Group-graded and group-bigraded rings

Let I be a nontrivial finite multiplicative group with the unit element e and A = ⨁x∈I Ax an I-graded ring. We construct a Frobenius extension Λ of A and study when the ring extension A of Ae can be a Frobenius extension. Also, formulating the ring structure of Λ, we introduce the notion of I-bigraded rings and show that every I-bigraded ring is isomorphic to the I-bigraded ring Λ constructed above.

Infinite-dimensional topological field theories from Hurwitz numbers

Classical Hurwitz numbers of a fixed degree together with Hurwitz numbers of seamed surfaces give rise to a Klein topological field theory (see [A. Alexeevski and S. Natanzon, The algebra of bipartite graphs and Hurwitz numbers of seamed surfaces, Izv. Math. 72(4) (2008) 627–646]). We extend this construction to Hurwitz numbers of all degrees simultaneously. The corresponding infinite-dimensional Cardy–Frobenius algebra is computed in terms of Young diagrams and bipartite graphs. This algebra turns out to be isomorphic to the algebra of differential operators introduced in [A. Mironov, A. Morozov and S. Natanzon, Cardy–Frobenius extension of algebra of cut-and-join operators, J. Geom. Phys. 73 (2012) 243–251, arXiv:1210.6955; A Hurwitz theory avatar of open-closed string, Eur. Phys. J. C 73(2) (2013) 1–10, arXiv:1208.5057], which serves a model for open-closed string theory. We prove that the operators corresponding to Young diagrams and bipartite graphs give rise to relations between Hurwitz numbers.

Subring Depth, Frobenius Extensions, and Towers

The minimum depthd(B,A)of a subringB⊆Aintroduced in the work of Boltje, Danz and Külshammer (2011) is studied and compared with the tower depth of a Frobenius extension. We show thatd(B,A)< ∞ ifAis a finite-dimensional algebra andBehas finite representation type. Some conditions in terms of depth and QF property are given that ensure that the modular function of a Hopf algebra restricts to the modular function of a Hopf subalgebra. IfA⊇Bis a QF extension, minimum left and right even subring depths are shown to coincide. IfA⊇Bis a Frobenius extension with surjective Frobenius, homomorphism, its subring depth is shown to coincide with its tower depth. Formulas for the ring, module, Frobenius and Temperley-Lieb structures are noted for the tower over a Frobenius extension in its realization as tensor powers. A depth 3 QF extension is embedded in a depth 2 QF extension; in turn certain depthnextensions embed in depth 3 extensions if they are Frobenius extensions or other special ring extensions with ring structures on their relative Hochschild bar resolution groups.

GENERALIZED SKEIN MODULES OF SURFACES

There is a one-to-one correspondence between 2-dimensional Topological quantum field theories and Frobenius extensions. Therefore each Frobenius extension defines an invariant of surfaces. We explore these invariants for a family of Frobenius extensions. In addition, we investigate the skein module of surfaces embedded in 3-manifolds corresponding to this family of Frobenius extensions.

A Remark on Frobenius Extensions and Endomorphism Rings

In his paper [1] F. Kasch developed a theory of Frobenius extensions as a generalization of the theory of Frobenius algebras. In it he established a very interesting relationship between the Frobenius property of an extension and that of its endomorphism ring [1, Satz 5], from which he further derived the Frobenius extension property of Galois extensions of simple rings [1, Satz 6]; with these results he kindly responded to what had been vaguely “conjectured” (as he wrote) by one of the writers on the connection between Galois theory and the theory of Frobenius algebras [2].