Finitely Deep Matrices

In the algebraic study of deep matrices ? _X () on a finite set of indices over a field, Christopher Kennedy has recently shown that there is a unique proper ideal  whose quotient is a central simple algebra. He showed that this ideal, which doesn't appear for infinite index sets, is itself a central simple algebra. In this article we extend the result to deep matrices with a finite set of 2 or more indices over an arbitrary coordinate algebra A, showing that when the coordinates are simple there is again such a unique proper ideal, and in general that the lattice of ideals of ? _X (A)/ and  are isomorphic to the lattice of ideals of the coordinate algebra A.     

On the structure of inner ideals of nest algebras

IfA is a nest algebra andA _s=A ∩ A* , whereA* is the set of the adjoints of the operators lying inA, then the pair (A, A _s) forms a partial Jordan *-triple. Important tools when investigating the structure of a partial Jordan *-triple are its tripotents. In particular, given an orthogonal family of tripotents of the partial Jordan *-triple (A, A _s), the nest algebraA splits into a direct sum of subspaces known as the Peirce decomposition relative to that family. In this paper, the Peirce decomposition relative to an orthogonal family of minimal tripotents is used to investigate the structure of the inner ideals of (A, A _s), whereA is a nest algebra associated with an atomic nest. A property enjoyed by inner ideals of the partial Jordan *-triple (A, A _s) is presented as the main theorem. This result is then applied in the final part of the paper to provide examples of inner ideals. A characterization of the minimal tripotents as a certain class of rank one operators is also obtained as a means to deduce the principal theorem.     

Co-Point Modules Over Frobenius Koszul Algebras

Let A be a Frobenius Koszul algebra such that its Koszul dual A ^! is a quantum polynomial algebra. Co-point modules over A were defined as dual notion of point modules over A ^! with respect to the Koszul duality. In this article, we will see that various important functors between module categories over A used in representation theory of finite dimensional algebras send co-point modules to co-point modules. As a consequence, we will show that if (E, σ) is a geometric pair associated to A ^!, then the map σ:E → E is an automorphism of the point scheme E of A ^!, so that there is a bijection between isomorphism classes of left point modules over A ^! and those of right point modules over A ^!.     

Quotient Rings and f-Radical Extensions of Rings

By testing quotient rings, we give another viewpoint concerning the relationship between PI and Goldie properties, etc., and f-radical extensions of rings. The main result proved here is as follows: Let R be a prime algebra without nonzero nil right ideals. Suppose that R is f-radical over a subalgebra A, where f(X ₁,…, X _t ) is a multilinear polynomial, not an identity for p × p matrices in case char R = p > 0. Suppose that f is not power-central valued in R. Then the maximal ring of right (left) quotients of A coincides with that of R. Moreover, R is right Goldie if and only if A is.     

Invariants of the action of a semisimple finite-dimensional Hopf algebra on special algebras

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra H on a special algebra A, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero H-stable ideals. We prove that the algebra A is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a k-algebra. We give a counterexample to the finite generation of a non-semisimple Hopf algebra.     

Quotient rings of finitely generated P.I. Algebras

This paper generalizes properties which hold for localization of Azumaya algebras, in two directions. Firstly, fully left bounded left Noetherian rings, especially finitely generated Noetherian algebras, are considered. It is noted that for such rings every idempotent kernel functor a is symmetric, i.e. the filter T(σ) of a-dense left ideals has a basis of a-dense ideals. A prime ideal P of a f.l.b.l.N. ring R is localizable if and only if it is the intersection of the P-critical left ideals. In case R is a finitely generated algebra over its (Noetherian) center C, we apply the technique of “descent” of kernel functors. If a is a symmetric kernel functor such that R(A n c) S T(σ) for every A G T(σ) and such that a has property (T) then there is a kernel functor a’ on C-modules such that Qσ (R) ?Q? ,(R). If P is a prime ideal of R then σ_- descends to C if and only if P is localizable. Secondly, a class of rings is described in terms of the Zariski topology on Spec. The imposed condition is weaker than maximal centrallity and does not imply fully left boundedness either, but the good properties of Spec R in case R is an Azumaya algebra are preserved.     

