The inflation class operator

We consider the inflation class operator, denoted by F, where for any class K of algebras, F(K) is the class of all inflations of algebras in K. We study the interaction of this operator with the usual algebraic operators H, S andP, and describe the partially-ordered monoid generated by H, S, P andF (with the isomorphism operator I as an identity). Received February 3, 2004; accepted in final form January 3, 2006.     

The partially ordered monoid generated by the operators H, S, P u , P f on classes of algebras

Let I, H, S, P _u , P _f denote the following operators on classes of algebras of the same type: I, H for isomorphic and homomorphic images of algebras, S for subalgebras and P _u , P _f for ultra and filtered products, respectively. In this paper, the monoid generated by the operators H, S, P _u , P _f with I as an identity is described. It turns out that there are 44 different operators such that every composite of H, S, P _u , P _f coincides with one of them (including the empty composite, the identity operator).     

SOME t-CLOSED PAIRS

Let A ? B be integral domains. (A, B) is called a t-closed pair if each subring of B containing A is t-closed. Let R be a t-closed domain containing a field K and let I be a nonzero proper ideal of R. Let D be a subring of K and let S = D + I. If D is a field then it is shown that (S, R) is a t-closed pair if and only if R is integral over S and I is a maximal ideal of R. If D is not a field then we prove in this note that (S, R) is a t-closed pair if and only if (D, K) is a t-closed pair and R = K + I.     

A note on homomorphic images, subalgebras and various products

Let I, H, S, P, P_fin, P_f , P_u, P_s be the usual operators on classes of algebras of the same type (P, P_fin, P_f, P_u, P_s are respectively for direct, finite, filtered, ultra and subdirect products). The partially ordered monoid generated by the operators H, S, P with respect to composition of operators, I as an identity element, and natural ordering between operators is described by Pigozzi (Algebra Universalis 2 (1972), 346–353). The aim of this note is to describe the partially ordered monoids generated by H, S, P_u and by H, S, P_s and as well to summarize known results on the partially ordered monoids of operators generated by H, S and some of the above introduced products.Dedicated to the memory of Aleksandar PopoviReceived April 1, 2001; accepted in final form July 11, 2004.     

Theta Series of Quaternion Algebras over Function Fields

Let K be the function field over a finite field of odd order, and let H be a definite quaternion algebra over K. If Λ is an order of level M in H, we define theta series for each ideal I of Λ using the reduced norm on H. Using harmonic analysis on the completed algebra H_∞ and the arithmetic of quaternion algebras, we establish a transformation law for these theta series. We also define analogs of the classical Hecke operators and show that in general, the Hecke operators map the theta series to a linear combination of theta series attached to different ideals, a generalization of the classical Eichler Commutation Relation.     

Asymptotic Primes of Delta Closures of Ideals

Abstract

For an ideal H in a Noetherian ring R let H? = ∪{H ⁱ⁺¹ : _R H ⁱ | i ≥ 0} and for a multiplicatively closed set Δ of nonzero ideals of R let H _Δ = ∪{HK: _R K | K ? Δ}. It is shown that four standard results concerning the associated prime ideals of the integral closure (bR)_a of a regular principal ideal bR do not hold for certain Δ closures (bR)_Δ of bR. To do this it is first shown that if I is an ideal in R such that height (I) ≥ 1, then each radical ideal J of R containing I is of the form J = K? :_R cR for some ideal K closely related to I, and if I _a :_R J ? U = ∪{I?R _P ∩ R | P is a minimal prime divisor of J} (where I _a is the integral closure of I), then J = I _Δ :_R CR and I ? I _Δ ? I _a).     

On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.     

Commutators in model spaces

Let θ be an inner function, let K _θ = H ² ⊖ θH ², and let S_θ : K_θ → S_θ be defined by the formula S_θf = P_θzf, where f ∈ K_θ is the orthogonal projection of H² onto K_θ. Consider the set A of all trace class operators L : K_θ → K_θ, L = ∑(·,u_n)v_n, ∑∥u_n∥∥v_n∥ < ∞ (u_n, v_n ∈ K_θ), such that ∑ū_n v_n ∈ H ₀ ¹ . It is shown that trace class commutators of the form XS_θ − S_θX (where X is a bounded linear operator on K_θ) are dense in A in the trace class norm. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 54–61.     

