Irreducible representations of the classical Lie superalgebra of type A(0,n) in prime characteristic

Let $\mathfrak{g }=\mathfrak{s }\mathfrak{l }(1|n+1)$ be the classical Lie superalgebra of type $A(0,n)$ over an algebraically closed field of prime characteristic $p>2$ . A sufficient condition is provided for baby Kac $\mathfrak{g }$ -modules to be simple. Moreover, simple $\mathfrak{g }$ -modules with (quasi) regular semisimple characters are classified. In particular, up to isomorphism, all the simple modules for $\mathfrak{s }\mathfrak{l }(1|2)$ are determined, and representatives and dimensions of simples are precisely given. As an application, simple modules for the general linear Lie superalgebra $\mathfrak{g }\mathfrak{l }(1|n+1)$ with certain $p$ -characters are classified. In particular, a complete classification of simple $\mathfrak{g }\mathfrak{l }(1|2)$ -modules is given.     

Computing α-Invariants of Singular del Pezzo Surfaces

We prove a new local inequality for divisors on surfaces and utilize it to compute α-invariants of singular del Pezzo surfaces, which implies that del Pezzo surfaces of degree one whose singular points are of type $\mathbb{A}_{1}$ , $\mathbb{A}_{2}$ , $\mathbb{A}_{3}$ , $\mathbb{A}_{4}$ , $\mathbb{A}_{5}$ , or $\mathbb{A}_{6}$ are Kähler-Einstein.     

Supports of (\mathfrak{g},\mathfrak{k})-modules of finite type

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{k}$ be a reductive subalgebra in $\mathfrak{g}$ . We say that a $\mathfrak{g}$ -module M is a $(\mathfrak{g},\mathfrak{k})$ -module if M, considered as a $\mathfrak{k}$ -module, is a direct sum of finite-dimensional $\mathfrak{k}$ -modules. We say that a $(\mathfrak{g},\mathfrak{k})$ -module M is of finite type if all $\mathfrak{k}$ -isotopic components of M are finite-dimensional. In this paper we prove that any simple $(\mathfrak{g},\mathfrak{k})$ -module of finite type is holonomic. A simple $\mathfrak{g}$ -module M is associated with the invariants V(M), V(LocM), and L(M) reflecting the ??directions of growth of M.?? We also prove that for a given pair $(\mathfrak{g},\mathfrak{k})$ the set of possible invariants is finite.     

Uniqueness of \mathbb{C}^{ * } - and \mathbb{C}_{{\text{ + }}} -actions on Gizatullin Surfaces

A Gizatullin surface is a normal affine surface V over $ \mathbb{C} $ , which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with $ \mathbb{C}_{{\text{ + }}} $ -actions on V considered up to a “speed change”. Non-Gizatullin surfaces are known to admit at most one $ \mathbb{A}^{1} $ -fibration V → S up to an isomorphism of the base S. Moreover, an effective $ \mathbb{C}^{ * } $ -action on them, if it does exist, is unique up to conjugation and inversion t $ \mapsto $ t ^?1 of $ \mathbb{C}^{ * } $ . Obviously, uniqueness of $ \mathbb{C}^{ * } $ -actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of $ \mathbb{C}^{ * } $ -actions and $ \mathbb{A}^{{\text{1}}} $ -fibrations, see, e.g., [FKZ1]. In the present paper we obtain a criterion as to when $ \mathbb{A}^{{\text{1}}} $ -fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base $ S \cong \mathbb{A}^{{\text{1}}} $ . We exhibit as well large subclasses of Gizatullin $ \mathbb{C}^{ * } $ -surfaces for which a $ \mathbb{C}^{ * } $ -action is essentially unique and for which there are at most two conjugacy classes of $ \mathbb{A}^{{\text{1}}} $ -fibrations over $ \mathbb{A}^{{\text{1}}} $ .     

