Morita context of weak Hopf algebras

Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A~H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context connecting the smash product A#H and the invariant subalgebra A~H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.     

On the number of divisors of x2 + x + A

Let A be a positive or negative rational integer such that integers in the field of √1 ? 4A have unique prime factorization. An elementary criterion will be obtained for x² + x + A to be a prime number, where x is a positive integer. The criterion implies that for positive A the polynomial x² + x + A is prime for x = 0, 1,…, A ? 2.     

Irreducibility of associated matrices

For an n by n matrix A, let K(A) be the associated matrix corresponding to a permutation group (of degree m) and one of its characters. Let D_r(A) be the coefficient of x^m?r in K(A+xI). If A is reducible, then D_r(A) is reducible. If A is irreducible and the character is identically one, then D₁(A) is irreducible. If A is row stochastic and the character is identically one, then D_r(A) is essentially row stochastic. Finally, the results motivate the definition of group induced diagraphs.     

On lifting and extending derivations of approximately finite-dimensional C1-algebras

It is shown that if A is the C¹-algebra inductive limit of a sequence of finite-dimensional C¹-algebras, then for each closed two-sided ideal J of A derivations can be lifted to A from $\text{A}\text{J}$, and for each projection e in A derivations can be extended to A from eAe. An application of the second result is given.     

Hereditary Endomorphism Rings of Mixed Abelian Groups

Let A be a finite-rank, torsion-free, self-small mixed abelian sp-group and let E(A) be the endomorphism ring of A. We give conditions for right and left heredity of E(A). A ring is right hereditary if each of its right ideals is projective. We also find the structure of one-sided ideals of E(A).     

The endomorphism spectrum of a monounary algebra

The endomorphism spectrum specA of an algebra A is defined as the set of all positive integers, which are equal to the number of elements in an endomorphic image of A, for all endomorphisms of A. In this paper we study finite monounary algebras and their endomorphism spectrum. If a finite set S of positive integers is given, one can look for a monounary algebra A with S = specA. We show that for countably many finite sets S, no such A exists. For some sets S, an appropriate A with spec A = S are described. For n ∈ ? it is easy to find a monounary algebra A with {1, 2, ..., n} = specA. It will be proved that if i ∈ ?, then there exists a monounary algebra A such that specA skips i consecutive (consecutive eleven, consecutive odd, respectively) numbers. Finally, for some types of finite monounary algebras (binary and at least binary trees) A, their spectrum is shown to be complete.     

The number of meets between two subsets of a lattice

Let L be a lattice of divisors of an integer (isomorphically, a direct product of chains). We prove |A| |B| ? |L| |A ∧ B| for any A, B ? L, where |·| denotes cardinality and A ∧ B = {a ∧ b: a?A, b?B}. |A ∧ B| attains its minimum for fixed |A|, |B| when A and B are ideals. |·| can be replaced by certain other weight functions. When the n chains are of equal size k, the elements may be viewed as n-digit k-ary numbers. Then for fixed |A|, |B|, |A ∧ B| is minimized when A and B are the |A| and |B| smallest n-digit k-ary numbers written backwards and forwards, respectively. |A ∧ B| for these sets is determined and bounded. Related results are given, and conjectures are made.     

Generalized Specker lattice-ordered groups and two types of distributivity

Let A be a lattice-ordered group, B a generalized Boolean algebra. The Boolean extension A _B of A has been investigated in the literature; we will refer to A _B as a generalized Specker lattice-ordered group (namely, if A is the linearly ordered group of all integers, then A _B is a Specker lattice-ordered group). The paper establishes that some distributivity laws extend from A _B to both A and B, and (under certain circumstances) also conversely.     

Sensitivity of convergent matrices and polynomials

A matrix A is said to be convergent if and only if all its characteristic roots have modulus less than unity. When A is real an explicit expression is given for real matrices B such that A + B is also convergent, this expression depending upon the solution of a quadratic matrix equation of Riccati type. If A and A + B are taken to be in companion form, then the result becomes one of convergent polynomials (i.e., polynomials whose roots have modulus less then unity), and is much easier to apply. A generalization is given for the case when A and A + B are complex and have the same number of roots inside and outside a general circle.     

Deletion-restriction for subspace arrangements

Let A be a subspace arrangement in V with a designated maximal affine subspace A₀. Let A^′=A?{A₀} be the deletion of A₀ from A and A^″={A∩A₀|A∩A₀≠∅} be the restriction of A to A₀. Let M=V?_?A∈AA be the complement of A in V. If A is an arrangement of complex affine hyperplanes, then there is a split short exact sequence, 0→Hk(M^′)→Hk(M)→Hk+1−codimR(A₀)(M^″)→0. In this paper, we determine conditions for when the triple (A,A^′,A^″) of arrangements of affine subspaces yields the above split short exact sequence. We then generalize the no-broken-circuit basis nbc of Hk(M) for hyperplane arrangements to deletion-restriction subspace arrangements.     