Partially Ordered Monoids Generated by Operators H , S , P , and by H , S , P f Are Isomorphic

Let I , H , S , P , P _f be the usual operators on classes of algebras of the same type (P _f for filtered products). The partially ordered monoid generated by the operators H , S , P with respect to composition of operators, I as an identity element, and a natural ordering between operators is described by Pigozzi (Algebra Universalis 2 (1972), 346—353). Let us denote by \cal M =\langle H, S, P\rangle and by \cal M _f =\langle H, S, P _f \rangle the partially ordered monoids generated by {H, S, P} and by {H, S, P _f } respectively. The aim of this paper is to prove that \cal M is isomorphic to \cal M _f . October 29, 1999     

Deddens Algebras and Shift

We consider Deddens algebras associated to operators of the form S−λI, where S is the unilateral shift and λ is a complex number. We show that such an algebra properly contains the commutant of S and that it is always weakly dense in L(H){{\mathcal L}({\mathcal H})}. Yet, it contains no rank one operators, unless λ = 0, in which case it equals L(H){{\mathcal L}({\mathcal H})}.     

Sur l’annulation du deuxième foncteur de (co)homologie d’André-Quillen

For a given idealI of a commutative ringA, B=A/I, the vanishing of the second André-Quillen (co)homology functorH ₂ (A, B, δ) is characterized in terms of the canonical homomorphism α:S(I)→R(I) from the symmetric algebra of the idealI onto its Rees algebra. This is done by introducing a Koszul complex that characterizes commutative graded algebras which are symmetric algebras.

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This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

The set of indeterminate rings of a normal pair as a partially ordered set

Let R ì S{R\subset S} be an extension of integral domains and let [R, S] be the set of intermediate rings between R and S ordered by inclusion. If (R, S) is normal pair and [R, S] is finite, we do prove that there exists a semi-local Prüfer ring T with quotient field K such that [R,S] @ [T,K]{[R,S]\cong \lbrack T,K]} (as a partially ordered set). Consequently, any problem relative to the finiteness conditions in [R, S] can be investigated in the particular case where R is a semi-local Prüfer ring with quotient field S.     

Semiprime Lie Rings of (Anti-)Symmetric Derivations of Commutative Rings

Let R be a 2-torsion free commutative ring with involution, and δ a nonzero derivation of R. Let S be the set of symmetric elements in R, and let K be the set of anti-symmetric elements in R. In this article, we investigate the semiprimeness of the Lie rings Sδ when δ is symmetric and Kδ when δ is anti-symmetric.     

Shuffles of permutations and the Kronecker product

The Kronecker product of two homogeneous symmetric polynomialsP ₁,P ₂ is defined by means of the Frobenius map by the formulaP ₁oP ₂=F(F ⁻¹ P ₁)(F ⁻¹ P ₂). WhenP ₁ andP ₂ are the Schur functionsS _I ,S _J then the resulting productS _I oS _J is the Frobenius characteristic of the tensor product of the two representations corresponding to the diagramsI andJ. Taking the scalar product ofS _I oS _J with a third Schur functionsS _K gives the so called Kronecker coefficientc _I,J,K =<S _I oS _J ,S _K >. In recent work lascoux [7] and Gessel [3] have given what appear to be two separate combinatorial interpretations for thec _I,J,K in terms of some classes of permutations. In Lascoux's workI andJ are restricted to be hooks and in Gessel's both have to be zigzag partitions. In this paper we give a general result relating shuffles of permutations and Kronecker products. This leads us to a combinatorial interpretation of <S _I oS _J ,S _K > forS _I a product of homogeneous symmetric functions andJ, K unrestricted skew shapes. We also show how Gessel's and Lascoux's results are related and show how they can be derived from a special case of our result. Work supported by NSF grant at the University of California, San Diego.     

