Hopf coactions on commutative algebras generated by a quadratically independent comodule

Let 𝒜 be a commutative unital algebra over an algebraically closed field k of characteristic ≠2, whose generators form a finite-dimensional subspace V, with no nontrivial homogeneous quadratic relations. Let 𝒬 be a Hopf algebra that coacts on 𝒜 inner-faithfully, while leaving V invariant. We prove that 𝒬 must be commutative when either: (i) the coaction preserves a non-degenerate bilinear form on V; or (ii) 𝒬 is co-semisimple, finite-dimensional, and char(k) = 0.     

TORSION THEORIES FOR FINITE VON NEUMANN ALGEBRAS

ABSTRACT

The study of modules over a finite von Neumann algebra 𝒜 can be advanced by the use of torsion theories. In this work, some torsion theories for 𝒜 are presented, compared, and studied. In particular, we prove that the torsion theory (T, P) (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for 𝒜.

Using torsion theories, we describe the injective envelope of a finitely generated projective 𝒜-module and the inverse of the isomorphism K ₀(𝒜) → K ₀ (𝒰), where 𝒰 is the algebra of affiliated operators of 𝒜. Then the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra ? of a finite von Neumann algebra 𝒜 to 𝒜. With these results, we prove that the capacity is invariant under the induction of a ?-module.     

Balanced pairs induce recollements

Let 𝒜 be an abelian category with arbitrary (set-indexed) coproducts and exact products. Let (𝒫,?) be a complete balanced pair. Then as in the classical case, we prove that there exists a recollement with the middle term K(𝒜), the homotopy category of 𝒜. In particular, this implies that the relative derived category exists. Two applications are given.     

The Lower Socle and Finite Rank Elementary Operators

Abstract

We define the lower socle of a semiprime algebra 𝒜 as the sum of all minimal left ideals 𝒜e where e is a minimal idempotent such that the division algebra e𝒜e is finite dimensional. We study the connection between the condition that the elements a _k , b _k , 1 ≤ k ≤ n, lie in the lower socle of 𝒜 and the condition that the elementary operator x ? a ₁ xb ₁ + ? + a _n xb _n has finite rank. As an application we obtain some results on derivations certain of whose powers have finite rank.     

Generic Picard–Vessiot Extensions for Nonconnected Groups

Let K be a differential field with algebraically closed field of constants 𝒞 and G a linear algebraic group over 𝒞. We provide a characterization of the K-irreducible G-torsors for nonconnected groups G in terms of the first Galois cohomology H¹(K, G) and use it to construct Picard–Vessiot extensions which correspond to nontrivial torsors for the infinite quaternion group, the infinite multiplicative and additive dihedral groups and the orthogonal groups. The extensions so constructed are generic for those groups.     

Trace Invariants of Finite-Dimensional Algebras

With every finite-dimensional algebra A over any field k we associate an 8-tuple of linear or bilinear forms on A, all of which are defined in terms of traces. For every groupoid 𝒞 formed by a class of k-algebras of fixed finite dimension, this passage is functorial and, when composed with any map that is constant on isoclasses, gives rise to an abundance of maps f: 𝒞 → I such that the fibres of f form a block decomposition of 𝒞. We study this decomposition for specific choices of 𝒞 and f, thereby putting established results from diverse algebraic theories into a unifying perspective, but also gaining new insight into classical groupoids of algebras such as associative unital algebras, division algebras, or composition algebras over general ground fields.     

THE STRUCTURE OF THE TANGENT CONE: AN INTERESTING BIJECTION#

ABSTRACT

Let X = Spec(R) be a reduced equidimensional algebraic variety over an algebraically closed field k. Let Y = Spec(R/𝔮) be a codimension one ordinary multiple subvariety, where 𝔮 is a prime ideal of height 1 of R. If U is a nonempty open subset of Y and 𝔪 a closed point of U, we denote by A ? R _𝔪 its local ring in X, by 𝔭 the extension of 𝔮 in A, and by K the algebraic closure of the residue field k(𝔭).