On Weak Crossed Products of Weak Hopf Algebras

Let H be a weak Hopf algebra in the sense of Böhm et al. (J Algebra 221:385–438, 1999) measuring an algebra A. Let A# _σ H be a weak crossed product with σ invertible. Then in this paper we first give some conditions for A# _σ H to be a weak Hopf algebra. Next the spectral sequence for Ext will be constructed which yields an estimate for the global dimension of A# _σ H in terms of the corresponding data for H and A. Furthermore, we will investigate when A???A# _σ H becomes a separable extension. Finally, we prove that if H and its dual H* are both semisimple, then the finitistic dimension of A# _σ H is equal to that of A.     

The local determination of Jordan bases for H-self-adjoint operators

Let H be an invertible self-adjoint operator on a finite dimensional Hilbert space $\text{X}$. A linear operator A is said to be H-self-adjoint (or self-adjoint relative to H) if HA = A¹H. Let σ(A) denote, as usual, the spectrum of A. If A is H-self-adjoint, then A is similar to A¹ and λ ∈ σ(A) implies λ̄ ∈ σ (A), so that the spectrum of A issymmetric with respect to the real axis. Given spectral information for A at an eigenvalue λ₀ (≠ λ̄₀), we investigate the corresponding information at λ̄₀ and, in particular, the unique pairing of Jordan bases for the root subspaces at λ₀ and λ̄₀.     

Representation dimension of m-replicated algebras

Let A be a finite-dimensional hereditary algebra over an algebraically closed field and A ^(m) be the m-replicated algebra of A. We prove that the representation dimension of A ^(m) is at most 3, and that the dominant dimension of A ^(m) is at least m.     

Tensor products of closed operators on Banach spaces

Let A and B be closed operators on Banach spaces X and Y. Assume that A and B have nonempty resolvent sets and that the spectra of A and B are unbounded. Let α be a uniform cross norm on X ? Y. Using the Gelfand theory and resolvent algebra techniques, a spectral mapping theorem is proven for a certain class of rational functions of A and B. The class of admissable rational functions (including polynomials) depends on the spectra of A and B. The theory is applied to the cases A ? I + I ? B and A ? B where A and B are the generators of bounded holomorphic semigroups.     

The order bidual of lattice ordered algebras

The main topic of the present paper is a systematic investigation of the second order dual A″ of an Archimedean ?-algebra A with point separating order dual A′. It is shown that in the case that A has a unit element, the equality A″ = (A′)′_n holds, where (A′)′_n is the collection of all order continuous linear functionals on A′. It turns out that in general (A′)′_n, equipped with the Arens multiplication is an ?-algebra again. Necessary and sufficient conditions are derived for (A′)′_n to be semiprime and for (A′)′_n to have a unit element with respect to this multiplication.     

LU decomposition of M-matrices by elimination without pivoting

It is shown that if A or ?A is a singular M-matrix satisfying the generalized diagonal dominance condition y^TA?0 for some vector y? 0, then A can be factored into A = LU by a certain elimination algorithm, where L is a lower triangular M-matrix with unit diagonal and U is an upper triangular M-matrix. The existence of LU decomposition of symmetric permutations of A and for irreducible M-matrices and symmetric M-matrices follow as colollaries. This work is motivated by applications to the solution of homogeneous systems of linear equations Ax = 0, where A or ?A is an M-matrix. These applications arise, e.g., in the analysis of Markov chains, input-output economic models, and compartmental systems. A converse of the theorem metioned above can be established by considering the reduced normal form of A.     

Rings of differential operators on curves

Let k be an algebraically closed field of characteristic 0 and let A be a finitely generated k-algebra that is a domain whose Gelfand-Kirillov dimension is in [2, 3). We show that if A has a nonzero locally nilpotent derivation then A has quadratic growth. In addition to this, we show that A either satisfies a polynomial identity or A is isomorphic to a subalgebra of D(X), the ring of differential operators on an irreducible smooth affine curve X, and A is birationally isomorphic to D(X).     

On the generalized Hyers-Ulam stability of module left derivations

Let A be a unital normed algebra and let M be a unitary Banach left A-module. If f:A→M is an approximate module left derivation, then f:A→M is a module left derivation. Moreover, if M=A is a semiprime unital Banach algebra and f(tx) is continuous in t∈R for each fixed x in A, then every approximately linear left derivation f:A→A is a linear derivation which maps A into the intersection of its center Z(A) and its Jacobson radical rad(A). In particular, if A is semisimple, then f is identically zero.     

Finite dimensional Wiener-Hopf equations and factorizations of matrices

Let A be the algebra of all n × n matrices over the real or complex numbers. Let A₊ be the subalgebra of upper triangular matrices, and A_- the subalgebra of strictly lower triangular matrices. Denote by P the projection of A onto A₊ with kernel A_-. In this paper we investigate the Wiener-Hopf equation P(ax₊) = y₊, where y₊ ∈ A₊ is given and x₊ ∈ A₊ is a solution.