Finitely Generated Flat Modules and a Characterization of Semiperfect Rings

Abstract

For a ring S, let K ₀(FGFl(S)) and K ₀(FGPr(S)) denote the Grothendieck groups of the category of all finitely generated flat S-modules and the category of all finitely generated projective S-modules respectively. We prove that a semilocal ring Ris semiperfect if and only if the group homomorphism K ₀(FGFl(R)) → K ₀(FGFl(R/J(R))) is an epimorphism and K ₀(FGFl(R)) = K ₀(FGPr(R)).     

More on the Hochschild and Cyclic Homologies of Crossed Modules of Algebras

We investigate the Hochschild and cyclic homologies of crossed modules of algebras in some special cases. We prove that the cotriple cyclic homology of a crossed module of algebras (I, A, ρ) is isomorphic to HC _*(ρ): HC _*(I) → HC _*(A), provided I is H-unital and the ground ring is a field with characteristic zero. We also calculate the Hochschild and cyclic homologies of a crossed module of algebras (R, 0, 0) for each algebra R with trivial multiplication. At the end, we give some applications proving a new five term exact sequence.     

Derived Equivalences of Upper Triangular Differential Graded Algebras

This paper generalises a result for upper triangular matrix rings to the situation of upper triangular matrix differential graded algebras. An upper triangular matrix DGA has the form (R, S, M) where R and S are differential graded algebras and M is a DG-left-R-right-S-bimodule. We show that under certain conditions on the DG-module M and with the existance of a DG-R-module X, from which we can build the derived category D(R), that there exists a derived equivalence between the upper triangular matrix DGAs (R, S, M) and (S, M′, R′), where the DG-bimodule M′ is obtained from M and X and R′ is the endomorphism differential graded algebra of a K-projective resolution of X.     

Weighted sums of squares in local rings and their completions, I

Let A be an excellent local ring of real dimension ≤2, let T be a finitely generated preordering in A, and let ${\widehat{T}}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if ${{\rm Ext}^{1}_{R}\,(M, T)\,=\,0}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if Ext¹_R (M, T) = 0{{\rm Ext}^{1}_{R}\,(M, T)\,=\,0} for all torsion modules T, and M is Mittag-Leffler in case the canonical map M?_R ?_{i ? I}Q_i? ?_{i ? I}(M?_RQ_i){M\otimes_R \prod _{i\in I}Q_i\to \prod _{i\in I}(M\otimes_RQ_i)} is injective where {Q_i}_{i ? I}{\{Q_i\}_{i\in I}} are arbitrary left R-modules. We show that a module M is Baer iff M is p-filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification of the Mittag-Leffler modules.     

Finite vs affine W-algebras

In Section 1 we review various equivalent definitions of a vertex algebra V. The main novelty here is the definition in terms of an indefinite integral of the λ-bracket. In Section 2 we construct, in the most general framework, the Zhu algebra Zhu_ΓV, an associative algebra which “controls” Γ-twisted representations of the vertex algebra V with a given Hamiltonian operator H. An important special case of this construction is the H-twisted Zhu algebra Zhu_H V. In Section 3 we review the theory of non-linear Lie conformal algebras (respectively non-linear Lie algebras). Their universal enveloping vertex algebras (resp. universal enveloping algebras) form an important class of freely generated vertex algebras (resp. PBW generated associative algebras). We also introduce the H-twisted Zhu non-linear Lie algebra Zhu_H R of a non-linear Lie conformal algebra R and we show that its universal enveloping algebra is isomorphic to the H-twisted Zhu algebra of the universal enveloping vertex algebra of R. After a discussion of the necessary cohomological material in Section 4, we review in Section 5 the construction and basic properties of affine and finite W-algebras, obtained by the method of quantum Hamiltonian reduction. Those are some of the most intensively studied examples of freely generated vertex algebras and PBW generated associative algebras. Applying the machinery developed in Sections 3 and 4, we then show that the H-twisted Zhu algebra of an affine W-algebra is isomorphic to the finite W-algebra, attached to the same data. In Section 6 we define the Zhu algebra of a Poisson vertex algebra, and we discuss quasiclassical limits. In the Appendix, the equivalence of three definitions of a finite W-algebra is established. “I am an old man, and I know that a definition cannot be so complicated.” I.M. Gelfand (after a talk on vertex algebras in his Rutgers seminar)