Decomposing Frobenius Heisenberg categories

Abstract

We give two alternate presentations of the Frobenius Heisenberg category, ${\mathcal{H}eis}_{F,k}$, defined by Savage, when the Frobenius algebra $F=F_1\oplus \cdots \oplus F_n$ decomposes as a direct sum of Frobenius subalgebras. In these alternate presentations, the morphism spaces of ${\mathcal{H}eis}_{F,k}$ are given in terms of planar diagrams consisting of strands "colored" by integers $i=1,\dots ,n$, where a strand of color $i$ carries tokens labelled by elements of $F_i.$ In addition, we prove that when $F$ decomposes this way, the tensor product of Frobenius Heisenberg categories, ${\mathcal{H}eis}_{F_1,k}\otimes \cdots \otimes {\mathcal{H}eis}_{F_n,k},$ is equivalent to a certain subcategory of the Karoubi envelope of ${\mathcal{H}eis}_{F,k}$ that we call the $\text{partial}$ Karoubi envelope of ${\mathcal{H}eis}_{F,k}$.

21 pages. v2: Some definitions and results in Section 4 generalized to strict k-linear monoidal categories, see Def. 4.4, Def. 4.5, Lem. 4.6, and Cor. 4.7; minor corrections/changes, results unchanged. arXiv admin note: text overlap with arXiv:1802.01626 by other authors