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  • universal enveloping algebra

Hopf algebra

• Comultiplication: \(\Delta: H\to H{ {}^{ \scriptstyle\otimes_{k}^{2} } }\)
• Counit \({\varepsilon}; H\to k\)
• Antipode \(S: H{\circlearrowleft}\)

16:16

• See Manin’s universal quantum groups.

  • Manin defines a universal bialgebra for \(A\), which coacts on \(A\) in a universal way.

  • See Unsorted/koszul duality

  • Forgetful functor from Hopf algebras to bialgebras has a left adjoint: the Hopf envelope.

  • Universal quantum group: take Hopf envelope of universal bialgebra.

• See quadratic algebra

• Twisting conditions for bialgebras: \(B\) is \({\mathbf{Z}}{\hbox{-}}\)graded and \(\Delta(B_n) \subseteq B{ {}^{ \scriptstyle\otimes_{k}^{2} } }\).

• Zhang twist: supplies a twisted multiplication.

  • Possibly related to alpha twisted vector space?

  • \({\mathsf{A}{\hbox{-}}\mathsf{grMod}} { \, \xrightarrow{\sim}\, }{\mathsf{A^{\phi}}{\hbox{-}}\mathsf{grMod}}\) for \(A^{\phi}\) a Zhang twist.

• Morita-Takeuchi equivalence: equivalence of categories of comodules.

• This talk compares cocycle twists to Zhang twists.

• For \({\mathcal{O}}(G)\) the coordinate ring of \(G\in \mathsf{Alg} {\mathsf{Grp}}\), elements \(g\in G\) induce automorphism \(r_g, \ell_g: {\mathcal{O}}(G){\circlearrowleft}\) by left/right translation, and every twisting pair is of the form \((r_g, \ell_g^{-1})\).

• Sovereign: equivalence between left and right duality functors.

• Pointed algebra: simple comodules are 1-dimensional

• Smash product of Hopf algebras: \(H_1\otimes H_2\) as a vector space, with a deformed multiplication.

  • Example: \(U({\mathfrak{g}})\wedge k[G]\).

• See quantum Yang Baxter equations. Solutions are \(R\in { \operatorname{End} }_k(V{ {}^{ \scriptstyle\otimes_{2}^{)} } }\) satisfying a tensor formula corresponding to moving strands in a braid.

  • Can be obtained from any braiding on \({\mathsf{H}{\hbox{-}}\mathsf{coMod}}\).

  • Use equivalence of braided monoidal cats to get new solutions: \(\cmods{H} { \, \xrightarrow{\sim}\, }\comod{A(R) \left[ { \scriptstyle { {g}^{-1}} } \right]} { \, \xrightarrow{\sim}\, }{\mathsf{A(R) \left[ { \scriptstyle { {g}^{-1}} } \right]^{\sigma}}{\hbox{-}}\mathsf{coMod}}\).

Examples

The group algebra:

• \(k[G]\) for \(g\in {\mathsf{Fin}}{\mathsf{Grp}}\) with \(g\to g\otimes g, g\to 1, g\to g^{-1}\)
• \(U({\mathfrak{g}})\) with \(x\to x\otimes 1 + 1\otimes x, x\to 0, x\to x^{-1}\).
• The coordinate ring of any \(X\in \mathsf{Alg} {\mathsf{Grp}}\).
• Actions: \(H\curvearrowright M\) where \(\lambda: H\otimes M\to M\) compatible with mult
• Coaction: \(\rho: M\to H\otimes M\) compatible with comult.
• Convolution algebra: \(\mathop{\mathrm{Hom}}_k(H, k)\) under \(fg \mapsto \mu_k \circ (f\otimes g) \circ \Delta_H\) for \(\Delta_H\) the comult on \(H\) and \(\mu_k\) the multiplication on \(k\).
• Can twist multiplication by a 2-cocycle \(\sigma: H{ {}^{ \scriptstyle\otimes_{2}^{\to } } }k\).