Equivariant map superalgebras

Suppose a group $\Gamma $ acts on a scheme $X$ and a Lie superalgebra $\mathfrak {g}$ . The corresponding equivariant map superalgebra is the Lie superalgebra of equivariant regular maps from $X$ to $\mathfrak {g}$ . We classify the irreducible finite dimensional modules for these superalgebras under the assumptions that the coordinate ring of $X$ is finitely generated, $\Gamma $ is finite abelian and acts freely on the rational points of $X$ , and $\mathfrak {g}$ is a basic classical Lie superalgebra (or $\mathfrak {sl}\,(n,n)$ , $n \ge 1$ , if $\Gamma $ is trivial). We show that they are all (tensor products of) generalized evaluation modules and are parameterized by a certain set of equivariant finitely supported maps defined on $X$ . Furthermore, in the case that the even part of $\mathfrak {g}$ is semisimple, we show that all such modules are in fact (tensor products of) evaluation modules. On the other hand, if the even part of $\mathfrak {g}$ is not semisimple (more generally, if $\mathfrak {g}$ is of type I), we introduce a natural generalization of Kac modules and show that all irreducible finite dimensional modules are quotients of these. As a special case, our results give the first classification of the irreducible finite dimensional modules for twisted loop superalgebras.     

A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras

We study cohomological induction for a pair $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ , $ \mathfrak{g} $ being an infinitedimensional locally reductive Lie algebra and $ \mathfrak{k} \subset \mathfrak{g} $ being of the form $ \mathfrak{k}_{0} \subset C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ , where $ \mathfrak{k}_{0} \subset \mathfrak{g} $ is a finite-dimensional reductive in $ \mathfrak{g} $ subalgebra and $ C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ is the centralizer of $ \mathfrak{k}_{0} $ in $ \mathfrak{g} $ . We prove a general nonvanishing and $ \mathfrak{k} $ -finiteness theorem for the output. This yields, in particular, simple $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ -modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when $ \mathfrak{g} $ is a root-reductive or diagonal locally simple Lie algebra.     

A zoo of diffeomorphism groups on \mathbb{R }^{n}

We consider the groups ${\mathrm{Diff }}_\mathcal{B }(\mathbb{R }^n)$ , ${\mathrm{Diff }}_{H^\infty }(\mathbb{R }^n)$ , and ${\mathrm{Diff }}_{\mathcal{S }}(\mathbb{R }^n)$ of smooth diffeomorphisms on $\mathbb{R }^n$ which differ from the identity by a function which is in either $\mathcal{B }$ (bounded in all derivatives), $H^\infty = \bigcap _{k\ge 0}H^k$ , or $\mathcal{S }$ (rapidly decreasing). We show that all these groups are smooth regular Lie groups.     

Value-sets of polynomials at p-adic integers

In this paper we are concerned with the classification of the subsets A of ${\mathbb{Z}_p}$ which occur as images ${f(\mathbb{Z}_p^r)}$ of polynomial functions ${f:\mathbb{Z}_p^r\to \mathbb{Z}_p}$ , limiting ourselves to compact-open subsets (i.e. finite unions of open balls). We shall prove three main results: (i) Every compact-open ${A\subset \mathbb{Z}_p}$ is of the shape ${A=f(\mathbb{Z}_p^r)}$ for suitable r and ${f\in\mathbb{Z}_p[X_1,\ldots ,X_r]}$ . (ii) For each r ₀ there is a compact-open A such that in (i) we cannot take r < r ₀. (iii) For any compact-open set ${A\subset \mathbb{Z}_p}$ there exists a polynomial ${f\in\mathbb{Q}_p[X]}$ such that ${f(\mathbb{Z}_p)=A}$ . We shall also discuss in more detail which sets A can be represented as ${f(\mathbb{Z}_p)}$ for a polynomial ${f\in\mathbb{Z}_p[X]}$ in a single variable.     

Postulation of Disjoint Unions of Lines and Double Lines in {\mathbb{P}^3}

A double line ${C \subset \mathbb{P}^3}$ is a connected divisor of type (2, 0) on a smooth quadric surface. Fix ${(a, c) \in \mathbb{N}^2\ \backslash\ \{(0, 0)\}}$ . Let ${X \subset \mathbb{P}^3}$ be a general disjoint union of a lines and c double lines. Then X has maximal rank, i.e. for each ${t \in \mathbb{Z}}$ either ${h^1(\mathcal{I}_X(t)) = 0}$ or ${h^0(\mathcal{I}_X(t)) = 0}$ .     

