A Galois connection between subalgebras and tensor subcategories

Abstract

Let $\mathcal{B}$ be a braided finite tensor category and let $A$ be a simple commutative algebra in $\mathcal{B}$. We construct an order-reversing Galois connection between subalgebras of $A$ and tensor subcategories of $\mathcal{B}_A$. Let $\mathcal{B}'$ denote the M\"uger center and set $A':=A\cap\mathcal{B}'$. The closure operators are $B\mapsto\langle B,A'\rangle_{\mathrm{alg}}$ and $\mathcal{E}\mapsto\langle\mathcal{E},\mathcal{B}_A^{\mathrm{loc}}\rangle_{\otimes}$. Thus the closed subalgebras are those containing $A'$, while the closed tensor subcategories are those containing $\mathcal{B}_A^{\mathrm{loc}}$; equivalently, the fixed-point intervals $[A',A]_{\mathrm{alg}}$ and $[\mathcal{B}_A^{\mathrm{loc}},\mathcal{B}_A]_{\otimes}$ are anti-isomorphic as lattices. For a finite tensor category $\mathcal{C}$, the canonical algebra in $\mathcal{Z}(\mathcal{C})$ yields an anti-isomorphism between its subalgebras and tensor subcategories of $\mathcal{C}$. When $\mathcal{B}$ is nondegenerate, Frobenius extensions correspond to unimodular tensor subcategories. For a Hopf algebra in $\mathcal{B}$, the correspondence specializes to an order-preserving bijection between Hopf ideals and normal left coideal subalgebras.