Simplicial complexes with rigid depth

We extend a result of Minh and Trung (Adv. Math. 226:1285–1306, 2011) to get criteria for depth ${I = \rm {depth}\sqrt{I}}$ , where I is an unmixed monomial ideal of the polynomial ring S?=?K[x ₁, . . . , x _n ]. As an application we characterize all the pure simplicial complexes Δ which have rigid depth, that is, which satisfy the condition that for every unmixed monomial ideal ${I\subset S}$ with ${\sqrt{I}=I_\Delta}$ one has depth(I)?=?depth(I _Δ).     

On ideals with the Rees property

A homogeneous ideal I of a polynomial ring S is said to have the Rees property if, for any homogeneous ideal ${J \subset S}$ which contains I, the number of generators of J is smaller than or equal to that of I. A homogeneous ideal ${I \subset S}$ is said to be ${\mathfrak{m}}$ -full if ${\mathfrak{m}I:y=I}$ for some ${y \in \mathfrak{m}}$ , where ${\mathfrak{m}}$ is the graded maximal ideal of ${S}$ . It was proved by one of the authors that ${\mathfrak{m}}$ -full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals which have the Rees property but are not ${\mathfrak{m}}$ -full in a polynomial ring with more than two variables. To prove this result, we also show that every Artinian monomial almost complete intersection in three variables has the Sperner property.     

A commutant realization of Odake’s algebra

The bcβγ-system $ \mathcal{W} $ of rank 3 has an action of the affine vertex algebra $ {V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ , and the commutant vertex algebra $ \mathcal{C}=\mathrm{Com}\left( {{V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right),\mathcal{W}} \right) $ contains copies of V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and Odake’s algebra $ \mathcal{O} $ . Odake’s algebra is an extension of the N = 2 super-conformal algebra with c = 9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and $ \mathcal{O} $ form a Howe pair (i.e., a pair of mutual commutants) inside $ \mathcal{C} $ . More generally, any finite-dimensional representation of a Lie algebra $ \mathfrak{g} $ gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of $ \mathfrak{s}{{\mathfrak{l}}_2} $ .     

Endomorphism algebras and Igusa–Todorov algebras

Let A be an Artin algebra. If $V\in \operatorname{mod} A$ such that the global dimension of  $\operatorname{End}_{A}V$ is at most 3, then for any ${M\in \operatorname{add}_{A}V}$ , both B and B ^op are 2-Igusa–Todorov algebras, where ${B=\operatorname{End}_{A}M}$ . Let ${P\in \operatorname{mod} A}$ be projective and ${B=\operatorname{End}_{A}P}$ such that the projective dimension of P as a right B-module is at most n(<∞). If A is an m-syzygy-finite algebra (resp. an m-Igusa–Todorov algebra), then B is an (m+n)-syzygy-finite algebra (resp. an (m+n)-Igusa–Todorov algebra); in particular, the finitistic dimension of B is finite in both cases. Some applications of these results are given.     

Hall Polynomials and Composition Algebra of Representation Finite Algebras

Let \(\mathcal{A}\) be a representation finite algebra over finite field k such that the indecomposable \(\mathcal{A}\) -modules are determined by their dimension vectors and for each \(M, L \in ind(\mathcal{A})\) and \(N\in mod(\mathcal{A})\) , either \(F^{M}_{N L}=0\) or \(F^{M}_{L N}=0\) . We show that \(\mathcal{A}\) has Hall polynomials and the rational extension of its Ringel–Hall algebra equals the rational extension of its composition algebra. This result extend and unify some known results about Hall polynomials. As a consequence we show that if \(\mathcal{A}\) is a representation finite simply-connected algebra, or finite dimensional k-algebra such that there are no short cycles in \(mod(\mathcal{A})\) , or representation finite cluster tilted algebra, then \(\mathcal{A}\) has Hall polynomials and \(\mathcal{H}(\mathcal{A})\otimes_\mathbb{Z}Q=\mathcal{C}(\mathcal{A})\otimes_\mathbb{Z}Q\) .     