On the topology of semi-algebraic functions on closed semi-algebraic sets

We consider a closed semi-algebraic set ${X \subset \mathbb{R}^n}$ and a C ² semi-algebraic function ${f : \mathbb{R}^n \rightarrow\mathbb{R}}$ such that ${f_{\vert X}}$ has a finite number of critical points. We relate the topology of X to the topology of the sets ${X \cap \{ f * \alpha \}}$ , where ${* \in \{\le,=,\ge \}}$ and ${\alpha \in \mathbb{R}}$ , and the indices of the critical points of ${f_{\vert X}}$ and ${-f_{\vert X}}$ . We also relate the topology of X to the topology of the links at infinity of the sets ${X \cap \{ f * \alpha\}}$ and the indices of these critical points. We give applications when ${X=\mathbb{R}^n}$ and when f is a generic linear function.     

On the Hall algebra of semigroup representations over \mathbb F _1

Let $\mathrm{A }$ be a finitely generated semigroup with 0. An $\mathrm{A }$ -module over $\mathbb F _1$ (also called an $\mathrm{A }$ -set), is a pointed set $(M,*)$ together with an action of $\mathrm{A }$ . We define and study the Hall algebra $\mathbb H _{\mathrm{A }}$ of the category $\mathcal C _{\mathrm{A }}$ of finite $\mathrm{A }$ -modules. $\mathbb H _{\mathrm{A }}$ is shown to be the universal enveloping algebra of a Lie algebra $\mathfrak n _{\mathrm{A }}$ , called the Hall Lie algebra of $\mathcal C _{\mathrm{A }}$ . In the case of $\langle t \rangle $ —the free monoid on one generator $\langle t \rangle $ , the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\langle t \rangle $ -modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when $\mathrm{A }$ is a quotient of $\langle t \rangle $ by a congruence, and the monoid $G \cup \{ 0\}$ for a finite group $G$ .     

Estimates in Beurling-Helson type theorems: Multidimensional case

We consider the spaces A _p ( $\mathbb{T}^m $ ) of functions f on the m-dimensional torus $\mathbb{T}^m $ such that the sequence of Fourier coefficients $\hat f = \{ \hat f(k),k \in \mathbb{Z}^m \} $ belongs to l ^p (? ^m ), 1 ≤ p < 2. The norm on A _p ( $\mathbb{T}^m $ ) is defined by $\left\| f \right\|_{A_p (\mathbb{T}^m )} = \left\| {\hat f} \right\|_{l^p (\mathbb{Z}^m )} $ . We study the rate of growth of the norms $\left\| {e^{i\lambda \phi } } \right\|_{A_p (\mathbb{T}^m )} $ as |λ| → ∞, λ ∈ ?, for C ¹-smooth real functions φ on $\mathbb{T}^m $ (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogs for the spaces A _p (? ^m ).     

On the Linear Dependence of a Finite Set of Additive Functions

In this note we prove the following: Let n?≥ 2 be a fixed integer. A system of additive functions ${A_{1},A_{2},\ldots,A_{n}:\mathbb{R} \to\mathbb{R}}$ is linearly dependent (as elements of the ${\mathbb{R}}$ vector space ${\mathbb{R}^{\mathbb{R}}}$ ), if and only if, there exists an indefinite quadratic form ${Q:\mathbb{R}^{n}\to\mathbb{R} }$ such that ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\geq 0}$ or ${Q(A_{1}(x),A_{2}(x),\ldots,A_{n}(x))\leq 0}$ holds for all ${x\in\mathbb{R}}$ .     

