791 research outputs found
Vertex operator algebras and operads
Vertex operator algebras are mathematically rigorous objects corresponding to
chiral algebras in conformal field theory. Operads are mathematical devices to
describe operations, that is, -ary operations for all greater than or
equal to , not just binary products. In this paper, a reformulation of the
notion of vertex operator algebra in terms of operads is presented. This
reformulation shows that the rich geometric structure revealed in the study of
conformal field theory and the rich algebraic structure of the theory of vertex
operator algebras share a precise common foundation in basic operations
associated with a certain kind of (two-dimensional) ``complex'' geometric
object, in the sense in which classical algebraic structures (groups, algebras,
Lie algebras and the like) are always implicitly based on (one-dimensional)
``real'' geometric objects. In effect, the standard analogy between
point-particle theory and string theory is being shown to manifest itself at a
more fundamental mathematical level.Comment: 16 pages. Only the definitions of "partial operad" and of "rescaling
group" have been improve
Algebraic Rieffel Induction, Formal Morita Equivalence, and Applications to Deformation Quantization
In this paper we consider algebras with involution over a ring C which is
given by the quadratic extension by i of an ordered ring R. We discuss the
*-representation theory of such *-algebras on pre-Hilbert spaces over C and
develop the notions of Rieffel induction and formal Morita equivalence for this
category analogously to the situation for C^*-algebras. Throughout this paper
the notion of positive functionals and positive algebra elements will be
crucial for all constructions. As in the case of C^*-algebras, we show that the
GNS construction of *-representations can be understood as Rieffel induction
and, moreover, that formal Morita equivalence of two *-algebras, which is
defined by the existence of a bimodule with certain additional structures,
implies the equivalence of the categories of strongly non-degenerate
*-representations of the two *-algebras. We discuss various examples like
finite rank operators on pre-Hilbert spaces and matrix algebras over
*-algebras. Formal Morita equivalence is shown to imply Morita equivalence in
the ring-theoretic framework. Finally we apply our considerations to
deformation theory and in particular to deformation quantization and discuss
the classical limit and the deformation of equivalence bimodules.Comment: LaTeX2e, 51pages, minor typos corrected and Note/references adde
Deformation theory of objects in homotopy and derived categories III: abelian categories
This is the third paper in a series. In part I we developed a deformation
theory of objects in homotopy and derived categories of DG categories. Here we
show how this theory can be used to study deformations of objects in homotopy
and derived categories of abelian categories. Then we consider examples from
(noncommutative) algebraic geometry. In particular, we study noncommutative
Grassmanians that are true noncommutative moduli spaces of structure sheaves of
projective subspaces in projective spaces.Comment: Alexander Efimov is a new co-author of this paper. Besides some minor
changes, a new part (part 3) about noncommutative Grassmanians was adde
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