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}}.     

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.     

Complex cycles on algebraic models of smooth manifolds

Every compact smooth manifold M is diffeomorphic to the set X(\mathbbR){X(\mathbb{R})} of real points of a nonsingular projective real algebraic variety X, which is called an algebraic model of M. Each algebraic cycle of codimension k on the complex variety X_\mathbbC=X×_\mathbbR\mathbbC{X_{\mathbb{C}}=X\times_{\mathbb{R}}\mathbb{C}} determines a cohomology class in H^2k(X(\mathbbR);\mathbbD){H^{2k}(X(\mathbb{R});\mathbb{D})} , where \mathbbD{\mathbb{D}} denotes \mathbbZ{\mathbb{Z}} or \mathbbQ{\mathbb{Q}} . We investigate the behavior of such cohomology classes as X runs through the class of algebraic models of M.     

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}.     

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) .     

Legendrian Links, Causality, and the Low Conjecture

Let (X ^m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of ${\mathbb{R}^{m}}Let (X ^m+1, g) be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of \mathbbR^m{\mathbb{R}^{m}} . The Legendrian Low conjecture formulated by Natário and Tod says that two events x, y ? X{x, y \in X} are causally related if and only if the Legendrian link of spheres \mathfrakS_x, \mathfrakS_y{{\mathfrak{S}_x,\,\mathfrak{S}_y}} whose points are light geodesics passing through x and y is non-trivial in the contact manifold of all light geodesics in X. The Low conjecture says that for m = 2 the events x, y are causally related if and only if \mathfrakS_x, \mathfrakS_y{{\mathfrak{S}_x,\,\mathfrak{S}_y}} is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements hold for any globally hyperbolic (X ^m+1, g) such that a cover of its Cauchy surface is diffeomorphic to an open domain in \mathbbR^m{\mathbb{R}^{m}} .     

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

On the rationality of the moduli space of Lüroth quartics

We prove that the moduli space \mathfrakM_L{\mathfrak{M}_L} of Lüroth quartics in \mathbbP²{\mathbb{P}^2}, i.e. the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of PGL₃ (\mathbbC){\mathrm{PGL}_3 (\mathbb{C})} is rational, as is the related moduli space of Bateman seven-tuples of points in \mathbbP²{\mathbb{P}^2}.     

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})} .     

Local–global divisibility by 4 in elliptic curves defined over {\mathbb{Q}}

Let ${\mathcal{E}}Let E{\mathcal{E}} be an elliptic curve defined over \mathbbQ{\mathbb{Q}} . Let P ? E(\mathbb Q){P\in {\mathcal{E}}(\mathbb {Q})} and let q be a positive integer. Assume that for almost all valuations v ? \mathbbQ{v\in \mathbb{Q}} , there exist points D_v ? E(\mathbb Q_v){D_v\in {\mathcal{E}}(\mathbb {Q}_v)} such that P = qD _v . Is it possible to conclude that there exists a point D ? E(\mathbb Q){D\in {\mathcal{E}}(\mathbb {Q})} such that P = qD? A full answer to this question is known when q is a power of almost all primes p ? \mathbbN{p\in \mathbb{N}} , but some cases remain open when p ? S={2,3,5,7,11,13,17,19,37,43,67,163}{p\in S=\{2,3,5,7,11,13,17,19,37,43,67,163\}} . We now give a complete answer in the case when q = 4.     

Simple and Nearly Simple Deep Matrix Algebras

Deep matrix algebras were originally created by Cuntz (Comm. Math. Phys. 57:173–185, 1977) and McCrimmon (2006). Further study of the associative case was done by the author in Kennedy (2004) and Kennedy (Algebr. Represent. Theory 9:525–537, 2006). In this paper, the associative algebra DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) based on a set X over a field \mathbbK{\mathbb{K}} and various of its subalgebras are studied for the purpose of determining the structure of the associated Lie algebra \mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}) and its subalgebras. Several key examples of deep matrix Lie algebras are constructed. These are shown to be either simple or nearly simple depending on the cardinality of the set X. Cartan subalgebras are constructed and two of the key Lie algebras are then decomposed with respect to the adjoint action of these subalgebras. In the process, an infinite dimensional analogue to \mathfraksl₂(\mathbbK)\mathfrak{sl}_2({\mathbb{K}}) is naturally realized as a key subalgebra in deep matrix Lie algebras.     

Topology and Smooth Structure for Pseudoframes

Given a closed subspace ${\mathcal{S}}Given a closed subspace S{\mathcal{S}} of a Hilbert space H{\mathcal{H}}, we study the sets F_S{\mathcal{F}_\mathcal{S}} of pseudo-frames, CF_S{\mathcal{C}\mathcal{F}_\mathcal{S}} of commutative pseudo-frames and \mathfrakX_S{\tiny{\mathfrak{X}}_{\mathcal{S}}} of dual frames for S{\mathcal{S}}, via the (well known) one to one correspondence which assigns a pair of operators (F, H) to a frame pair ({f_n}_{n ? \mathbbN},{h_n}_{n ? \mathbbN}){(\{f_n\}_{n\in\mathbb{N}},\{h_n\}_{n\in\mathbb{N}})},