Infinite-dimensional quasigroups of finite orders

Let Σ be a finite set of cardinality k > 0, let $\mathbb{A}$ be a finite or infinite set of indices, and let $\mathcal{F} \subseteq \Sigma ^\mathbb{A}$ be a subset consisting of finitely supported families. A function $f:\Sigma ^\mathbb{A} \to \Sigma$ is referred to as an $\mathbb{A}$ -quasigroup (if $\left| \mathbb{A} \right| = n$ , then an n-ary quasigroup) of order k if $f\left( {\bar y} \right) \ne f\left( {\bar z} \right)$ for any ordered families $\bar y$ and $\bar z$ that differ at exactly one position. It is proved that an $\mathbb{A}$ -quasigroup f of order 4 is reducible (representable as a superposition) or semilinear on every coset of $\mathcal{F}$ . It is shown that the quasigroups defined on Σ^?, where ? are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1].     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

Positive definite superfunctions and unitary representations of lie supergroups

For a broad class of Fréchet-Lie supergroups $ \mathcal{G} $ , we prove that there exists a correspondence between positive definite smooth (resp., analytic) superfunctions on $ \mathcal{G} $ and matrix coefficients of smooth (resp., analytic) unitary representations of the Harish-Chandra pair (G, $ \mathfrak{g} $ ) associated to $ \mathcal{G} $ . As an application, we prove that a smooth positive definite superfunction on $ \mathcal{G} $ is analytic if and only if it restricts to an analytic function on the underlying manifold of $ \mathcal{G} $ . When the underlying manifold of $ \mathcal{G} $ is 1-connected we obtain a necessary and sufficient condition for a linear functional on the universal enveloping algebra U( $ {{\mathfrak{g}}_{\mathbb{C}}} $ ) to correspond to a matrix coefficient of a unitary representation of (G, $ \mathfrak{g} $ ). The class of Lie supergroups for which the aforementioned results hold is characterised by a condition on the convergence of the Trotter product formula. This condition is strictly weaker than assuming that the underlying Lie group of $ \mathcal{G} $ is a locally exponential Fréchet-Lie group. In particular, our results apply to examples of interest in representation theory such as mapping supergroups and diffeomorphism supergroups.     

Linear recurring sequences and subfield subcodes of cyclic codes

Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .     

Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Consider a finite dimensional complex Hilbert space ${\mathcal{H}}$ , with ${dim(\mathcal{H}) \geq 3}$ , define ${\mathbb{S}(\mathcal{H}):= \{x\in \mathcal{H} \:|\: \|x\|=1\}}$ , and let ${\nu_\mathcal{H}}$ be the unique regular Borel positive measure invariant under the action of the unitary operators in ${\mathcal{H}}$ , with ${\nu_\mathcal{H}(\mathbb{S}(\mathcal{H}))=1}$ . We prove that if a complex frame function ${f : \mathbb{S}(\mathcal{H})\to \mathbb{C}}$ satisfies ${f \in \mathbb{L}^2(\mathbb{S}(\mathcal{H}), \nu_\mathcal{H})}$ , then it verifies Gleason’s statement: there is a unique linear operator ${A: \mathcal{H} \to \mathcal{H}}$ such that ${f(u) = \langle u| A u\rangle}$ for every ${u \in \mathbb{S}(\mathcal{H}).\,A}$ is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.     

Sets Computing the Symmetric Tensor Rank

Let ${\nu_{d} : \mathbb{P}^{r} \rightarrow \mathbb{P}^{N}, N := \left( \begin{array}{ll} r + d \\ \,\,\,\,\,\, r \end{array} \right)- 1,}$ denote the degree d Veronese embedding of ${\mathbb{P}^{r}}$ . For any ${P\, \in \, \mathbb{P}^{N}}$ , the symmetric tensor rank sr(P) is the minimal cardinality of a set ${\mathcal{S} \subset \nu_{d}(\mathbb{P}^{r})}$ spanning P. Let ${\mathcal{S}(P)}$ be the set of all ${A \subset \mathbb{P}^{r}}$ such that ${\nu_{d}(A)}$ computes sr(P). Here we classify all ${P \,\in\, \mathbb{P}^{n}}$ such that sr(P) <  3d/2 and sr(P) is computed by at least two subsets of ${\nu_{d}(\mathbb{P}^{r})}$ . For such tensors ${P\, \in\, \mathbb{P}^{N}}$ , we prove that ${\mathcal{S}(P)}$ has no isolated points.