Then there exists a bijection γ^𝔪:Proj(G _𝔭(A) ?  _A/𝔭 k) → Proj(G(A _𝔭) ?  _k(𝔭)K) such that for every subset Σ of Proj(G _𝔭(A) ?  _A/𝔭 k), the Hilbert function of Σ coincides with the Hilbert function of γ^𝔪(Σ). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety.     

The Coradical Filtration for Drinfeld Double Ringel-Hall Algebras

Abstract

Let 𝒟(Λ) be the Drinfeld double Ringel-Hall algebra with Λ being any finite dimensional hereditary algebra over a finite field k. We determine the coradical filtration for 𝒟(Λ). As an application, we describe the group of Hopf algebra automorphisms of the Drinfeld double Ringel composition algebra of Λ.     

2-Colored triangles in edge-colored complete graphs, II

The largest number n = n(k) for which there exists a k-coloring of the edges of k_n with every triangle 2-colored is found to be n(k) = 2^r5^m, where k = 2m + r and r = 0 or 1, and all such colorings are given. We also prove the best possible result that a k-colored K_p satisfying 1 < k < 1 + √p contains at most k − 2 vertices not in a bichromatic triangle.     

The structure of the algebraic eigenspace to the spectral radius of eventually compact, nonnegative integral operators

Let K be an eventually compact linear integral operator on L^p(Ω, μ), 1 p < ∞, with nonnegative kernel k(x, y), where the underlying measure μ is totally σ-finite on the domain set Ω when P = 1. This work extends the previous analysis of the author who characterized the distinguished eigenvalues of K and K^*, and the support sets for the eigenfunctions and generalized eigenfunctions belonging to the spectral radius of K or K^*. The characterizations of the support sets for the algebraic eigenspaces of K or K^* are phrased in terms of significant k-components which are maximal irreducible subsets of Ω and which yield a positive spectral radius for the integral operator defined by the restriction of k(x, y) to the Cartesian product of such sets. In this paper, we show that a basis for the functions, constituting the algebraic eigenspaces of K and K^* belonging to the spectral radius of K, can be chosen to consist of elements which are positive on their sets of support, except possibly on sets of measure less than some arbitrarily specified positive number. In addition, we present necessary and sufficient conditions, in terms of the significant k-components, for both K and K^* to possess a positive eigenfunction (a.e. μ) corresponding to the spectral radius, as well as necessary and sufficient conditions for the sequence γⁿKⁿg _p to converge whenever g 0, where − _p denotes the norm in L^p(Ω, μ), and γ₁ the smallest (in modulus) characteristic value of K. This analysis is made possible by introducing the concepts of chains, lengths of chains, height, and depth of a significant k-component as was done by U. Rothblum [Lin. Alg. Appl. 12 (1975), 281–292] for the matrix setting.     

Additivity of Lie multiplicative maps on triangular algebras

Let 𝒜 and ? be unital algebras over a commutative ring ?, and ? be a (𝒜,??)-bimodule, which is faithful as a left 𝒜-module and also as a right ?-module. Let 𝒰?=?Tri(𝒜,??,??) be the triangular algebra and 𝒱 any algebra over ?. Assume that Φ?:?𝒰?→?𝒱 is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST???TS)?=?Φ(S)Φ(T)???Φ(T)Φ(S) for all S, T?∈?𝒰. Then Φ(S?+?T)?=?Φ(S)?+?Φ(T)?+?Z _S,T for all S, T?∈?𝒰, where Z _S,T is an element in the centre 𝒵(𝒱) of 𝒱 depending on S and T.     

A Generalization of Noncommuting Graph via Automorphisms of a Group

Let G be a group. By using a family 𝒜 of subsets of automorphisms of G, we introduced a simple graph Γ_𝒜(G), which is a generalization of the non-commuting graph. In this paper, we study the combinatorial properties of Γ_𝒜(G).     

