Quick question: When $A$ and $B$ are Hopf algebras, what does $A /\!\!/ B$ stand for exactly? In context, it seems to be a type of quotient, but I need to be sure.
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Are they just rings? If they are Hopf algebras and $B$ is a normal subalgebra of $A$, then $A//B$ usually denotes the Hopf quotient, which as an algebra is the quotient of $A$ by the ideal generated by the augmentation ideal of B. This can be generalized to augmented algebras —it is done in Cartan-Eilenberg, for example.
If $A$ and $B$ are the group algebras of a group and a normal subgroup, then the Hopf quotient is isomorphic to the group algebra of the quotient group; a similar observation works for enveloping algebras of Lie algebras and their ideals. This motivates the construction.