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\begin{document}

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\section*{equivariant noncommutative algebraic geometry}

\hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation}

One would like to have symmetry objects like [[algebraic group]]s and [[Lie algebra]]s in [[noncommutative geometry]] (including algebraic flavour). The group-like objects should be noncommutative spaces themselves, they should have representation theory, they should act on other noncommutative spaces, define quotients and so on. A first massive appearance were quantum groups, and one should be warned that quantum groups are not cogroup objects in the category of noncommutative rings, because they are Hopf algebras with respect to the tensor product rather than categorical coproduct of algebras.

\hypertarget{main_characters}{}\subsection*{{Main characters}}\label{main_characters}

Here one should write about [[quantum groups]] (Drinfeld, Manin, Woronowicz, Jimbo, Lusztig, Faddeev-Reshetikin-Tahtajan, Majid), [[Hopf algebras]], Hopf algebroids (quantum groupoids), quantum Lie algebras, entwinings/[[distributive law]]s, quantum flag varieties, (co)module (co)algebras, quantum principal bundles, associated bundles, Drinfel'd center, equivariant cyclic homology etc.

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

\begin{itemize}%
\item V. G. Drinfel'd, \emph{Quantum groups}, Proc. Int. Cong. Math. 1986, Vol. 1, 2 798--820, AMS 1987.


\item S. Majid, \emph{Foundations of quantum group theory}, Cambridge University Press 1995, 2000.


\item B. Parshall, J.Wang, \emph{Quantum linear groups}, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.


\item Z. \v{S}koda, \emph{Some equivariant constructions in noncommutative geometry}, Georgian Math. J. 16 (2009) 1; 183--202 (\href{http://front.math.ucdavis.edu/0811.4770}{arXiv:0811.4770})


\item Yu. Manin, \emph{Quantum groups and noncommutative geometry}, CRM, Montreal (1988)


\item L. Faddeev, N. Reshetikin, L. Tahtajan, \emph{Quantization of Lie groups and Lie algebras}, Algebra i Analiz 1 (1989) 178 (transl. Leningrad Math. J. 1 (1990), 193-225)


\item S. Montgomery, \emph{Hopf algebras and their action on rings}, AMS 1994, 240p.


\item Y. Soibelman, \emph{On the quantum flag manifold}, Funk. analiz i ego pril. \textbf{26}, 3, 90--92, 1992


\item Z. \v{S}koda, \emph{Localizations for construction of quantum coset spaces}, \href{http://front.math.ucdavis.edu/math.QA/0301090}{math.QA/0301090}, Banach Center Publ. vol.61, pp. 265--298, Warszawa 2003.


\item N. Reshetikhin, A. A. Voronov, A. Weinstein. \emph{Semiquantum geometry}, \href{http://arxiv.org/abs/q-alg/9606007}{math.q-alg/9606007}


\item T. Brzeziski, S. Majid, \emph{Coalgebra bundles}, Comm. Math. Phys. 191 (1998), no. 2, 467--492 (\href{http://arxiv.org/abs/q-alg/9602022}{arXiv version}).


\item B. Day, R. Street, \emph{Monoidal bicategories and Hopf algebroids}, Adv. Math. \textbf{129}, (1997) 99-157 \href{http://dx.doi.org/10.1006/aima.1997.1649}{doi}



\end{itemize}
category: algebraic geometry, noncommutative geometry



\end{document}