On duality for weakly minimized vector valued optimization problems

An ordering that accords with the definition of a weak minimum is used to establish quasiduality, duality and converse duality theorems for optimization problems where the objective function takes values in real normed spaces of any dimension.     

Free group functors

We construct group functors whose Lie algebras are free.     

Fourier transform of the additive group in algebraically closed valued fields

We continue the study of the Hrushovski–Kazhdan integration theory and consider exponential integrals. The Grothendieck ring is enlarged via a tautological additive character and hence can receive such integrals. We then define the Fourier transform in our integration theory and establish some fundamental properties of it. Thereafter, a basic theory of distributions is also developed. We construct the Weil representations in the end as an application. The results are completely parallel to the classical ones.     

On a duality between Boolean valued analysis and topological Reduction Theory

By creating an unbounded topological reduction theory for complex Hilbert spaces over Stonean spaces, we can give a category-theoretic duality between Boolean valued analysis and topological reduction theory for complex Hilbert spaces. MSC: 03C90, 03E40, 06E15, 46M99.     

Schur functors for the symplectic group

The purpose of this work is to describe some new connections between the characteristic-free representation theories of the symplectic group and the corresponding general linear group (Theorem 2.2 and Theorem 2.6).     

On cohomotopy-type functors

This article deals with Chogoshvili cohomotopy functors which are defined by extending a cohomology functor given on some special auxiliary subcategories of the category of topological spaces. The question of choosing these subcategories is discussed. In particular, it is shown that in the singular case to define absolute groups it is sufficient that auxiliary subcategories should have as objects only spheresS ⁿ, Moore spacesP ⁿ(t)=S^n–1 U_t eⁿ, and one-point unions of these spaces.     

Reflection functors for quiver varieties and Weyl group actions

We introduce reflectionfunctors on quiver varieties. They are hyper-Kähler isometries between quiver varieties with different parameters, related by elements in the Weyl group. The definition is motivated by the origial reflection functor given by Bernstein-Gelfand-Ponomarev [1], but they behave much nicely. They are isomorphisms and satisfy the Weyl group relations. As an application, we define Weyl group representations of homology groups of quiver varieties. They are analogues of Slodowys construction of Springer representations of the Weyl group. Mathematics Subject Classification (2000):Primary 53C26; Secondary 14D21, 16G20, 20F55, 33D80Supported by the Grant-in-aid for Scientific Research (No.11740011), the Ministry of Education, Japan.     

Sequences of reflection functors and the preprojective component of a valued quiver

This paper concerns preprojective representations of a finite connected valued quiver without oriented cycles. For each such representation, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the representation. The set of equivalence classes of the above sequences is a partially ordered set that contains a great deal of information about the preprojective component of the Auslander-Reiten quiver. The results apply to the study of reduced words in the Weyl group associated with an indecomposable symmetrizable generalized Cartan matrix.     

Compatibility of quantization functors of Lie bialgebras with duality and doubling operations

We study the behavior of the Etingof–Kazhdan quantization functors under the natural duality operations of Lie bialgebras and Hopf algebras. In particular, we prove that these functors are “compatible with duality”, i.e., they commute with the operation of duality followed by replacing the coproduct by its opposite. We then show that any quantization functor with this property also commutes with the operation of taking doubles. As an application, we show that the Etingof–Kazhdan quantizations of some affine Lie superalgebras coincide with their Drinfeld–Jimbo-type quantizations. To the memory of Paulette Libermann (1919–2007)     

Extensions of semigroup valued,finitely additive measures

LetC be a field of subsets of a non-empty setX and let μ:C→E be a finitely additive measure (a “charge”) taking values in a commutative semigroupE. We consider the problem of extending μ to a charge defined on the power set and we say thatE has the charge extension property (CEP) if such extensions always exist. Los and Marczewski proved [4] that the semigroup of non-negative reals has CEP, and Carlson and Prikry [2] have shown that everygroup has CEP. We prove that every compact semigroup has CEP and show that CEP follows from certain completeness and distributivity conditions. Specializing to the case of lattices considered as semigroups under the operation of supremum, we characterize the class of lattices with CEP. An application to closure operators in general topology is also discussed.