When is a Coalgebra a Generator?

Let C be a coalgebra over a field k. The aim of this paper is to study the following problem : (P) If C is a k-coalgebra such that C is a generator for the category of left comodules, is C a left quasi-co-Frobenius coalgebra ? The converse always holds. We show that if C has a finite coradical series, the answer is positive.     

When is a Quasi-p-injective Module Continuous?

It is well-known that every quasi-p-injective module has C2-condition. In this note, it is shown that for a quasi-p-injective module M which is a self-generator, if M is projective, duo and semiperfect, then M is continuous. As a special case we re-obtain a result of Puninski-Wisbauer-Yousif saying that, a semiperfect ring R is right continuous if it is right duo, right p-injective.AMS Subject Classifications (1991): 16D50, 16D70, 16D80Supported by The Royal Golden Jubilee Project     

When is a Cleft Extension H-Azumaya?

We show which H ^op -cleft extensions of k for a dual quasi-triangular Hopf algebra (H, r) are H-Azumaya. The result is given in terms of bijectivity of a map defined in terms of the universal r-form r and the 2-cocycle σ, generalizing a well-known result for the commutative and co-commutative case. We illustrate the Theorem with an explicit computation for the Hopf algebras of type E(n).Presented by A. Verschoren     

When is a symplectic quotient an orbifold?

Let K be a compact Lie group of positive dimension. We show that for most unitary K-modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a finite group). When K is connected, we show that even a symplectomorphism to a linear symplectic orbifold does not exist. Our results yield conditions that preclude the symplectic quotient of a Hamiltonian K  -manifold from being locally isomorphic to an orbifold. As an application, we determine which unitary _SU2${\mathrm{SU}}_2$-modules yield symplectic quotients that are Z⁺${\mathbb{Z}}^{+}$-graded regularly symplectomorphic to a linear symplectic orbifold. We similarly determine which unitary circle representations yield symplectic quotients that admit a regular diffeomorphism to a linear symplectic orbifold.     

When is a Markovian Semigroup Induced by a Semiflow?

t , for t ≥ 0, be a strongly continuous Markovian semigroup acting on C(X), where X is a compact Hausdorf space, and let D denote the domain of its infinitesimal generator Z. Suppose D contains a (perhaps finite) family of functions f separating the points of X and satisfying Zf² = 2fZf. If either (1) there exists δ > 0 such that (T^t f)²∈ D if 0 ≤ t ≤δ for each f in this family; or (1′) for some core D′ of Z, g ∈ D′ implies g²∈ D, then the underlying Markoff process on X is deterministic. That is, there exists a semiflow — a semigroup (under composition) of continuous functions φ_t from X into X — such that T^tf(x) = f(φ_t (x)). If the domain D should be an algebra then conditions (1) and (1′) hold trivially. Conversely, if we have a separating family satisfying Zf² = 2fZf then each of these conditions implies that D is an algebra. It is an open question as to whether these conditions are redundant. If the functions φ_t are homeomorphisms from X onto X, then of course we have a Markovian group induced by a flow. This result is obtained by first providing general results about the null-space N of the (function-valued) positive semidefinite quadratic form defined by < f, g > = Z(fg) - fZg - gZf. The set N can be defined for any generator Z of a strongly continuous Markovian semigroup and is equivalently given by N = {f ∈ D| f²∈ D and Zf² = 2fZf} = {f ∈ D| T^t(f²)-(T^tf)² is o(t²) in C(X)}. In the general case N is an algebra closed under composition with any C¹-function φ from the reals to the reals, and Z(φ[f]) = (Zf)φ′[f] if f ∈ N. This "chain rule" on N (on which Z must act as a derivation) is a special case of a theorem for C²-functions φ which holds more generally for all f in d, viz., Z(φ[f] = (Zf) φ′[f] + ? <f, f> φ″[f], Provided Z is a local operator and D is an algebra. In this case the form < f, g > itself enjoys the relation < φ[f], ψ[g] > = φ′ [f] ψ′[g] < f, g >, for C²functions φ and ψ. Some of the results and their proofs continue to hold when the setting is switched from the commutative C*-algebra C(X) to a general (noncommutative) C*-algebra A. In the norm continuous case we obtain a sharp characterization of Markovian semigroups that are groups: Let T^t = e^tz , defined for t ≥ 0, be a Markovian semigroup acting on a C*-algebra A that is norm continuous, i.e., ||T^t - I|| ⇒ 0 as t ⇒ 0 +. Assume Z(a²) = a(Za) + (Za) a for some (perhaps finite) set of self-adjoint elements a that generate a Jordan algebra dense among the self-adjoint elements of A. The e^tz , -∞ < t < ∞, is a group of Markovian operators.     