F:l2? H,     F({cn}n ? \mathbbN )=?n cn fn,F:\ell^2\to\,\mathcal{H}, \quad F\left(\{c_n\}_{n\in\mathbb{N}} \right)=\sum_n c_n f_n,      14. Composition Followed by Differentiation Between H ∞ and Zygmund Spaces      Yongmin Liu  Yanyan Yu 《Complex Analysis and Operator Theory》2012,6(1):121-137 Let j{\varphi} be an analytic self-map of the unit disk \mathbbD{\mathbb{D}}, H(\mathbbD){H(\mathbb{D})} the space of analytic functions on \mathbbD{\mathbb{D}} and g ? H(\mathbbD){g \in H(\mathbb{D})}. The boundedness and compactness of the operator DCj : H￥ ? Z{DC_\varphi : H^\infty \rightarrow { \mathcal Z}} are investigated in this paper.      15. Some low-density parity-check codes derived from finite geometries      Peter Vandendriessche 《Designs, Codes and Cryptography》2010,54(3):287-297 We look at low-density parity-check codes over a finite field \mathbbK{\mathbb{K}} associated with finite geometries T2*(K){T_2^*(\mathcal{K})}, where K{\mathcal{K}} is a sufficiently large k-arc in PG(2, q), with q = p h . The code words of minimum weight are known. With exception of some choices of the characteristic of \mathbbK{\mathbb{K}} we compute the dimension of the code and show that the code is generated completely by its code words of minimum weight.      16. Deep Matrix Modules      Christopher Kennedy 《Algebras and Representation Theory》2011,14(2):317-339 A deep matrix algebra, DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}), is a unital associative algebra over a field \mathbbK\mathbb{K} with basis all deep matrix units, \mathfrake(h,k)\mathfrak{e}(h,k), indexed by pairs of elements h and k taken from a free monoid generated by a set X. After briefly describing the construction of DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}), we determine necessary and sufficient conditions for constructing representations for DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}). With these conditions in place, we define null modules and give three canonical examples of such. A classification of general null modules is then given in terms of the canonical examples along with their submodules and quotients. In the final section, additional examples of natural actions for DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}) are given and their submodules determined depending on the cardinality of the set X.      17. Character formulas for Feigin–Stoyanovsky’s type subspaces of standard $\mathfrak{sl}(3, \mathbb{C})^{\widetilde{}}$-modules      Miroslav Jerković 《The Ramanujan Journal》2012,27(3):357-376 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.      18. Metrizability of the space of {\mathbb{R}} -places of a real function field      Michał Machura  Murray Marshall  Katarzyna Osiak 《Mathematische Zeitschrift》2010,266(1):237-242 For n = 1, the space of ${\mathbb{R}}For n = 1, the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x1,?, xn){\mathbb{R}(x_1,\ldots, x_n)} is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x, y){\mathbb{R}(x, y)} is not metrizable. We explain how the proof generalizes to show that the space of \mathbbR{\mathbb{R}} -places of any finitely generated formally real field extension of \mathbbR{\mathbb{R}} of transcendence degree ≥ 2 is not metrizable. We also consider the more general question of when the space of \mathbbR{\mathbb{R}} -places of a finitely generated formally real field extension of a real closed field is metrizable.      19. A modified BLM approach to quantum affine {\mathfrak{gl}_n}      Jie Du  Qiang Fu 《Mathematische Zeitschrift》2010,266(4):747-781 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\vartriangleh,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 \mathfrakgln{\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)} .      20. An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}      Vitaly Bergelson  Terence Tao  Tamar Ziegler 《Geometric And Functional Analysis》2010,19(6):1539-1596 Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U k (X) for an ergodic action (Tg)g ? \mathbb Fw{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb Fw{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S1{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.

Composition Followed by Differentiation Between H ∞ and Zygmund Spaces

Let j{\varphi} be an analytic self-map of the unit disk \mathbbD{\mathbb{D}}, H(\mathbbD){H(\mathbb{D})} the space of analytic functions on \mathbbD{\mathbb{D}} and g ? H(\mathbbD){g \in H(\mathbb{D})}. The boundedness and compactness of the operator DC_j : H^￥ ? Z{DC_\varphi : H^\infty \rightarrow { \mathcal Z}} are investigated in this paper.     

Some low-density parity-check codes derived from finite geometries

We look at low-density parity-check codes over a finite field \mathbbK{\mathbb{K}} associated with finite geometries T₂^*(K){T_2^*(\mathcal{K})}, where K{\mathcal{K}} is a sufficiently large k-arc in PG(2, q), with q = p ^h . The code words of minimum weight are known. With exception of some choices of the characteristic of \mathbbK{\mathbb{K}} we compute the dimension of the code and show that the code is generated completely by its code words of minimum weight.     

Deep Matrix Modules

A deep matrix algebra, DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}), is a unital associative algebra over a field \mathbbK\mathbb{K} with basis all deep matrix units, \mathfrake(h,k)\mathfrak{e}(h,k), indexed by pairs of elements h and k taken from a free monoid generated by a set X. After briefly describing the construction of DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}), we determine necessary and sufficient conditions for constructing representations for DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}). With these conditions in place, we define null modules and give three canonical examples of such. A classification of general null modules is then given in terms of the canonical examples along with their submodules and quotients. In the final section, additional examples of natural actions for DM(X,\mathbbK)\mathcal{DM}(X,\mathbb{K}) are given and their submodules determined depending on the cardinality of the set X.     

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.     

Metrizability of the space of {\mathbb{R}} -places of a real function field

For n = 1, the space of ${\mathbb{R}}For n = 1, the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x₁,?, x_n){\mathbb{R}(x_1,\ldots, x_n)} is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x, y){\mathbb{R}(x, y)} is not metrizable. We explain how the proof generalizes to show that the space of \mathbbR{\mathbb{R}} -places of any finitely generated formally real field extension of \mathbbR{\mathbb{R}} of transcendence degree ≥ 2 is not metrizable. We also consider the more general question of when the space of \mathbbR{\mathbb{R}} -places of a finitely generated formally real field extension of a real closed field is metrizable.     

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)} .     

An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}

Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U ^k (X) for an ergodic action (T_g)_{g ? \mathbb F^w}{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb F^w{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S¹{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.