The monoidal center construction and bimodules

✍ Scribed by Peter Schauenburg

• Book ID: 104152293
• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 185 KB
• Volume: 158
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(00)00040-2

✦ Synopsis

Let C be a cocomplete monoidal category such that the tensor product in C preserves colimits in each argument. Let A be an algebra in C. We show (under some assumptions including "faithful atness" of A) that the center of the monoidal category (ACA; ⊗A) of A-A-bimodules is equivalent to the center of C (hence in a sense trivial): Z(ACA) ∼ = Z(C). Assuming A to be a commutative algebra in the center Z(C), we compute the center Z(CA) of the category of right A-modules (considered as a subcategory of ACA using the structure of A ∈ Z(C). We ÿnd Z(CA) ∼ = dys Z(C)A, the category of dyslectic right A-modules in the braided category Z(C).