C (n)-cardinals

For each natural number n, let C ⁽ⁿ⁾ be the closed and unbounded proper class of ordinals α such that V _α is a Σ _n elementary substructure of V. We say that κ is a C ⁽ⁿ⁾ -cardinal if it is the critical point of an elementary embedding j : V → M, M transitive, with j(κ) in C ⁽ⁿ⁾. By analyzing the notion of C ⁽ⁿ⁾-cardinal at various levels of the usual hierarchy of large cardinal principles we show that, starting at the level of superstrong cardinals and up to the level of rank-into-rank embeddings, C ⁽ⁿ⁾-cardinals form a much finer hierarchy. The naturalness of the notion of C ⁽ⁿ⁾-cardinal is exemplified by showing that the existence of C ⁽ⁿ⁾-extendible cardinals is equivalent to simple reflection principles for classes of structures, which generalize the notions of supercompact and extendible cardinals. Moreover, building on results of Bagaria et?al. (2010), we give new characterizations of Vopeňka’s Principle in terms of C ⁽ⁿ⁾-extendible cardinals.     

Splittings of spaces CX

The n-fold free loop space ⁿSⁿX is for connected spaces X weakly equivalent to a simpler space C_nX, which has a natural filtration F_inr C_nX. It is well known that there is a splitting S^tF_r(C_nX) V _m=1 ^p S^t(F_m(C_nX)¦F_m–1(C_nX) inducing a stable splitting of C_nX. We give a simple construction for such a splitting with comparatively low estimates for the number t of necessary suspension coordinates.     

Analysis on discrete cocompact subgroups of the generic filiform Lie groups

Summary Let F_n, n≧ 1, denote the sequence of generic filiform (connected, simply connected) Lie groups. Here we study, for each F_n, the infinite dimensional simple quotients of the group C*-algebra of (the most obvious) one of its discrete cocompact subgroups D_n. For D_n, the most attractive concrete faithful representations are given in terms of Anzai flows, in analogy with the representations of the discrete Heisenberg group H₃ ⊆G₃ on L²(T) that result from the irrational rotation flows on T; the representations of D_n generate infinite-dimensional simple quotients A_n,θ of the group C*-algebra C*(D_n). For n>1, there are other infinite-dimensional simple quotients of C*(D_n) arising from non-faithful representations of D_n. Flows for these are determined, and they are also characterized and represented as matrix algebras over simple affine Furstenberg transformation group C*-algebras of the lower dimensional tori.     

Dimensions of Cantor and post varieties

A dimension of a finitely based variety V of algebras is the greatest length of a basis (that is, an independent generating set) for the SC-theory SC(V) with the strong Mal'tsev conditions satisfied in V. A dimension is said to be infinite if the lengths of bases in SC(V) are unbounded. We prove that the dimension of a Cantor variety C_m,n in the general form, i.e., with n>m≥1, is infinite. The algorithm of constructing a basis of any given length in SC(C_m,n) is presented. By contrast, any Post variety P_n generated by a primal algebra of order n>1 is shown to have a finite dimension not exceeding the number of maximal subalgebras in the iterative Post algebra over the set {0,1,…,n−1}. Specifically, the dimension of the variety of Boolean algebras is at most four. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 359–369, May–June, 1996.     

The Embedding Theorem for Cantor Varieties

Let m and n be fixed integers, with 1 m < n. A Cantor variety C _m,n is a variety of algebras with m n-ary and n m-ary basic operations which is defined in a signature ={g₁,...,g_m,f₁,...,f_n} by the identities f_ig₁x₁,...,x_n),...,g_mx₁,...,x_n) = x_i, _i=1,...,n, g_jf₁x₁,...,x_m),...,f_nx₁,...,x_m)) = x_j, _j=1,...,m. We prove the following: (a) every partial C _m,n-algebra A is isomorphically embeddable in the algebra G= A; S(A) of C _m,n; (b) for every finitely presented algebra G= A; S in C _m,n, the word problem is decidable; (c) for finitely presented algebras in _{C m}, the occurrence problem is decidable; (d) _{C m,n} has a hereditarily undecidable elementary theory.     

Graded polynomial identities for matrices with the transpose involution over an infinite field

Let F be an infinite field and let M_n(F) be the algebra of n×n matrices over F endowed with an elementary grading whose neutral component coincides with the main diagonal. In this paper, we find a basis for the graded polynomial identities of M_n(F) with the transpose involution. Our results generalize for infinite fields of arbitrary characteristic previous results in the literature, which were obtained for the field of complex numbers and for a particular class of elementary G-gradings.     

On the abelianizations of congruence subgroups of Aut(<Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>)

Let F _n be the free group of rank n, and let Aut⁺(F _n ) be its special automorphism group. For an epimorphism π : F _n → G of the free group F _n onto a finite group G we call the standard congruence subgroup of Aut⁺(F _n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ⁺(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ⁺(G, π) ≤ Aut⁺(F ₂) has infinite abelianization.     

