Simple current extensions beyond semi-simplicity

Semi-simple Extensions of Rings

In this paper we investigate the conditions under which a given ring is a sub-ring of a semi-simple ring. For convenience, we say that a ring A is an extension of a ring B, if B is a sub-ring of A. It is found that the existence of a semi-simple extension is equivalent to the vanishing of the extension radical, a two-sided ideal defined analogously to the ordinary radical. In Theorem II we give...

Establishing Advanced Technologies beyond Extensions to Current Systems

Module categories of simple current extensions of vertex operator algebras

In this article, we study module categries of simple current extensions of vertex operator algebras. Under certain assumptions, we show that every module for a rational vertex operator algebra be lifted to a twisted module for an extended algebra with an abelian symmetry.

Noncommutative Motives, Numerical Equivalence, and Semi-simplicity

In this article we further the study of the relationship between pure motives and noncommutative motives, initiated in [25]. Making use of Hochschild homology, we introduce the category NNum(k)F of noncommutative numerical motives (over a base ring k and with coefficients in a field F ). We prove that NNum(k)F is abelian semi-simple and that Grothendieck’s category Num(k)Q of numerical motives ...

Simple Extensions of Polytopes

We introduce the simple extension complexity of a polytope P as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto P . We devise a combinatorial method to establish lower bounds on the simple extension complexity and show for several polytopes that they have large simple extension complexities. These example...