a finitary factor power of a finitary symmetric group of countable order. It is proved that all automorphisms of FP_\textfin⁺ ( \mathfrakS_\textfin( \mathbbN ) ) FP_{\text{fin}}^{+} \left( {{\mathfrak{S}_{\text{fin}}}\left( \mathbb{N} \right)} \right) are induced by permutations from \mathfrakS( \mathbbN ) \mathfrak{S}\left( \mathbb{N} \right) .     

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 one class of extreme extensions of a measure

We consider a relationship between two sets of extensions of a finite finitely additive measure μ defined on an algebra \mathfrakB \mathfrak{B} of sets to a broader algebra \mathfrakA \mathfrak{A} . These sets are the set ex S _μ of all extreme extensions of the measure μ and the set H _μ of all extensions defined as l(A) = [^(m)]( h(A) ),   A ? \mathfrakA \lambda (A) = \hat{\mu }\left( {h(A)} \right),\,\,\,A \in \mathfrak{A} , where [^(m)] \hat{\mu } is a quotient measure on the algebra \mathfrakB

A semilattice of numberings. II

\mathfrakc \mathfrak{c} -Universal semilattices \mathfrakA \mathfrak{A} of the power of the continuum (of an upper semilattice of m-degrees ) on admissible sets are studied. Moreover, it is shown that a semilattice of \mathbbH\mathbbF( \mathfrakM ) \mathbb{H}\mathbb{F}\left( \mathfrak{M} \right) -numberings of a finite set is \mathfrakc \mathfrak{c} -universal if \mathfrakM \mathfrak{M} is a countable model of a c-simple theory.     

Balanced Deep Matrix Algebras

This paper continues the study of associative and Lie deep matrix algebras, DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) and \mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras, BDM(X,\mathbbK){\mathcal{BDM}}(X,{\mathbb{K}}) and \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}), are defined and studied for an infinite set X. The global structures of these two algebras are studied, devising a depth grading on both as well as determining their ideal lattices. In particular, \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) is shown to be semisimple. The Lie algebra \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) possesses a deep Cartan decomposition and is locally finite with every finite subalgebra naturally enveloped by a semi-direct product of \mathfraksl_n{\mathfrak{{sl}_n}}’s. We classify all associative bilinear forms on \mathfraksl₂\mathfrakd{\mathfrak{sl}_2\mathfrak{d}} (a natural depth analogue of \mathfraksl₂{\mathfrak{{sl}_2}}) and \mathfrakbld{\mathfrak{bld}}.     

An $ \mathfrak{X} $-crown of a finite soluble group

Let G be a finite soluble group and F_\mathfrakX(G) {\Phi_\mathfrak{X}}(G) an intersection of all those maximal subgroups M of G for which 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

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.     

Local limit theorem for the first crossing time of a fixed level by a random walk

Let X,X(1),X(2),... be independent identically distributed random variables with mean zero and a finite variance. Put S(n) = X(1) + ... + X(n), n = 1, 2,..., and define the Markov stopping time η _y = inf {n ≥ 1: S(n) ≥ y} of the first crossing a level y ≥ 0 by the random walk S(n), n = 1, 2,.... In the case $ \mathbb{E} $ \mathbb{E} |X|³ < ∞, the following relation was obtained in [8]: $ \mathbb{P}\left( {\eta _0 = n} \right) = \frac{1} {{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right) $ \mathbb{P}\left( {\eta _0 = n} \right) = \frac{1} {{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right) as n → ∞, where the constant R and the bounded sequence ν _n were calculated in an explicit form. Moreover, there were obtained necessary and sufficient conditions for the limit existence $ H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) $ H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) for every fixed y ≥ 0, and there was found a representation for H(y). The present paper was motivated by the following reason. In [8], the authors unfortunately did not cite papers [1, 5] where the above-mentioned relations were obtained under weaker restrictions. Namely, it was proved in [5] the existence of the limit $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right) for every fixed y ≥ 0 under the condition $ \mathbb{E} $ \mathbb{E} X ² < ∞ only; In [1], an explicit form of the limit $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right) $ \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right) was found under the same condition $ \mathbb{E} $ \mathbb{E} X ² < ∞ in the case when the summand X has an arithmetic distribution. In the present paper, we prove that the main assertion in [5] fails and we correct the original proof. It worth noting that this corrected version was formulated in [8] as a conjecture.     

Soluble minimal non-(finite-by-Baer)-groups

Let \mathfrakX{\mathfrak{X}} be a class of groups. A group G is called a minimal non- \mathfrakX{\mathfrak{X}}-group if it is not an \mathfrakX{\mathfrak{X}}-group but all of whose proper subgroups are \mathfrakX{\mathfrak{X}}-groups. In [16], Xu proved that if G is a soluble minimal non-Baer-group, then G/G ^′′ is a minimal non-nilpotent-group which possesses a maximal subgroup. In the present note, we prove that if G is a soluble minimal non-(finite-by-Baer)-group, then for all integer n ≥ 2, G/γ _n (G′) is a minimal non-(finite-by-abelian)-group.