Quasi-bialgebras, quasi-antipodes, and quasi-Hopf algebras

Fix a field $k$ and use algebraic tensor products over $k$ as in Scalars, tensor powers, the empty tensor, opposite algebras and finite sums. A quasi-bialgebra is an associative unital $k$-algebra $H$ (in the sense of Algebras over a commutative ring, central structure maps, and algebra homomorphisms) with unital algebra maps $\mathrm{\Delta }:H\to H\otimes H$ and $\epsilon :H\to k$, and an invertible element $\mathrm{\Phi }\in H^{\otimes 3}$, satisfying the counit identities $(\epsilon \otimes \mathrm{i}\mathrm{d})\mathrm{\Delta }=\mathrm{i}\mathrm{d}=(\mathrm{i}\mathrm{d}\otimes \epsilon )\mathrm{\Delta }$, the quasi-coassociativity identity $(\mathrm{i}\mathrm{d}\otimes \mathrm{\Delta })\mathrm{\Delta }(h)=\mathrm{\Phi }(\mathrm{\Delta }\otimes \mathrm{i}\mathrm{d})\mathrm{\Delta }(h){\mathrm{\Phi }}^{-1}(h\in H),$ the pentagon identity $(1\otimes \mathrm{\Phi })(\mathrm{i}\mathrm{d}\otimes \mathrm{\Delta }\otimes \mathrm{i}\mathrm{d})(\mathrm{\Phi })(\mathrm{\Phi }\otimes 1)=(\mathrm{i}\mathrm{d}\otimes \mathrm{i}\mathrm{d}\otimes \mathrm{\Delta })(\mathrm{\Phi })(\mathrm{\Delta }\otimes \mathrm{i}\mathrm{d}\otimes \mathrm{i}\mathrm{d})(\mathrm{\Phi }),$ and the normalization $(\mathrm{i}\mathrm{d}\otimes \epsilon \otimes \mathrm{i}\mathrm{d})(\mathrm{\Phi })=1\otimes 1$. Here $\mathrm{\Phi }$ is the reassociator. With this convention, $\mathrm{\Phi }$ acts from the left-parenthesized tensor product to the right-parenthesized one. When $\mathrm{\Phi }=1$, these conditions give the ordinary bialgebra axioms of Bialgebras and bialgebra maps.

A quasi-antipode for a quasi-bialgebra is a triple $(S,\alpha ,\beta )$ where $S:H\to H$ is a unital algebra antihomomorphism and $\alpha ,\beta \in H$ satisfy $\sum S(h_{(1)})\alpha h_{(2)}=\epsilon (h)\alpha ,\sum h_{(1)}\beta S(h_{(2)})=\epsilon (h)\beta ,$ $\sum X^1\beta S(X^2)\alpha X^3=1,\sum S(x^1)\alpha x^2\beta S(x^3)=1,$ where $\mathrm{\Delta }(h)=\sum h_{(1)}\otimes h_{(2)}$, $\mathrm{\Phi }=\sum X^1\otimes X^2\otimes X^3$, and ${\mathrm{\Phi }}^{-1}=\sum x^1\otimes x^2\otimes x^3$. A quasi-Hopf algebra is a quasi-bialgebra with a quasi-antipode whose map $S$ is bijective. A quasi-bialgebra need not have a quasi-antipode, and a quasi-antipode is additional data rather than part of the quasi-bialgebra structure.