2021-10-25

Tags: #web/quick-notes

Refs: Advice

00:00

  • #resources/advice Don’t put off problem-solving until another day! Work on something small today rather than waiting for “enough” time to devote some grand problem. Part of the trick is constant progress and assembling small results into large ones (and conversely breaking large problems into such small pieces)

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 \({\mathbb{Z}}{\hbox{-}}\)graded and \(\Delta(B_n) \subseteq B\tensorpower{2}\).

  • Zhang twist: supplies a twisted multiplication.

    • Possibly related to alpha twisted vector space?

    • \(\grmod{A} { \, \xrightarrow{\sim}\, }\grmod{A^{\phi}}\) 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 \(R\in \mathop{\mathrm{End}}_k(V{ {}^{ \scriptstyle\otimes_{2}^{)} } }\) 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: \(\cmods{H} { \, \xrightarrow{\sim}\, }{\mathsf{A(R) \left[ { \scriptstyle { {g}^{-1}} } \right]}{\hbox{-}}\mathsf{coMod}} { \, \xrightarrow{\sim}\, }{\mathsf{A(R) \left[ { \scriptstyle { {g}^{-1}} } \right]^{\sigma}}{\hbox{-}}\mathsf{coMod}}\).

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