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Orbifolding Frobenius algebras. (English) Zbl 1083.57037

It is well-known that \(1+1\)-dimensional topological field theories are parametrized by commutative Frobenius algebras [see M. Atiyah, Publ. Math., Inst. Hautes Étud. Sci. 68, 175–186 (1988; Zbl 0692.53053), R. Dijkgraaf, PhD Thesis, Utrecht (1989), B. Dubrovin, Lect. Notes Math. 1620, 120–348 (1996; Zbl 0841.58065)]. The purpose of this paper is to extend this result to orbifolds. More precisely, let \(G\) be a finite group. First, the author introduces the notion of \(G\)-Frobenius algebras together with graded and super versions. Essentially a \(G\)-Frobenius algebra (with respect to a character \(\chi\) of \(G\)) is an algebra \(A\) in the category of Yetter-Drinfeld modules over the group algebra of \(G\) together with an associative bilinear form on \(A\) (so that \(A\) is a Frobenius algebra), subject to a number of axioms, including braided commutativity and twisted invariance of the bilinear form with respect to \(\chi\). Then the author proves that \(G\)-Frobenius algebras classify functors from some geometric cobordism theories to the category of vector spaces over the base field. Finally particular families of \(G\)-Frobenius algebras are studied in detail; for instance, special \(G\)-Frobenius algebras – those with \(\dim A_g = 1\) for all \(g\in G\). These are in particular strongly \(G\)-graded algebras. The author provides an explicit construction of special \(G\)-Frobenius algebras in terms of 2-cocycles, generalizing the well-known description of strongly graded algebras.

MSC:

57R56 Topological quantum field theories (aspects of differential topology)
14N99 Projective and enumerative algebraic geometry
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
58D29 Moduli problems for topological structures
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16W50 Graded rings and modules (associative rings and algebras)
16L60 Quasi-Frobenius rings
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