Revisiting the Farey AF Algebra

In a recent paper, F. Boca investigates the AF algebra \mathfrakA{{\mathfrak{A}}} associated with the Farey-Stern-Brocot sequence. We show that \mathfrakA{{\mathfrak{A}}} coincides with the AF algebra \mathfrakM₁{{\mathfrak{M_{1}}}} introduced by the present author in 1988. As proved in that paper (Adv. Math., vol.68.1), the K ₀-group of \mathfrakA{\mathfrak{A}} is the lattice-ordered abelian group M₁{\mathcal{M}_{1}} of piecewise linear functions on the unit interval, each piece having integer coefficients, with the constant 1 as the distinguished order unit. Using the elementary properties of M₁{\mathcal{M}_{1}} we can give short proofs of several results in Boca’s paper. We also prove many new results: among others, \mathfrakA{{\mathfrak{A}}} is a *-subalgebra of Glimm universal algebra, tracial states of \mathfrakA{{\mathfrak{A}}} are in one-one correspondence with Borel probability measures on the unit real interval, all primitive ideals of \mathfrakA{{\mathfrak{A}}} are essential. We describe the automorphism group of \mathfrakA{{\mathfrak{A}}} . For every primitive ideal I of \mathfrakA{{{\mathfrak{A}}}} we compute K ₀(I) and K₀(\mathfrakA/I){{K_{0}(\mathfrak{A}/I)}}.     

Yangians and cohomology rings of Laumon spaces

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL _n . We construct the action of the Yangian of \mathfraksl_n{\mathfrak{sl}_n} in the cohomology of Laumon spaces by certain natural correspondences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of \mathfraksl_n[s^±1,t]{\mathfrak{sl}_n[s^{\pm1},t]}) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of cohomology. This basis is an affine analog of the Gelfand-Tsetlin basis. The affine analog of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space \mathfrakM_n,d{\mathfrak{M}_{n,d}} of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is naturally embedded into the cohomology ring of certain affine Laumon space. It is the image of the center Z of the Yangian of \mathfrakgl_n{\mathfrak{gl}_n} naturally embedded into the affine Yangian. In particular, the first Chern class of the determinant line bundle on \mathfrakM_n,d{\mathfrak{M}_{n,d}} is the image of a noncommutative power sum in Z.     

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}}.     

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.     

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})} .     

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.     

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 )

Absolutely Convergent Extensions of Nonclosable Positive Linear Functionals

The existence of extensions of a positive linear functional ω defined on a dense *-subalgebra \mathfrakA₀{\mathfrak{A}_0} of a topological *-algebra \mathfrakA{\mathfrak{A}}, satisfying certain regularity conditions, is examined. The main interest is focused on the case where ω is nonclosable and sufficient conditions for the existence of an absolutely convergent extension of ω are given.     

On Natural Superhomogeneous Models for Minkowski Superspacetime

It is shown how, the classification of those Lie superalgebras having \mathfraku₂{\mathfrak{u}_2} as its underlying Lie algebra, and further restricted by the fact that the representation of \mathfraku₂{\mathfrak{u}_2} into its odd module is the adjoint representation, leads to at least ten different natural superhomogeneous models for Minkowski superspacetime.     

