Irreducible algebraic sets over divisible decomposed rigid groups

A soluble group G is said to be rigid if it contains a normal series of the form G = G ₁ > G ₂ > …> G _p > G _p+1 = 1, whose quotients G _i /G _i+1 are Abelian and are torsion-free when treated as right ℤ[G/G _i ]-modules. Free soluble groups are important examples of rigid groups. A rigid group G is divisible if elements of a quotient G _i /G _i+1 are divisible by nonzero elements of a ring ℤ[G/G _i ], or, in other words, G _i /G _i+1 is a vector space over a division ring Q(G/G _i ) of quotients of that ring. A rigid group G is decomposed if it splits into a semidirect product A ₁ A ₂…A _p of Abelian groups A _i ≅ G _i /G _i+1. A decomposed divisible rigid group is uniquely defined by cardinalities α _i of bases of suitable vector spaces A _i , and we denote it by M(α₁,…, α _p ). The concept of a rigid group appeared in [arXiv:0808.2932v1 [math.GR], ], where the dimension theory is constructed for algebraic geometry over finitely generated rigid groups. In [11], all rigid groups were proved to be equationally Noetherian, and it was stated that any rigid group is embedded in a suitable decomposed divisible rigid group M(α₁,…, α _p ). Our present goal is to derive important information directly about algebraic geometry over M(α₁,… α _p ). Namely, irreducible algebraic sets are characterized in the language of coordinate groups of these sets, and we describe groups that are universally equivalent over M(α₁,…, α _p ) using the language of equations.     

On thin-complete ideals of subsets of groups

Let F ì P_G \mathcal{F} \subset {\mathcal{P}_G} be a left-invariant lower family of subsets of a group G. A subset A ⊂ G is called F \mathcal{F} -thin if xA ?yA ? F xA \cap yA \in \mathcal{F} for any distinct elements x, y ∈ G. The family of all F \mathcal{F} -thin subsets of G is denoted by t( F ) \tau \left( \mathcal{F} \right) . If t( F ) = F \tau \left( \mathcal{F} \right) = \mathcal{F} , then F \mathcal{F} is called thin-complete. The thin-completion t^*( F ) {\tau^*}\left( \mathcal{F} \right) of F \mathcal{F} is the smallest thin-complete subfamily of P_G {\mathcal{P}_G} that contains F \mathcal{F} . Answering questions of Lutsenko and Protasov, we prove that a set A ⊂ G belongs to τ*(G) if and only if, for any sequence (g _n ) _n∈ω of nonzero elements of G, there is n ∈ ω such that

Amalgamated free products of weakly rigid factors and calculation of their symmetry groups

We consider amalgamated free product II₁ factors M = M _1*B M _2*B … and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra Q ⊂ M can be unitarily conjugated into one of the M _i ’s. We apply this to the case where the M _i ’s are w-rigid II₁ factors, with B equal to either C, to a Cartan subalgebra A in M _i , or to a regular hyperfinite II₁ subfactor R in M _i , to obtain the following type of unique decomposition results, àla Bass–Serre: If M = (N _1 * CN_2*C …) ^t , for some t > 0 and some other similar inclusions of algebras C ⊂ N _i then, after a permutation of indices, (B ⊂ M _i ) is inner conjugate to (C ⊂ N _i ) ^t , for all i. Taking B = C and , with {t _i } _i⩾1 = S a given countable subgroup of R ₊ ^*, we obtain continuously many non-stably isomorphic factors M with fundamental group equal to S. For B = A, we obtain a new class of factors M with unique Cartan subalgebra decomposition, with a large subclass satisfying and Out(M) abelian and calculable. Taking B = R, we get examples of factors with , Out(M) = K, for any given separable compact abelian group K.     