The Algebra of Polynomial Integro-Differential Operators is a Holonomic Bimodule over the Subalgebra of Polynomial Differential Operators

In contrast to its subalgebra $A_n:=K\langle x_1, \ldots , x_n, \frac{\partial}{\partial x_1}, \ldots ,\frac{\partial}{\partial x_n}\rangle $ of polynomial differential operators (i.e. the n’th Weyl algebra), the algebra ${\mathbb{I}}_n:=K\langle x_1, \ldots ,$ $ x_n, \frac{\partial}{\partial x_1}, \ldots ,\frac{\partial}{\partial x_n}, \int_1, \ldots , \int_n\rangle $ of polynomial integro-differential operators is neither left nor right Noetherian algebra; moreover it contains infinite direct sums of nonzero left and right ideals. It is proved that ${\mathbb{I}}_n$ is a left (right) coherent algebra iff n?=?1; the algebra ${\mathbb{I}}_n$ is a holonomic A _n -bimodule of length 3 ⁿ and has multiplicity 3 ⁿ with respect to the filtration of Bernstein, and all 3 ⁿ simple factors of ${\mathbb{I}}_n$ are pairwise non-isomorphic A _n -bimodules. The socle length of the A _n -bimodule ${\mathbb{I}}_n$ is n?+?1, the socle filtration is found, and the m’th term of the socle filtration has length ${n\choose m}2^{n-m}$ . This fact gives a new canonical form for each polynomial integro-differential operator. It is proved that the algebra ${\mathbb{I}}_n$ is the maximal left (resp. right) order in the largest left (resp. right) quotient ring of the algebra ${\mathbb{I}}_n$ .     

Covering properties of ideals

Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set of indices of the required subsequence should be in a fixed ideal ${{\mathcal{J}}}$ on ω. We introduce the notion of the ${{\mathcal{J}}}$ -covering property of a pair ${({\mathcal{A}}, I)}$ where ${{\mathcal{A}}}$ is a σ-algebra on a set X and ${{I \subseteq \mathcal{P}(X)}}$ is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal ${\mathcal{N}}$ and the meager ideal ${\mathcal{M}}$ . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals ${{\mathcal{J}}}$ on ω such that ${\mathcal{M}}$ has the ${{\mathcal{J}}}$ -covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katětov–Blass order, in the family of those ideals ${\mathcal{J}}$ on ω such that ${\mathcal{N}}$ or ${\mathcal{M}}$ has the ${\mathcal{J}}$ -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails.     

Forbidden Configurations and Product Constructions

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F, we say that a (0,1)-matrix A has F as a configuration if there is a submatrix of A which is a row and column permutation of F (trace is the set system version of a configuration). Let \({\|A\|}\) denote the number of columns of A. We define \({{\rm forb}(m, F) = {\rm max}\{\|A\| \,:\, A}\) is m-rowed simple matrix and has no configuration F. We extend this to a family \({\mathcal{F} = \{F_1, F_2, \ldots , F_t\}}\) and define \({{\rm forb}(m, \mathcal{F}) = {\rm max}\{\|A\| \,:\, A}\) is m-rowed simple matrix and has no configuration \({F \in \mathcal{F}\}}\) . We consider products of matrices. Given an m ₁ × n ₁ matrix A and an m ₂ × n ₂ matrix B, we define the product A × B as the (m ₁ + m ₂) × n ₁ n ₂ matrix whose columns consist of all possible combinations obtained from placing a column of A on top of a column of B. Let I _k denote the k × k identity matrix, let \({I_k^{c}}\) denote the (0,1)-complement of I _k and let T _k denote the k × k upper triangular (0,1)-matrix with a 1 in position i, j if and only if i ≤ j. We show forb(m, {I ₂ × I ₂, T ₂ × T ₂}) is \({\Theta(m^{3/2})}\) while obtaining a linear bound when forbidding all 2-fold products of all 2 × 2 (0,1)-simple matrices. For two matrices F, P, where P is m-rowed, let \({f(F, P) = {\rm max}_{A} \{\|A\| \,:\,A}\) is m-rowed submatrix of P with no configuration F}. We establish f(I ₂ × I ₂, I _m/2 × I _m/2) is \({\Theta(m^{3/2})}\) whereas f(I ₂ × T ₂, I _m/2 × T _m/2) and f(T ₂ × T ₂, T _m/2 × T _m/2) are both \({\Theta(m)}\) . Additional results are obtained. One of the results requires extensive use of a computer program. We use the results on patterns due to Marcus and Tardos and generalizations due to Klazar and Marcus, Balogh, Bollobás and Morris.     

Stanley depth of weakly polymatroidal ideals

Let \({\mathbb{K}}\) be a field and \({S = \mathbb{K}[x_1,\ldots,x_n]}\) be the polynomial ring in n variables over the field \({\mathbb{K}}\) . In this paper, it is shown that Stanley’s conjecture holds for S/I if I is a weakly polymatroidal ideal.     

