We consider Frobenius algebras and their bimodules in certain abelian monoidal categories. In particular we study the Picard group of the category of bimodules over a Frobenius algebra, i.e. the group of isomorphism classes of invertible bimodules. The Rosenberg-Zelinsky sequence describes a homomorphism from the group of algebra automorphisms to the Picard group, which however is typically not surjective. We investigate under which conditions there exists a Morita equivalent Frobenius algebra for which the corresponding homomorphism is surjective. One motivation for our considerations is the orbifold construction in conformal field theory.
Deep Dive into On the Rosenberg-Zelinsky sequence in abelian monoidal categories.
We consider Frobenius algebras and their bimodules in certain abelian monoidal categories. In particular we study the Picard group of the category of bimodules over a Frobenius algebra, i.e. the group of isomorphism classes of invertible bimodules. The Rosenberg-Zelinsky sequence describes a homomorphism from the group of algebra automorphisms to the Picard group, which however is typically not surjective. We investigate under which conditions there exists a Morita equivalent Frobenius algebra for which the corresponding homomorphism is surjective. One motivation for our considerations is the orbifold construction in conformal field theory.
In the study of associative algebras it is often advantageous to collect algebras into a category whose morphisms are not algebra homomorphisms, but instead bimodules. One motivation for this is provided by the following observation. Let k be a field and consider finite-dimensional unital associative k-algebras. The condition on a k-linear map to be an algebra morphism is obviously not linear. As a consequence the category of algebras and algebra homomorphisms has the unpleasant feature of not being additive.
On the other hand, instead of an algebra homomorphism ϕ: A → B one can equivalently consider the B-A-bimodule B ϕ which as a k-vector space coincides with B and whose left action is given by the multiplication of B while the right action is application of ϕ composed with multiplication in B. This is consistent with composition in the sense that given another algebra homomorphism ψ: B → C there is an isomorphism C ψ ⊗ B B ϕ ∼ = C ψ•ϕ of C-A-bimodules. It is then natural not to restrict one’s attention to such special bimodules, but to allow all B-A-bimodules as morphisms from A to B [Be,sect. 5.7]. Of course, as bimodules come with their own morphisms, one then actually deals with the structure of a bicategory. The advantage is that the 1-morphism category A → B, i.e. the category of B-A-bimodules, is additive and even abelian.
Taking bimodules as morphisms has further interesting consequences. First of all, the concept of isomorphy of two algebras A and B is now replaced by Morita equivalence, which requires the existence of an invertible A-B-bimodule. Indeed, in applications involving associative algebras one often finds that not only isomorphic but also Morita equivalent algebras can be used for a given purpose. The classical example is the equivalence of the category of left (or right) modules over Morita equivalent algebras. Another illustration is the Morita equivalence between invariant subalgebras and crossed products, see e.g. [Ri]. Examples in the realm of mathematical physics include the observations that matrix theories on Morita equivalent noncommutative tori are physically equivalent [Sc], and that Morita equivalent symmetric special Frobenius algebras in modular tensor categories describe equivalent rational conformal field theories [FFRS1,FFRS3].
As a second consequence, instead of the automorphism group Aut(A) one now deals with the invertible A-bimodules. The isomorphism classes of these particular bimodules form the Picard group Pic(A-Bimod) of A-bimodules. While Morita equivalent algebras may have different automorphism groups, the corresponding Picard groups are isomorphic. One finds that for any algebra A the groups Aut(A) and Pic(A-Bimod) are related by the exact sequence 0 -→ Inn(A) -→ Aut(A)
which is a variant of the Rosenberg-Zelinsky [RZ, KO] sequence. Here Inn(A) denotes the inner automorphisms of A, and the group homomorphism Ψ A is given by assigning to an automorphism ω of A the bimodule A ω obtained from A by twisting the right action of A on itself by ω. In other words, Pic(A-Bimod) is the home for the obstruction to a Skolem-Noether theorem.
It should be noticed that the group homomorphism Ψ A in (1.1) is not necessarily a surjection. But for practical purposes in concrete applications it can be of interest to have an explicit realisa-tion of the Picard group in terms of automorphisms of the algebra available. This leads naturally to the following questions:
• Does there exist another algebra A ′ , Morita equivalent to A, such that the group homomorphism Ψ A ′ : Aut(A ′ ) → Pic(A ′ -Bimod) in (1.1) is surjective?
• And, once such an algebra A ′ has been constructed: Does this surjection admit a section, i.e. can the group Pic(A-Bimod) be identified with a subgroup of the automorphism group of the Morita equivalent algebra A ′ ?
We will investigate these questions in a more general setting, namely we consider algebras in k-linear monoidal categories more general than the one of k-vector spaces. Like many other results valid for vector spaces, also the sequence (1.1) continues to hold in this setting, see [VZ,prop. 3.14] and [FRS3,prop. 7].
We start in section 2 by collecting some aspects of algebras and Morita equivalence in monoidal categories and review the definition of invertible objects and of the Picard category. Section 3 collects information about fixed algebras under some subgroup of algebra automorphisms. In section 4 we answer the questions raised above for the special case that the algebra A is the tensor unit of the monoidal category D under consideration. As recalled in section 2, the categorical dimension provides a character on the Picard group with values in k × . The main result of section 4, Proposition 4.3, supplies, for any finite subgroup H of the Picard group on which this character is trivial, an algebra A ′ that is Morita equivalent to the tensor unit such that the elements of H can be identified with automorphisms of A. Theorem 4.12, in
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