Group measure space decomposition of II1 factors and W*-superrigidity

We prove a “unique crossed product decomposition” result for group measure space II₁ factors L ^∞(X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family G\mathcal{G}, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T _n denotes the group of upper triangular matrices in PSL (n,ℤ), then any free, mixing p.m.p. action of G = \operatornamePSL(n,\mathbbZ)*_Tn\operatornamePSL(n,\mathbbZ)\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z}) is W^∗-superrigid, i.e. any isomorphism between L ^∞(X)⋊Γ and an arbitrary group measure space factor L ^∞(Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family G\mathcal{G}, the Bernoulli actions of Γ are W^∗-superrigid.     

Relatively thin and sparse subsets of groups

Let G be a group with identity e and let I \mathcal{I} be a left-invariant ideal in the Boolean algebra P_G {\mathcal{P}_G} of all subsets of G. A subset A of G is called I \mathcal{I} -thin if gA ?A ? I gA \cap A \in \mathcal{I} for every g ∈ G\{e}. A subset A of G is called I \mathcal{I} -sparse if, for T every infinite subset S of G, there exists a finite subset F ⊂ S such that ?_{g ? F} gA ? F \bigcap\nolimits_{g \in F} {gA \in \mathcal{F}} . An ideal I \mathcal{I} is said to be thin-complete (sparse-complete) if every I \mathcal{I} -thin (I \mathcal{I} -sparse) subset of G belongs to I \mathcal{I} . We define and describe the thin-completion and the sparse-completion of an ideal in P_G {\mathcal{P}_G} .     

Σ-Uniform structures and Σ-functions. II

We construct a family of Σ-uniform Abelian groups and a family of Σ-uniform rings. Conditions are specified that are necessary and sufficient for a universal Σ-function to exist in a hereditarily finite admissible set over structures in these families. It is proved that there is a set S of primes such that no universal Σ-function exists in hereditarily finite admissible sets \mathbbH\mathbbF(G) \mathbb{H}\mathbb{F}(G) and \mathbbH\mathbbF(K) \mathbb{H}\mathbb{F}(K) , where G = ⊕{Z _p | p ∈ S} is a group, Z _p is a cyclic group of order p, K = ⊕{F _p | p ∈ S} is a ring, and F _p is a prime field of characteristic p.     

Logarithmic Dimension Bounds for the Maximal Function Along a Polynomial Curve

Let ℳ denote the maximal function along the polynomial curve (γ ₁ t,…,γ _d t ^d ):

A Hilbert theorem for vertex algebras

Given a simple vertex algebra A \mathcal{A} and a reductive group G of automorphisms of A \mathcal{A} , the invariant subalgebra A^G {\mathcal{A}^G} is strongly finitely generated in most examples where its structure is known. This phenomenon is subtle, and is generally not true of the classical limit of A^G {\mathcal{A}^G} , which often requires infinitely many generators and infinitely many relations to describe. Using tools from classical invariant theory, together with recent results on the structure of the W_{1 + ￥} {\mathcal{W}_{{1 + }\infty }} algebra, we establish the strong finite generation of a large family of invariant subalgebras of βγ-systems, bc-systems, and bcβγ-systems.     

On modules over integer-valued group rings of locally soluble groups with rank restrictions imposed on subgroups

We study a \mathbbZG \mathbb{Z}G -module A such that \mathbbZ \mathbb{Z} is the ring of integer numbers, the group G has an infinite sectional p-rank (or an infinite 0-rank), C _G (A) = 1, A is not a minimax \mathbbZ \mathbb{Z} -module, and, for any proper subgroup H of infinite sectional p-rank (or infinite 0-rank, respectively), the quotient module A/C _A (H) is a minimax \mathbbZ \mathbb{Z} -module. It is shown that if the group G is locally soluble, then it is soluble. Some properties of soluble groups of this kind are discussed.     