Indices of Toeplitz tuples on pseudoregular domains

This paper proves an index theorem of Toeplitz tuples on pseudoregular domains in Cn. Geometrically, the index of Toeplitz tuple TΦn is (-1)n time wrapping number of Φn around the origin. As one of the applications of the index theorem, we completely characterize the automorphism groups of Toeplitz algebras on Poincaré domain. As another application, it is shown that C*(Ω)C*(Bn) for every Poincare domain Ω in Cn(n≠2). It is also noticed that C*(Ω)C*(B2) if and only if the Poincaré conjecture is true for Ω.     

Free algebras in discriminator varieties

We investigate V-free algebras onn generators,F _n =F _r (V, n), where V is a discriminator variety and, more specifically, where V is a variety of relation algebras or of cylindricalgebras. Sample questions are: (a) IsF _n+1 embeddable inF _n ? (b) DoesF _n contain an n-element set that generates it non-freely? The answer to (a) is affirmative in some varieties of relation algebras, but it is negative in every congruence extensile variety in which some nontrivial finite member is an absolute retract. The answer to (b) is affirmative in every variety of relation algebras that contains the full algebra of relations on an infinite set.     

Separable algebras over infinite fields are 2-generated and finitely presented

We prove that every separable algebra over an infinite field F admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix algebras over F and for group algebras FG, where G is a finite group such that the order of G is invertible in F. We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2 × 2 matrices (A, B) and (A′, B′) with entries in a commutative ring R are automorphically conjugate over the matrix algebra M ₂(R), under an additional assumption that both pairs generate M ₂(R) as an R-algebra.     

On the maximal subgroup of the sandwich semigroup of generalized circulant Boolean matrices

Let n be a positive integer, and C _n (r) the set of all n × n r-circulant matrices over the Boolean algebra B = {0, 1}, . For any fixed r-circulant matrix C (C ≠ 0) in G _n , we define an operation “*” in G _n as follows: A * B = ACB for any A, B in G _n , where ACB is the usual product of Boolean matrices. Then (G _n , *) is a semigroup. We denote this semigroup by G _n (C) and call it the sandwich semigroup of generalized circulant Boolean matrices with sandwich matrix C. Let F be an idempotent element in G _n (C) and M(F) the maximal subgroup in G _n (C) containing the idempotent element F. In this paper, the elements in M(F) are characterized and an algorithm to determine all the elements in M(F) is given.     

Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.     

Torsion in Boundary Coinvariants and K -Theory for Affine Buildings

Let (G, I, N, S) be an affine topological Tits system, and let Γ be a torsion-free cocompact lattice in G. This article studies the coinvariants H ₀(Γ; C(Ω,Z)), where Ω is the Furstenberg boundary of G. It is shown that the class [1] of the identity function in H ₀(Γ; C(Ω, Z)) has finite order, with explicit bounds for the order. A similar statement applies to the K ₀ group of the boundary crossed product C ^*-algebra C(Ω)Γ. If the Tits system has type ? ₂, exact computations are given, both for the crossed product algebra and for the reduced group C ^*-algebra.     

The algebra ofG-relations

In this paper, we study a tower {A _n ^G: n} ≥ 1 of finite-dimensional algebras; here, G represents an arbitrary finite group,d denotes a complex parameter, and the algebraA _n ^G(d) has a basis indexed by ‘G-stable equivalence relations’ on a set whereG acts freely and has 2n orbits. We show that the algebraA _n ^G(d) is semi-simple for all but a finite set of values ofd, and determine the representation theory (or, equivalently, the decomposition into simple summands) of this algebra in the ‘generic case’. Finally we determine the Bratteli diagram of the tower {A _n ^G(d): n} ≥ 1 (in the generic case).     

Composition Operators on the Bloch Space of Several Complex Variables

Abstract In this paper, we study the boundedness and compactness of composition operator C _φ on the Bloch space β(Ω), Ω being a bounded homogeneous domain. For Ω = B _n, we give the necessary and sufficient conditions for a composition operator C _φ to be compact on β(B _n) or β ₀(B _n). Supported by the National Natural Science Foundation and the National Education Committee Doctoral Foundation     

On Whittaker modules over a class of algebras similar to U(sl 2)

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A ₁ and finding interesting families of new noetherian rings, a class of algebras similar to U(sl ₂) was introduced and studied by Smith. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained. But it seems that not much has been done on the part of nonweight modules. In this paper, we generalize Kostant’s results on the Whittaker model for the universal enveloping algebras U(g) of finite dimensional semisimple Lie algebras g to Smith’s algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith’s algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.      

Construction of representation-finite selfinjective artin algebras of classB n andC n

In this paper, we discuss the representation-finite selfinjective artin algebras of classB _n andC _n and obtain the following main results: For any fieldk, let Λ be a representation-finite selfinjective artin algebras of classB _n orC _n overk.

Classification of the isomorphic types of martingale-H 1 spaces

LetF _n be an increasing sequence of finite fields on a probability space (Ω,F _n,P) whereF denotes the σ-algebra generated by ∪F _n. ThenF _n is isomorphic to one of the following spaces:H ¹(δ), ΣH _n ¹ ,l ^l.