Characterizations of Jordan derivations and Jordan homomorphisms

Let 𝒜 be a unital Banach algebra and ? be a unital 𝒜-bimodule. We show that if δ is a linear mapping from 𝒜 into ? satisfying δ(ST)?=?δ(S)T?+Sδ(T) for any S, T?∈?𝒜 with ST?=?W, where W is a left or right separating point of ?, then δ is a Jordan derivation. Also, it is shown that every linear mapping h from 𝒜 into a unital Banach algebra ? which satisfies h(S)h(T)?=?h(ST) for any S,?T?∈?𝒜 with ST?=?W is a Jordan homomorphism if h(W) is a separating point of ?.     

On reducible matrix patterns

A tool to study the inertias of reducible nonzero (resp. sign) patterns is presented. Sumsets are used to obtain a list of inertias attainable by the pattern 𝒜 ⊕ ? dependent upon inertias attainable by patterns 𝒜 and ?. It is shown that if ? is a pattern of order n, and 𝒜 is an inertially arbitrary pattern of order at least 2(n ? 1), then 𝒜 ⊕ ? is inertially arbitrary if and only if ? allows the inertias (0, 0, n), (0, n, 0) and (n, 0, 0). We illustrate how to construct other reducible inertially (resp. spectrally) arbitrary patterns from an inertially (resp. spectrally) arbitrary pattern 𝒜 ⊕ ?, by replacing 𝒜 with an inertially (resp. spectrally) arbitrary pattern 𝒮. We identify reducible inertially (resp. spectrally) arbitrary patterns of the smallest orders that contain some irreducible components that are not inertially (resp. spectrally) arbitrary. It is shown there exist nonzero (resp. sign) patterns 𝒜 and ? of orders 4 and 5 (resp. 4 and 4) such that both 𝒜 and ? are non-inertially-arbitrary, and 𝒜 ⊕ ? is inertially arbitrary.     

Maps preserving product X*Y + YX* on factor von Neumann algebras

Let 𝒜 and ? be two factor von Neumann algebras. In this article, we prove that a nonlinear bijective map Φ?:?𝒜?→?? satisfies Φ(X*?Y?+?YX*)?=Φ(X)*Φ(Y)?+?Φ(Y)Φ(X)* (?X,?Y?∈?𝒜), if and only if Φ is a *-ring isomorphism. In particular, if 𝒜 and ? are type I factors, then Φ is a unitary isomorphism or conjugate unitary isomorphism.     

On the composition subalgebras of Bridgeland’s Hall algebras

Let A be a finite dimensional hereditary algebra over a finite field k and 𝒫 the category consisting of finite dimensional projective (left) A-modules. In this paper, we consider the composition subalgebra of Bridgeland’s Hall algebra of the category 𝒞_m(𝒫) of m-cyclic complexes for any positive integer m≥2 and determine its generating relations.     

Generalized skew derivations on triangular algebras determined by action on zero products

For a triangular algebra 𝒜 and an automorphism σ of 𝒜, we describe linear maps F,G:𝒜→𝒜 satisfying F(x)y+σ(x)G(y) = 0 whenever x,y∈𝒜 are such that xy = 0. In particular, when 𝒜 is a zero product determined triangular algebra, maps F and G satisfying the above condition are generalized skew derivations of the form F(x) = F(1)x+D(x) and G(x) = σ(x)G(1)+D(x) for all x∈𝒜, where D:𝒜→𝒜 is a skew derivation. When 𝒜 is not zero product determined, we show that there are also nonstandard solutions for maps F and G.     

Coherent Continuation on the Category of (𝔤, K)-Modules

Abstract

Let 𝔤 be a complex semisimple Lie algebra. Let K be an algebraic group acting on the flag variety of 𝔤 with finitely many orbits. We give a geometric interpretation of the coherent continuation on the category of finitely generated (𝔤, K )-modules in terms of the intertwining functors on the category of K-equivariant 𝒟-modules.