Resolutions of subsets of finite sets of points in projective space

Abstract

Eisenbud et al. proved a number of results regarding Gröbner bases and initial ideals of those ideals J in the free associative algebra K ?X ₁,…, X _n ? which contain the commutator ideal. We prove similar results for ideals which contains the anti-commutator ideal (the defining ideal of the exterior algebra). We define one weak notion of generic initial ideals in K ?X ₁,…, X _n ?, and show that generic initial ideals of ideals containing the anti-commutator ideal, or the commutator ideal, are finitely generated.     

On subgroups normalized by EO(2<Emphasis Type="Italic">l,R</Emphasis>)

It is shown that the problem of describing those subgroups in the general linear group GL(n, R) which are normalized by a classical group is much more difficult than believed previously. For the case of even orthogonal groups, a thorough level calculation is performed, which shows that, even under the assumption 2 ∈ R*, the level of a subgroup H ≤ GL(2l, R), l ≥ 3, normalized by EO(2l, R), is determined by three ideals (A, B, C) in R rather than by two ideals, as was generally believed. These ideals are related by C ² ≤ A = B ∩ C, and triples of such ideals are said to be admissible. Here, A is the level of H with respect to the linear transvections t _ij (ξ), and B is the level of H with respect to the orthogonal transvections T _ij (ξ). The definition of the third level component is a little more complicated. In an appropriate realization, the Lie algebra of the even orthogonal group consists of matrices antisymmetric with respect to the skew diagonal. The component C is the level of H with respect to the complementary invariant subspace, which consists of matrices symmetric with respect to the skew diagonal. With any admissible triple (A, B, C) we associate a relative elementary subgroup EEO(2l, R, A, B, C), which is normalized by EO(2l, R) and, moreover, is EO(2l, R)-perfect.     

Hyperbolic Involutions and Quadratic Extensions

This is a variation on a theme of Bayer-Fluckiger, Shapiro, and Tignol related to hyperbolic involutions. More precisely, criteria for the hyperbolicity of involutions of quadratic extensions of simple algebras and involutions of the form σ ? τ and σ ? ρ, where σ is an involution of a central simple algebra A, τ is the nontrivial automorphism of a quadratic extension of the center of A, and ρ is an involution of a quaternion algebra are obtained.     

GOOD IDEALS IN MATRIX RINGS OVER COMMUTATIVE PIR's

Abstract

An ideal A of a ring R is called a good ideal if the coset product r ₁ r ₂ + A of any two cosets r ₁ + A and r ₂ + A of A in the factor ring R/A equals their set product (r ₁ + A) º (r ₂ + A): = {(r ₁ + a)(r ₂ + a ₂): a ₁, a ₂ ε A}. Good ideals were introduced in [3] to give a characterization of regular right duo rings. We characterize the good ideals of blocked triangular matrix rings over commutative principal ideal rings and show that the condition A º A = A is sufficient for A to be a good ideal in this class of matrix rings, none of which are right duo. It is not known whether good ideals in a base ring carries over to good ideals in complete matrix rings over the base ring. Our characterization shows that this phenomenon occurs indeed for complete matrix rings of certain sizes if the base ring is a blocked triangular matrix ring over a commutative principal ideal ring.     

An Algebraic Framework for Group Duality

A Hopf algebra is a pair (A, Δ) whereAis an associative algebra with identity andΔa homomorphism formAtoA⊗Asatisfying certain conditions. If we drop the assumption thatAhas an identity and if we allowΔto have values in the so-called multiplier algebraM(A⊗A), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A, Δ) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (Â, Δ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (Â, Δ) is canonically isomorphic with the original multiplier Hopf algebra (A, Δ). It is possible to generalize many aspects of abstract harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting.     

The Maximal Graded Left Quotient Algebra of a Graded Algebra1)

We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms （of left A-modules） from graded dense left ideals of A into a graded left quotient algebra of A. In the case of a superalgebra, and with some extra hypothesis, we prove that the component in the neutral element of the group of the maximal graded left quotient algebra coincides with the maximal left quotient algebra of the component in the neutral element of the group of the superalgebra.     

On a Banach Algebra Generated in its Multiplier Space and Applications to Locally Compact Groups

In this paper, we present a general version of the algebra A_M(G) which was introduced by B. Forrest. Indeed, for a faithful commutative Banach algebra A, we embed it in ?(A), the multiplier algebra of A, and obtain Banach algebra A_M. Then, we study the spaceability of A_M? A and A_M (G) ? ?A(G). These results give some characterizations of compactness and discreteness of locally compact groups. Also, we show that A_M(G) is an ideal in its second dual if and only if G is discrete. Finally, we study the BSE-property of A_M(G).