A modified BLM approach to quantum affine {\mathfrak{gl}_n}

We introduce a spanning set of Beilinson–Lusztig–MacPherson type, {A(j, r)} _A,j , for affine quantum Schur algebras S_\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and construct a linearly independent set {A(j)} _A,j for an associated algebra [^(K)]_\vartriangle(n){{{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n)} . We then establish explicitly some multiplication formulas of simple generators E^\vartriangle_h,h+1(0){E^\vartriangle_{h,h+1}}(\mathbf{0}) by an arbitrary element A(j) in [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle(n)}} via the corresponding formulas in S_\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle(n, r)}} , and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the Ringel–Hall algebras \mathfrak H_\vartriangle(n){{{\boldsymbol{\mathfrak H}_\vartriangle(n)}}} associated with cyclic quivers. This allows us to use the triangular relation between monomial and PBW type bases for \mathfrak H_\vartriangle(n){{\boldsymbol{\mathfrak H}}_\vartriangle}(n) established in Deng and Du (Adv Math 191:276–304, 2005) to derive similar triangular relations for S_\vartriangle(n, r){{{\boldsymbol{\mathcal S}}_\vartriangle}(n, r)} and [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{\mathcal K}}}_\vartriangle}(n) . Using these relations, we then show that the subspace \mathfrak A_\vartriangle(n){{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)} of [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} spanned by {A(j)} _A,j contains the quantum enveloping algebra U_\vartriangle(n){{{\mathbf U}_\vartriangle}(n)} of affine type A as a subalgebra. As an application, we prove that, when this construction is applied to quantum Schur algebras S(n,r){\boldsymbol{\mathcal S}(n,r)} , the resulting subspace \mathfrak A_\vartriangle(n){{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}} is in fact a subalgebra which is isomorphic to the quantum enveloping algebra of \mathfrakgl_n{\mathfrak{gl}_n} . We conjecture that \mathfrak A_\vartriangle(n){{{{{\boldsymbol{\mathfrak A}}_\vartriangle}(n)}}} is a subalgebra of [^(K)]_\vartriangle(n){{\boldsymbol{\widehat{{{\mathcal K}}}}_\vartriangle}(n)} .     

Stability of unitary SK<Subscript>1</Subscript> of Azumaya algebras

For an Azumaya algebra A with center C of rank n ² and a unitary involution τ, we study the stability of the unitary SK₁ under reduction. We show that if R = C ^τ is a Henselian ring with maximal ideal \mathfrakm{\mathfrak{m}} and 2 and n are invertible in R then SK₁(A, t) @ SK₁(A/ \mathfrakm A, overline t){{{\rm SK}_1}(A, \tau) \cong {{\rm SK}_1}(A/ \mathfrak{m} A, overline \tau)}.     

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

A Selberg integral for the Lie algebra A<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A new q-binomial theorem for Macdonald polynomials is employed to prove an A _n analogue of the celebrated Selberg integral. This confirms the \mathfrakg = A_n \mathfrak{g} ={\rm{A}}_{n} case of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg integral for every simple Lie algebra \mathfrakg \mathfrak{g} .     

Localization algebras and deformations of Koszul algebras

We show that the center of a flat graded deformation of a standard Koszul algebra A behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed point set. In particular, the center of A acts by characters on the deformed standard modules, providing a “localization map”. We construct a universal graded deformation of A and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming from the algebra Koszul dual to A. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal deformation for the ring governing a block of parabolic category O{\mathcal{O}} for \mathfrakgl_n{\mathfrak{gl}_n} is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the “category O{\mathcal{O}}” of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.     

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.     

The affine preservers of non-singular matrices

When \mathbbK{\mathbb{K}} is an arbitrary field, we study the affine automorphisms of M_n(\mathbbK){{\rm M}_n(\mathbb{K})} that stabilize GL_n(\mathbbK){{\rm GL}_n(\mathbb{K})}. Using a theorem of Dieudonné on maximal affine subspaces of singular matrices, this is easily reduced to the known case of linear preservers when n > 2 or # ${\mathbb{K} > 2}${\mathbb{K} > 2}. We include a short new proof of the more general Flanders theorem for affine subspaces of M_p,q(\mathbbK){{\rm M}_{p,q}(\mathbb{K})} with bounded rank. We also find that the group of affine transformations of M₂(\mathbbF₂){{\rm M}_2(\mathbb{F}_2)} that stabilize GL₂(\mathbbF₂){{\rm GL}_2(\mathbb{F}_2)} does not consist solely of linear maps. Using the theory of quadratic forms over \mathbbF₂{\mathbb{F}_2}, we construct explicit isomorphisms between it, the symplectic group Sp₄(\mathbbF₂){{\rm Sp}_4(\mathbb{F}_2)} and the symmetric group \mathfrakS₆{\mathfrak{S}_6}.     

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.     

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.