When is the self-intersection of a subvariety a fibration?

We provide a necessary and sufficient condition for the derived self-intersection of a smooth subscheme inside a smooth scheme to be a fibration over the subscheme. As a consequence we deduce a generalized HKR isomorphism. We also investigate the relationship of our result to path spaces in homotopy theory, Buchweitz–Flenner formality in algebraic geometry, and draw parallels with similar results in Lie theory and symplectic geometry.     

When is a stochastic integral a time change of a diffusion?

We give necessary and sufficient conditions that a time change of ann-dimensional Ito stochastic integralX _t of the form

When is the Isbell topology a group topology?

Conditions on a topological space X under which the space C(X,R) of continuous real-valued maps with the Isbell topology κ is a topological group (topological vector space) are investigated. It is proved that the addition is jointly continuous at the zero function in Cκ(X,R) if and only if X is infraconsonant. This property is (formally) weaker than consonance, which implies that the Isbell and the compact-open topologies coincide. It is shown the translations are continuous in Cκ(X,R) if and only if the Isbell topology coincides with the fine Isbell topology. It is proved that these topologies coincide if X is prime (that is, with at most one non-isolated point), but do not even for some sums of two consonant prime spaces.     

When is a -block?

Let and be distinct prime numbers and let be a finite group. If is a -block of and is a -block, we study when the set of ordinary irreducible characters in the blocks and coincide.

     

When is a centralizer near-ring isomorphic to a matrix near-ring?

Necessary conditions are found for a centralizer near-ring M_A(G) to be isomorphic to a matrix near-ring, where G is a finite group which is cyclic as an M_A(G)-module There are centralizer near-rings which are matrix near-rings. A class of such near-rings is exhibited. Examples of centralizer near-rings which are not matrix near-rings are given.     

When is a right orderable group locally indicable?

If a group has an ascending series of subgroups such that for each ordinal , and has no non-abelian free subsemigroup, then is right orderable if and only if it is locally indicable. In particular if is a radical-by-periodic group, then it is right orderable if and only if it is locally indicable.

     

When is the commutant of a Bol loop a subloop?

A left Bol loop is a loop satisfying . The commutant of a loop is the set of elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order , odd, the commutant is a subloop. We investigate conditions under which the commutant of a Bol loop is not a subloop. In a finite Bol loop of order relatively prime to , the commutant generates an abelian group of order dividing the order of the loop. This generalizes a well-known result for Moufang loops. After describing all extensions of a loop such that is in the left and middle nuclei of the resulting loop, we show how to construct classes of Bol loops with a non-subloop commutant. In particular, we obtain all Bol loops of order with a non-subloop commutant.

     

When a word in Riordan involutions is a Riordan involution?

Aequationes mathematicae - In this note it is discussed when a word in Riordan involutions is also a Riordan involution.     

When is Flatness Coherent?

ABSTRACT

We characterize those small categories with the property that flat (contravariant) functors on them are coherently axiomatized in the language of presheaves on them. They are exactly the categories with the property that every finite diagram into them has a finite set of (weakly) initial cocones.     

When is tit-for-tat unbeatable?

We characterize the class of symmetric two-player games in which tit-for-tat cannot be beaten even by very sophisticated opponents in a repeated game. It turns out to be the class of exact potential games. More generally, there is a class of simple imitation rules that includes tit-for-tat but also imitate-the-best and imitate-if-better. Every decision rule in this class is essentially unbeatable in exact potential games. Our results apply to many interesting games including all symmetric 2 $\times $ 2 games, and standard examples of Cournot duopoly, price competition, public goods games, common pool resource games, and minimum effort coordination games.