Intertwining operator superalgebras and vertex tensor categories for superconformal algebras, II

We construct the intertwining operator superalgebras and vertex tensor categories for the superconformal unitary minimal models and other related models.

     

Bosonic Formulas for \widehat{\mathfrak{s}\mathfrak{l}} 2 Coinvariants

We derive bosonic-type formulas for the characters of ₂ coinvariants.     

The Two-Step Nilpotent Representations of the Extended Affine Hecke Algebra of Type A

We define a set of cell modules for the extended affine Hecke algebra of type A which are parametrised by SL_n()-conjugacy classes of pairs (s, N), where s SL_n() is semisimple and N is a nilpotent element of the Lie algebra which has at most two Jordan blocks and satisfies Ad(s)·N=q ² N. When q ²–1, each of these has irreducible head, and the irreducible representations of the affine Hecke algebra so obtained are precisely those which factor through its Temperley–Lieb quotient. When q ²=–1, the above remarks apply to a subset of the cell modules. Using our work on the cellular nature of those quotients, we are able to obtain complete information on the decomposition of the cell modules in all cases, even when q is a root of unity. They turn out to be multiplicity free, and the composition factors may be precisely described in terms of a partial order on the pairs (s, N). These results give explicit formulae for the dimensions of the irreducibles. Assuming our modules are identified with the standard modules earlier defined by Bernstein–Zelevinski, Kazhdan–Lusztig and others, our results may be interpreted as the determination of certain Kazhdan–Lusztig polynomials. [This has now been proved and will appear in a subsequent work of the authors.]The second author thanks the Australian Research Council and the Alexander von Humboldt Stiftung for support and the Universität Bielefeld for hospitality during the preparation of this work.     

Sur l'' application de réciprocité pour une surface fibrée en coniques définie sur un corps local

For a proper smooth variety X defined over a local field k, unramified class field theory investigates the reciprocity map _X: SK₁(X) ^ab ₁(X) as introduced by S. Saito. We study this map in the case when X is a surface admitting a proper surjection onto a smooth geometrically connected curve C with a smooth conic as generic fibre. Without any assumption on the reduction of C, we prove that _X is injective modulo n for all n invertible in k and its cokernel is the same as that of _C.     

Constructive decomposition of functions of finite central mean oscillation

The space CMO of functions of finite central mean oscillation is an analogue of BMO where the condition that the sharp maximal function is bounded is replaced by the convergence of the sharp function at the origin. In this paper it is shown that each element of CMO is a singular integral image of an element of the Beurling space of functions whose Hardy-Littlewood maximal function converges at zero. This result is an analogue of Uchiyama's constructive decomposition of BMO in terms of singular integral images of bounded functions. The argument shows, in fact, that to each element of CMO one can construct a vector Calderón-Zygmund operator that maps that element into the proper subspace .

     

Topological density of ccc Boolean algebras--Every cardinality occurs

For every uncountable cardinal there is a ccc Boolean algebra whose topological density is .

     

Homotopy Structures for Algebras over a Monad

Let T be a monad over a category A. Then a homotopy structure for A, defined by a cocylinder P : A A, or path-endofunctor, can be lifted to the category A ^T of Eilenberg–Moore algebras over T, provided that P is consistent with T in a natural sense, i.e. equipped with a natural transformation : T P P T satisfying some obvious axioms. In this way, homotopy can be lifted from well-known, basic situations to various categories of algebras for instance, from topological spaces to topological semigroups, or spaces over a fixed space (fibrewise homotopy), or actions of a fixed topological group (equivariant homotopy); from categories to strict monoidal categories; from chain complexes to associative chain algebras. The interest is given by the possibility of lifting the homotopy operations (as faces, degeneracy, connections, reversion, interchange, vertical composition, etc.) and their axioms from A to A ^T , just by verifying the consistency between these operations and : T P P T. When this holds, the structure we obtain on our category of algebras is sufficiently powerful to ensure the main general properties of homotopy.     

External Homogenization for Hopf Algebras: Applications to Maschke's Theorem

We apply to Hopf algebras a construction from graded rings, called the group ring of a graded ring. By using this construction we study the transfer of properties between certain categories of relative Hopf modules. As another application, we obtain a Maschke-type theorem for a Galois extension over a semisimple Hopf algebra.     

Separability and the Twisted Frobenius Bimodule

This paper begins with an introduction to -Frobenius structure on a finite-dimensional Hopf subalgebra pair. In Section 2 a study is made of a generalization of Frobenius bimodules and -Frobenius extensions. Also a special type of twisted Frobenius bimodule which gives an endomorphism ring theorem and converse is studied. Section 3 brings together material on separable bimodules, the dual definitions of split and separable extension, and a theorem of Sugano on endomorphism rings of separable bimodules. In Section 4, separable twisted Frobenius bimodules are characterized in terms of data that generalizes a Frobenius homomorphism and a dual base. In the style of duality, two corollaries characterizing split -Frobenius and separable -Frobenius extensions are proven. Sugano"s theorem is extended to -Frobenius extensions and their endomorphism rings. In Section 5, the problem of when separable extensions are Frobenius extensions is discussed. A Hopf algebra example and a matrix example are given of finite rank free separable -Frobenius extensions which are not Frobenius in the ordinary sense.