     

Lie ideals of graded associative algebras

We study the Lie structure of graded associative algebras. Essentially, we analyze the relation between Lie and associative graded ideals, and between Lie and associative graded derivations. Gathering together results on both directions, we compute maximal graded algebras of quotients of graded Lie algebras that arise from associative algebras. We also show that the Lie algebra Der _gr (A) of graded derivations of a graded semiprime associative algebra is strongly non-degenerate (modulo a certain ideal containing the center of Der _gr (A)).     

Hyper MV ‐ideals in hyper MV ‐algebras

In this paper we define the hyper operations ?, ∨ and ∧ on a hyper MV ‐algebra and we obtain some related results. After that by considering the notions ofhyper MV ‐ideals and weak hyper MV ‐ideals, we prove some theorems. Then we determine relationships between (weak) hyper MV ‐ideals in a hyper MV ‐algebra (M, ⊕, *, 0) and (weak) hyper K ‐ideals in a hyper K ‐algebra (M, °, 0). Finally we give a characterization of hyper MV ‐algebras of order 3 or 4 based on the (weak) hyper MV ‐ideals (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Closed ideals of a quasianalytic Fréchet algebra

In this note, we examine the structure of closed ideals of a quasianalytic weighted Beurling algebra A\cal {A}. This algebra is contained in C^￥ (G){\cal C}^\infty (\mit\Gamma) and contains the set A^￥ (D)A^\infty (D). Like in a previous article (see [6]), we use division properties and we give a characterization of closed ideals I such that I?A^￥ 1 { 0}I\cap A^\infty\! \ne \{ 0\} . Then, we use a factorization property proved in [2], which allows us to describe all the closed ideals of A\cal {A}.     

Ideal Structure of Leavitt Path Algebras with Coefficients in a Unital Commutative Ring

For a (countable) graph E and a unital commutative ring R, we analyze the ideal structure of the Leavitt path algebra L _R (E) introduced by Mark Tomforde. We first modify the definition of basic ideals and then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of L _R (E), and we then determine prime graded basic ideals and left (or right) primitive graded ideals of L _R (E). In particular, when E satisfies Condition (K) and R is a field, they imply that the set of prime ideals and the set of primitive ideals of L _R (E) coincide.     

Topological algebras with maximal regular ideals closed

It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex.     

Normal Subalgebras, I

We extend the notions of normal subalgebras, clots and ideals of an algebra A in a variety of (universal) algebras, from the familiar case of a single constant to the case of any number of constants. The first idea is that a subalgebra of A is normal when it is the inverse image under some morphism of the subalgebra generated by constants in the target. We argue that a better approach is obtained by considering pullbacks of γ _B and g?:?A?→?B, where g?:?A?→?B is some morphism and γ _B is the morphism from the initial algebra of the variety to B. Examples are shown in Heyting algebras, boolean algebras and unitary rings. Ideals and clots are generalizations of this notion, defined instead by closure under derived operations which have the right behavior on constants. There are several characterizations of these notions; some of them aiming at a categorical generalization. We deal with an (extended) notion of subtractivity, showing that it implies that ideals coincide with normal subalgebras, and it is connected with notions of coherence of congruences, allowing a characterization of protomodular varieties.     

REPRESENTATIONS OF A CLASS OF DRINFELD'S DOUBLES

Let k be a field and A_n(ω) be the Taft's n²-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(A_n(ω)) of A_n(ω) is a ribbon Hopf algebra. In the previous articles, we constructed an n₄-dimensional Hopf algebra H_n(p, q) which is isomorphic to D(A_n(ω)) if p ≠ 0 and q = ω^?1 , and studied the irreducible representations of H_n(1, q) and the finite dimensional representations of H₃(1, q). In this article, we examine the finite-dimensional representations of H_n(l q), equivalently, of D(A_n(ω)) for any n ≥ 2. We investigate the indecomposable left H_n(1, q)-module, and describe the structures and properties of all indecomposable modules and classify them when k is algebraically closed. We also give all almost split sequences in mod H_n(1, q), and the Auslander-Reiten-quiver of H_n(1 q).