Projection decomposition in multiplier algebras

In this paper we present new structural information about the multiplier algebra M (A ){\mathcal M (\mathcal A )} of a σ-unital purely infinite simple C*-algebra A{\mathcal {A}}, by characterizing the positive elements A ? M (A ){A\in \mathcal M (\mathcal A )} that are strict sums of projections belonging to A{\mathcal A } . If A ? A{A\not\in \mathcal {A}} and A itself is not a projection, then the necessary and sufficient condition for A to be a strict sum of projections belonging to A{\mathcal {A} } is that ${\|A\| >1 }${\|A\| >1 } and that the essential norm ||A||_ess 3 1{\|A\|_{ess} \geq 1}. Based on a generalization of the Perera–Rordam weak divisibility of separable simple C*-algebras of real rank zero to all σ-unital simple C*-algebras of real rank zero, we show that every positive element of A{\mathcal {A}} with norm >1 can be approximated by finite sums of projections. Based on block tri-diagonal approximations, we decompose any positive element A ? M (A ){A\in \mathcal M (\mathcal {A} )} with ${\| A\| >1 }${\| A\| >1 } and || A||_ess 3 1{\| A\|_{ess} \geq 1} into a strictly converging sum of positive elements in A{\mathcal A} with norm >1.     

Finite Schur filtration dimension for modules over an algebra with Schur filtration

Let G = GL _N or SL _N as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2  ^N : Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H ^*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the H ⁱ (G, M) are Noetherian A ^G -modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product of Grassmannians.     

Differential operators and BV structures in noncommutative geometry

We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck’s notion of differential operators on a commutative algebra in such a way that derivations of the commutative algebra are replaced by \mathbbDer(A){\mathbb{D}{\rm er}(A)}, the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A){\mathcal{F}(A)}, a certain ‘Fock space’ associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)){\mathcal{D}(\mathcal{F}(A))} of differential operators is filtered and gr D(F(A)){\mathcal{D}(\mathcal{F}(A))}, the associated graded algebra, is commutative in some ‘wheeled’ sense. The resulting ‘wheeled’ Poisson structure on gr D(F(A)){\mathcal{D}(\mathcal{F}(A))} is closely related to the double Poisson structure on T_A \mathbbDer(A){T_{A} \mathbb{D}{\rm er}(A)} introduced by Van den Bergh. Specifically, we prove that gr D(F(A)) @ F(T_A(\mathbbDer(A)),{\mathcal{D}(\mathcal{F}(A))\cong\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)),} provided the algebra A is smooth. Our construction is based on replacing vector spaces by the new symmetric monoidal category of wheelspaces. The Fock space F(A){\mathcal{F}(A)} is a commutative algebra in this category (a “commutative wheelgebra”) which is a structure closely related to the notion of wheeled PROP. Similarly, we have Lie, Poisson, etc., wheelgebras. In this language, D(F(A)){\mathcal{D}(\mathcal{F}(A))} becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper, we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule connection on \mathbbDer(A){\mathbb{D}{\rm er}(A)} gives rise to a second-order (wheeled) differential operator, a noncommutative analogue of the Batalin-Vilkovisky (BV) operator, that makes F(T_A(\mathbbDer(A))){\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)))} a BV wheelgebra. In the final section, we explain how the wheeled differential operators D(F(A)){\mathcal{D}(\mathcal{F}(A))} produce ordinary differential operators on the varieties of n-dimensional representations of A for all n ≥ 1.     

On a graph related to permutability in finite groups

For a finite group G we define the graph Γ(G) to be the graph whose vertices are the conjugacy classes of cyclic subgroups of G and two conjugacy classes ${\mathcal {A}, \mathcal {B}}For a finite group G we define the graph Γ(G) to be the graph whose vertices are the conjugacy classes of cyclic subgroups of G and two conjugacy classes A, B{\mathcal {A}, \mathcal {B}} are joined by an edge if for some A ? A, B ? B A{A \in \mathcal {A},\, B \in \mathcal {B}\, A} and B permute. We characterise those groups G for which Γ(G) is complete.     

