On representations of affine Kac-Moody groups and related loop groups

We demonstrate a one to one correspondence between the irreducible projective representations of an affine Kac-Moody group and those of the related loop group, which leads to the results that every non-trivial representation of an affine Kac-Moody group must have its degree greater than or equal to the rank of the group and that the equivalence appears if and only if the group is of type for some . Moreover the characteristics of the base fields for the non-trivial representations are found being always zero.

     

Quantum superalgebra representations on cohomology groups of non-commutative bundles

Quantum homogeneous supervector bundles arising from the quantum general linear supergoup are studied. The space of holomorphic sections is promoted to a left exact covariant functor from a category of modules over a quantum parabolic sub-supergroup to the category of locally finite modules of the quantum general linear supergroup. The right derived functors of this functor provides a form of Dolbeault cohomology for quantum homogeneous supervector bundles. We explicitly compute the cohomology groups, which are given in terms of well understood modules over the quantized universal enveloping algebra of the general linear superalgebra.     

Dirac cohomology of unitary representations of equal rank exceptional groups

In this paper, we consider the unitary representations of equal rank exceptional groups of type E with a regular lambda-lowest K-type and classify those unitary representations with the nonzero Dirac cohomology.     

Vanishing of some cohomology groups and bounds for the Shafarevich-Tate groups of elliptic curves

Let E be an elliptic curve over Q and ? be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at ?. Under certain assumptions, we prove the vanishing of the Galois cohomology group H¹(Gal(K(E[?ⁱ])/K),E[?ⁱ]) for all i?1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyvagin, which finds a bound for the order of the ?-primary part of Shafarevich-Tate groups of E over K. This bound is consistent with the prediction of Birch and Swinnerton-Dyer conjecture.     

The cohomology of the Steenrod algebra and representations of the general linear groups

Let be the algebraic transfer that maps from the coinvariants of certain -representations to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer . It has been shown that the algebraic transfer is highly nontrivial, more precisely, that is an isomorphism for and that is a homomorphism of algebras.

In this paper, we first recognize the phenomenon that if we start from any degree and apply repeatedly at most times, then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the -representations. As a consequence, every finite -family in the coinvariants has at most nonzero elements. Two applications are exploited.

The first main theorem is that is not an isomorphism for . Furthermore, for every 5$">, there are infinitely many degrees in which is not an isomorphism. We also show that if detects a nonzero element in certain degrees of , then it is not a monomorphism and further, for each \ell$">, is not a monomorphism in infinitely many degrees.

The second main theorem is that the elements of any -family in the cohomology of the Steenrod algebra, except at most its first elements, are either all detected or all not detected by , for every . Applications of this study to the cases and show that does not detect the three families , and , and that does not detect the family .

     

Projective representations in algebras and cohomology

We give some conditions under which no two non-conjugate projective representations, in an algebra, of a given group can become conjugate over a separable extension of the base field. In particular, we show this is always the case for groups with trivial abelianization.     

Partial Dirac cohomology and tempered representations

The tempered representations of a real reductive Lie group G are naturally partitioned into series associated with conjugacy classes of Cartan subgroups H of G. We define partial Dirac cohomology, apply it for geometric construction of various models of these H–series representations, and show how this construction fits into the framework of geometric quantization and symplectic reduction.     

Conformal field theory and elliptic cohomology

In this paper, we use conformal field theory to construct a generalized cohomology theory which has some properties of elliptic cohomology theory which was some properties of elliptic cohomology. A part of our presentation is a rigorous definition of conformal field theory following Segal's axioms, and some examples, such as lattice theories associated with a unimodular even lattice. We also include certain examples and formulate conjectures on modular forms and Monstrous Moonshine related to the present work.     

Brauer groups and cohomology

We explain in an elementary way an example showing that the Brauer group of a scheme X does not always coincide with the torsion of Received: 22 June 2004     

Higher order operations in Deligne cohomology

Oblatum 10-VII-1993 & 26-IX-1994     

The Todd character and cohomology operations

In “On the Conflict of Bordism of Finite Complexes” [J. Differential Geometry], Conner and Smith introduced a homomorphism called the Todd character, relating complex bordism theory to rational homology. Specifically the Todd character consists of a family of homomorphisms

$\text{th}{}_{\mathrm{r}}\text{: MU}{}_{\mathrm{s}}\text{(X) → H}{}_{\mathrm{s\to r}}\text{(X;}\text{Q}\text{)}$

.In L. Smith, The Todd character and the integrality theorem for the Chern character, Ill. J. Math. it was shown (note that the indexing of the Todd character is somewhat different here) that there was an integrality theorem for th analogous to the Adams integrality theorem for the Chern character J. F. Adams, On the Chern character and the structure of the unitary group, Proc. Cambridge Philos. Soc.57 (1961), 189–199; On the Chern character revisted, Ill. J. Math. Now Adams' first paper contains a wealth of information about the Chern character in addition to the integrality theorem already mentioned. Our objective in the present note is to derive analogous results for the Todd character. As in Smith these may then be used to deduce the results of Adams for the Chern character.