Split Malcev algebras

We study the structure of split Malcev algebras of arbitrary dimension over an algebraically closed field of characteristic zero. We show that any such algebras M is of the form $M={\mathcal U} +\sum_{j}I_{j}$ with ${\mathcal U}$ a subspace of the abelian Malcev subalgebra H and any I _j a well described ideal of M satisfying [I _j ,I _k ]?=?0 if j????k. Under certain conditions, the simplicity of M is characterized and it is shown that M is the direct sum of a semisimple split Lie algebra and a direct sum of simple non-Lie Malcev algebras.     

Step Refinable Functions and Orthogonal MRA on Vilenkin Groups

We find necessary and sufficient conditions on refinable step function under which this function generates an orthogonal MRA in the $L_{2}(\mathfrak{G})$ -spaces on Vilenkin group $\mathfrak{G}$ . We consider a class of refinable step functions for which the mask m ₀(χ) is constant on cosets $\mathfrak{G}_{-1}^{\bot}\chi$ and its modulus |m ₀(χ)| has two values only: 0 and 1. We prove that any refinable step function φ from this class that generates an orthogonal MRA on Vilenkin group $\mathfrak{G}$ has Fourier transform with condition $\operatorname{supp}\hat{\varphi}(\chi)\subset\mathfrak{G}_{p-2}^{\bot}$ . We show the sharpness of this result, too.     

Poisson kernel and Cauchy formula of a non-symmetric transitive domain

In 1965, Lu Yu-Qian discovered that the Poisson kernel of the homogenous domain S m,p,q={Z∈Cm×m, Z1∈Cm×p,Z2 ∈Cq×m|2i1( Z-Z+)-Z1Z1′-Z2′Z20} does not satisfy the Laplace-Beltrami equation associated with the Bergman metric when S m,p,q is not symmetric. However the map T0:Z→Z, Z1→Z1 , Z2→Z2 transforms S m,p,q into a domain S I (m, m + p + q) which can be mapped by the Cayley transformation into the classical domains R I (m, m + p + q). The pull back of the Bergman metric of R I (m, m + p + q) to S m,p,q is a Riemann metric ds 2 which is not a Khler metric and even not a Hermitian metric in general. It is proved that the Laplace-Beltrami operator associated with the metric ds 2 when it acts on the Poisson kernel of S m,p,q equals 0. Consequently, the Cauchy formula of S m,p,q can be obtained from the Poisson formula.     

Unitary SK1 of semiramified graded and valued division algebras

We prove formulas for SK₁(E, τ), which is the unitary SK₁ for a graded division algebra E finite-dimensional and semiramified over its center T with respect to a unitary involution τ on E. Every such formula yields a corresponding formula for SK₁(D, ρ) where D is a division algebra tame and semiramified over a Henselian valued field and ρ is a unitary involution on D. For example, it is shown that if ${\sf{E} \sim \sf{I}_0 \otimes_{\sf{T}_0}\sf{N}}$ where I ₀ is a central simple T ₀-algebra split by N ₀ and N is decomposably semiramified with ${\sf{N}_0 \cong L_1\otimes_{\sf{T}_0} L_2}$ with L ₁, L ₂ fields each cyclic Galois over T ₀, then $${\rm SK}_1(\sf{E}, \tau) \,\cong\ {\rm Br}(({L_1}\otimes_{\sf{T}_0} {L_2})/\sf{T}_0;\sf{T}_0^\tau)\big/ \left[{\rm Br}({L_1}/\sf{T}_0;\sf{T}_0^\tau)\cdot {\rm Br}({L_2}/\sf{T}_0;\sf{T}_0^\tau) \cdot \langle[\sf{I}_0]\rangle\right].$$      

On the Stanley depth of weakly polymatroidal ideals

Let ${\mathbb{K}}$ be a field and ${S = \mathbb{K}[x_1,\dots,x_n]}$ be the polynomial ring in n variables over the field ${\mathbb{K}}$ . In this paper, it is shown that Stanley’s conjecture holds for I and S/I if I is a product of monomial prime ideals or I is a high enough power of a polymatroidal or a stable ideal generated in a single degree.     

Between\widehat{gl}(\infty ) and\widehat{sl}_N affine algebras. I. Geometrical actions

We consider the central extended $\widehat{gl}(\infty )$ Lie algebra and a set of its subalgebras parametrized by |q|=1, which coincides with the embedding of the quantum tori Lie algebras (QTLA) in $\widehat{gl}(\infty )$ . Forq ^N=1 there exists an ideal, and a factor over this ideal is isomorphic to an $\widehat{sl}_{N(z)} $ affine algebra. For a generic valueq the corresponding subalgebras are dense in $\widehat{gl}(\infty )$ . Thus, they interpolate between $\widehat{gl}(\infty )$ and $\widehat{sl}_{N(z)} $ . All these subalgebras are fixed points of automorphism of $\widehat{gl}(\infty )$ . Using the automorphisms, we construct geometrical actions for the subalgebras, starting from the Kirillov-Kostant form and the corresponding geometrical action for $\widehat{gl}(\infty )$ .     