On Defining Ideals or Subrings of Hall Algebras

Let A be a finitary algebra over a finite field k, and A- \textmod\text{mod} the category of finite dimensional left A-modules. Let H(A)\mathcal{H}(A) be the corresponding Hall algebra, and for a positive integer r let D _r (A) be the subspace of H(A)\mathcal{H}(A) which has a basis consisting of isomorphism classes of modules in A- \textmod\text{mod} with at least r + 1 indecomposable direct summands. If A is the path algebra of the quiver of type A _n with linear orientation, then D _r (A) is known to be the kernel of the map from the twisted Hall algebra to the quantized Schur algebra indexed by n + 1 and r. For any A, we determine necessary and sufficient conditions for D _r (A) to be an ideal and some conditions for D _r (A) to be a subring of H(A)\mathcal{H}(A). For A the path algebra of a quiver, we also determine necessary and sufficient conditions for D _r (A) to be a subring of H(A)\mathcal{H}(A).     

An extension of almost sure central limit theorem for order statistics

Let {X _n ,n ≥ 1} be a sequence of i.i.d. random variables. Let M _n and m _n denote the first and the second largest maxima. Assume that there are normalizing sequences a _n  > 0, b _n and a nondegenerate limit distribution G, such that . Assume also that {d _k ,k ≥ 1} are positive weights obeying some mild conditions. Then for x > y we have

Equational Noetherianness of rigid soluble groups

A group G is said to be rigid if it contains a normal series of the form G = G ₁ > G ₂ > … > G _m  > G _m + 1 = 1, whose quotients G _i /G _i + 1 are Abelian and are torsion free as right Z[G/G _i ]-modules. In studying properties of such groups, it was shown, in particular, that the above series is defined by the group uniquely. It is known that finitely generated rigid groups are equationally Noetherian: i.e., for any n, every system of equations in x ₁, …, x _n over a given group is equivalent to some of its finite subsystems. This fact is equivalent to the Zariski topology being Noetherian on G ⁿ , which allowed the dimension theory in algebraic geometry over finitely generated rigid groups to have been constructed. It is proved that every rigid group is equationally Noetherian. Supported by RFBR (project No. 09-01-00099) and by the Russian Ministry of Education through the Analytical Departmental Target Program (ADTP) “Development of Scientific Potential of the Higher School of Learning” (project No. 2.1.1.419). Translated from Algebra i Logika, Vol. 48, No. 2, pp. 258–279, March–April, 2009.     

Constant-Sign Solutions for Systems of Fredholm and Volterra Integral Equations: The Singular Case

We consider the system of Fredholm integral equations

On Bernstein–Walsh-type lemmas in regions of the complex plane

Let G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to [`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A _p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition

Quantum Spaces Associated to Multipermutation Solutions of Level Two

We study finite set-theoretic solutions (X,r) of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over ℂ with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra A(\mathbbC,X,r)\mathcal{A}(\mathbb{C},X,r) having a q-commutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots of a prescribed form appear as determined by the representation theory of the finite abelian group G\mathcal{G} of left actions on X. We study the structure of A(\mathbbC,X,r)\mathcal{A}(\mathbb{C},X,r) and show that they have a ∙-product form ‘quantizing’ the commutative algebra of polynomials in |X| variables. We obtain the ∙-product both as a Drinfeld cotwist for a certain canonical 2-cocycle and as a braided-opposite product for a certain crossed G\mathcal{G}-module (over any field k). We provide first steps in the noncommutative differential geometry of A(k,X,r)\mathcal{A}(k,X,r) arising from these results. As a byproduct of our work we find that every such level 2 solution (X,r) factorises as r = f ∘ τ ∘ f ^− 1 where τ is the flip map and (X,f) is another solution coming from X as a crossed G\mathcal{G}-set.     

Weak infinite powers of Blaschke products

Letb be a Blaschke product with zeros {z _n } in the open unit disk Δ. Let be the set of sequences of non-negative integersp=(p ₁,p ₂,…) such that ∑ _n=1 ^∞ p _n (1 − |z _n |) < ∞ andp _n →∞ asn→∞. We study the class of weak infinite powers ofb, Properties of these classes depend on the setS(b) of the cluster points in ∂Δ of {z _n }. It is proved thatS(b)=∂Δ if and only if , the Douglas algebra generated by . Also, it is proved thatdθ(S(b))=0 if and only if there exists an interpolating Blaschke productB such that .