Partition models for the crystal of the basic $U_{q}(\widehat {\mathfrak{sl}}_{n})$-module

For each n ≥3, we construct an uncountable family of models of the crystal of the basic U_q([^(\mathfrak sl)]_n)U_{q}(\widehat {\mathfrak {sl}}_{n})-module. These models are all based on partitions, and include the usual n-regular and n-restricted models, as well as Berg’s ladder crystal, as special cases.     

Module categories over pointed Hopf algebras

We develop some techniques for studying exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the small quantum groups u_q(\mathfraksl₂){u_q(\mathfrak{sl}_2)}.     

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.     

Reflection equation algebras, coideal subalgebras, and their centres

Reflection equation algebras and related U_q(\mathfrak g){U{_q}(\mathfrak g)} -comodule algebras appear in various constructions of quantum homogeneous spaces and can be obtained via transmutation or equivalently via twisting by a cocycle. In this paper we investigate algebraic and representation theoretic properties of such so called ‘covariantized’ algebras, in particular concerning their centres, invariants, and characters. The locally finite part F_l(U_q (\mathfrak g)){F_l(U{_q} (\mathfrak g))} of U_q(\mathfrak g){U{_q}(\mathfrak g)} with respect to the left adjoint action is a special example of a covariantized algebra. Generalising Noumi’s construction of quantum symmetric pairs we define a coideal subalgebra B _f of U_q(\mathfrak g){U{_q}(\mathfrak g)} for each character f of a covariantized algebra. We show that for any character f of F_l(U_q(\mathfrak g)){F_l(U{_q}(\mathfrak g))} the centre Z(B _f ) canonically contains the representation ring Rep(\mathfrak g){{\rm Rep}(\mathfrak g)} of the semisimple Lie algebra \mathfrak g{\mathfrak g} . We show moreover that for \mathfrak g = \mathfrak sl_n(\mathbb C){\mathfrak g = {\mathfrak sl}_n(\mathbb C)} such characters can be constructed from any invertible solution of the reflection equation and hence we obtain many new explicit realisations of Rep(\mathfrak sl_n(\mathbb C)){{\rm Rep}({\mathfrak sl}_n(\mathbb C))} inside U_q(\mathfrak sl_n(\mathbb C)){U_q({\mathfrak sl}_n(\mathbb C))} . As an example we discuss the solutions of the reflection equation corresponding to the Grassmannian manifold Gr(m,2m) of m-dimensional subspaces in \mathbb C^2m{{\mathbb C}^{2m}}.     

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}}.     

A simple bijectio n betwee n sta ndard 3× n tableaux a nd irreducible webs for $\mathfrak{sl}_{3}$

Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and \mathfrak sl₂\mathfrak {sl}_{2}. Building on a result of Kuperberg, Khovanov–Kuperberg found a recursive algorithm giving a bijection between standard Young tableaux of shape 3×n and irreducible webs for \mathfraksl₃\mathfrak{sl}_{3} whose boundary vertices are all sources.     

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}}.     

Character formulas for Feigin–Stoyanovsky’s type subspaces of standard $\mathfrak{sl}(3, \mathbb{C})^{\widetilde{}}$-modules

Exact sequences of Feigin–Stoyanovsky’s type subspaces for affine Lie algebra \mathfraksl(l+1,\mathbbC)^[\tilde]\mathfrak{sl}(l+1,\mathbb{C})^{\widetilde{}} lead to systems of recurrence relations for formal characters of those subspaces. By solving the corresponding system for \mathfraksl(3,\mathbbC)^[\tilde]\mathfrak{sl}(3,\mathbb{C})^{\widetilde{}}, we obtain a new family of character formulas for all Feigin–Stoyanovsky’s type subspaces at the general level.     

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})}.     

Expected degree of weights in Demazure modules of {\hat{\mathfrak{sl}}_2}

We compute the expected degree of a randomly chosen element in a basis of weight vectors in the Demazure module V _w (Λ) of [^(\mathfraksl)]₂ {\hat{\mathfrak{sl}}_2} . We obtain en passant a new proof of Sanderson's dimension formula for these Demazure modules.     

Quantum affine Gelfand�CTsetlin bases and quantum toroidal algebra via K-theory of affine 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 quantum loop algebra U_v(L\mathfraksl_n){U_v({\bf L}\mathfrak{sl}_n)} in the K-theory of Laumon spaces by certain natural correspondences. Also we construct the action of the quantum toroidal algebra ü _v ([^(\mathfraksl)]_n){(\widehat{\mathfrak{sl}}_n)} in the K-theory of the affine version of Laumon spaces.     

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.     

Approximate Subgroups of Linear Groups

We establish various results on the structure of approximate subgroups in linear groups such as SL _n (k) that were previously announced by the authors. For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we show that any approximate subgroup of SL_n(\mathbb F_q){{\rm SL}_{n}({\mathbb {F}}_{q})} which generates the group must be either very small or else nearly all of SL_n(\mathbb F_q){{\rm SL}_{n}({\mathbb {F}}_{q})}. The argument generalises to other absolutely almost simple connected (and non-commutative) algebraic groups G over an arbitrary field k and yields a classification of approximate subgroups of G(k). In a subsequent paper, we will give applications of this result to the expansion properties of Cayley graphs.     

Quantum symmetric pairs and representations of double affine Hecke algebras of type {C^\vee C_n}

We build representations of the affine and double affine braid groups and Hecke algebras of type C^ú C_n{C^\vee C_n} based upon the theory of quantum symmetric pairs (U, B). In the case U=U_q(\mathfrakgl_N){{\bf U}=\mathcal{U}_{\rm q}(\mathfrak{gl}_N)}, our constructions provide a quantization of the representations constructed by Etingof, Freund, and Ma in [15], and also a type C^ú C_n{C^\vee C_n} generalization of the results in [19].     

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})} .     

( q,t)-analogues and $GL_{n}({\ma thbb{F}}_{ q})$

We start with a (q,t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These (q,t)-binomial coefficients and their interpretations generalize further in two directions, one relating to column-strict tableaux and Macdonald’s “7 ^th variation” of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of GL_n(\mathbbF_q)GL_{n}({\mathbb{F}}_{q}) .