A construction of cylindric and polyadic algebras from atomic relation algebras

Given a simple atomic relation algebra ${\mathcal{A}}$ and a finite n ?? 3, we construct effectively an atomic n-dimensional polyadic equality-type algebra ${\mathcal{P}}$ such that for any subsignature L of the signature of ${\mathcal{P}}$ that contains the boolean operations and cylindrifications, the L-reduct of ${\mathcal{P}}$ is completely representable if and only if ${\mathcal{A}}$ is completely representable. If ${\mathcal{A}}$ is finite then so is ${\mathcal{P}}$ . It follows that there is no algorithm to determine whether a finite n-dimensional cylindric algebra, diagonal-free cylindric algebra, polyadic algebra, or polyadic equality algebra is representable (for diagonal-free algebras this was known). We also obtain a new proof that the classes of completely representable n-dimensional algebras of these types are non-elementary, a result that remains true for infinite dimensions if the diagonals are present, and also for infinite-dimensional diagonal-free cylindric algebras.     

Zhelobenko invariants, Bernstein-Gelfand-Gelfand operators and the analogue Kostant Clifford algebra conjecture

Let $ \mathfrak{g} $ be a complex simple Lie algebra and $ \mathfrak{h} $ a Cartan subalgebra. The Clifford algebra C( $ \mathfrak{g} $ ) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2?m?+?1 is just the zero weight vector of the simple (2?m?+?1)-dimensional module of the principal s-triple obtained from the Langlands dual $ {\mathfrak{g}^\vee } $ . Bazlov [1] settled this conjecture positively in type A. The hard part of the Kostant Clifford algebra conjecture is a question concerning the Harish-Chandra map for the enveloping algebra U( $ \mathfrak{g} $ ) composed with evaluation at the half sum ?? of the positive roots. The analogue Kostant conjecture is obtained by replacing the Harish-Chandra map by a ??generalized Harish-Chandra?? map. This map had been studied notably by Zhelobenko [15]. The proof given here involves a symmetric algebra version of the Kostant conjecture, the Zhelobenko invariants in the adjoint case, and, surprisingly, the Bernstein-Gelfand-Gelfand operators introduced in their study [3] of the cohomology of the flag variety.     

A base-matrix lemma for sets of rationals modulo nowhere dense sets

We study some properties of the quotient forcing notions ${Q_{tr(I)} = \wp(2^{< \omega})/tr(I)}$ and P _I ?= B(2 ^ω )/I in two special cases: when I is the σ-ideal of meager sets or the σ-ideal of null sets on 2 ^ω . We show that the remainder forcing R _I =?Q _tr(I)/P _I is σ-closed in these cases. We also study the cardinal invariant of the continuum ${\mathfrak{h}_{\mathbb{Q}}}$ , the distributivity number of the quotient ${Dense(\mathbb{Q})/nwd}$ , in order to show that ${\wp(\mathbb{Q})/nwd}$ collapses ${\mathfrak{c}}$ to ${\mathfrak{h}_{\mathbb{Q}}}$ , thus answering a question addressed in Balcar et?al. (Fundamenta Mathematicae 183:59–80, 2004).     

Constant-Sign Periodic and Almost Periodic Solutions for a System of Integral Equations

We consider the following system of integral equations $${u_{i}(t)=\int\nolimits_{I} g_{i}(t, s)f(s, u_{1}(s), u_{2}(s), \cdots, u_{n}(s))ds, \quad t \in I, \ 1 \leq i\leq n}$$ where I is an interval of $\mathbb{R}$ . Our aim is to establish criteria such that the above system has a constant-sign periodic and almost periodic solution (u ₁, u ₂,…,u _n ) when I is an infinite interval of $\mathbb{R}$ , and a constant-sign periodic solution when I is a finite interval of $\mathbb{R}$ . The above problem is also extended to that on $\mathbb{R}$ $$u_{i} {\left( t \right)} = {\int_\mathbb{R} {g_{i} {\left( {t,s} \right)}f_{i} {\left( {s,u_{1} {\left( s \right)},u_{2} {\left( s \right)}, \cdots ,u_{n} {\left( s \right)}} \right)}ds\quad t \in \mathbb{R},\quad 1 \leqslant i \leqslant n.} }$$