非可換幾何学に関する解説

非可換幾何学に関する解説書もいろいろ出ている。 年代順に挙げると以下の通り。

• Connes の解説書 [Con90] (フランス語)
• Manin の [Man91]
• Connes の解説書 [Con94] (英語)
• Landi の [Lan97] (物理の人向け) と [Lan07] (gauge theory が目的)
• Gracia-Bondia と Varilly と Figueroa の本 [GVF01]
• Connes の [Con00]
• Varilly の [Var]
• Khalkhali の解説 [Kha]
• Marcolli の [Mar05]
• Ginzburg の [Gin]
• Connes と Marcolli の [CM08]
• Khalkhali の [Kha08]

Ginzburg のものは, Atiyah algebra や Morita同値などの代数的な基礎から書いてあって, 読みやすい。

解説書とは言えないが, Skoda の [Ško06] は, 非可換 (代数) 幾何学の様々なアイデアを概観するのによいと思う。

Euclid 空間の幾何学を非可換化しようという試みについては, Banica の [Ban] がある。

References

[Ban]

Teo Banica. Affine noncommutative geometry. arXiv: 2012.10973.

[CM08]

Alain Connes and Matilde Marcolli. “A walk in the noncommutative garden”. In: An invitation to noncommutative geometry. World Sci. Publ., Hackensack, NJ, 2008, pp. 1–128. arXiv: math/0601054. url: http://dx.doi.org/10.1142/9789812814333_0001.

[Con00]

Alain Connes. “Noncommutative geometry—year 2000”. In: Geom. Funct. Anal. Special Volume, Part II (2000). GAFA 2000 (Tel Aviv, 1999), pp. 481–559. arXiv: math.QA/0011193.

[Con90]

Alain Connes. Géométrie non commutative. Paris: InterEditions, 1990, p. 240. isbn: 2-7296-0284-4.

[Con94]

Alain Connes. Noncommutative geometry. San Diego, CA: Academic Press Inc., 1994, pp. xiv+661. isbn: 0-12-185860-X.

[Gin]

Victor Ginzburg. Lectures on Noncommutative Geometry. arXiv: math/0506603.

[GVF01]

José M. Gracia-Bondía, Joseph C. Várilly, and Héctor Figueroa. Elements of noncommutative geometry. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Boston, Inc., Boston, MA, 2001, pp. xviii+685. isbn: 0-8176-4124-6. url: http://dx.doi.org/10.1007/978-1-4612-0005-5.

[Kha]

Masoud Khalkhali. Very Basic Noncommutative Geometry. arXiv: math/0408416.

[Kha08]

Masoud Khalkhali. “Lectures on noncommutative geometry”. In: An invitation to noncommutative geometry. World Sci. Publ., Hackensack, NJ, 2008, pp. 169–273. arXiv: math/0702140. url: http://dx.doi.org/10.1142/9789812814333_0003.

[Lan07]

Giovanni Landi. “Examples of noncommutative instantons”. In: Geometric and topological methods for quantum field theory. Vol. 434. Contemp. Math. Providence, RI: Amer. Math. Soc., 2007, pp. 39–72. arXiv: math/0603426.

[Lan97]

Giovanni Landi. An introduction to noncommutative spaces and their geometries. Vol. 51. Lecture Notes in Physics. New Series m: Monographs. Berlin: Springer-Verlag, 1997, pp. xii+200. isbn: 3-540-63509-2.

[Man91]

Yuri I. Manin. Topics in noncommutative geometry. M. B. Porter Lectures. Princeton, NJ: Princeton University Press, 1991, pp. viii+164. isbn: 0-691-08588-9.

[Mar05]

Matilde Marcolli. Arithmetic noncommutative geometry. Vol. 36. University Lecture Series. With a foreword by Yuri Manin. American Mathematical Society, Providence, RI, 2005, pp. xii+136. isbn: 0-8218-3833-4. url: http://dx.doi.org/10.1090/ulect/036.

[Ško06]

Zoran Škoda. “Noncommutative localization in noncommutative geometry”. In: Non-commutative localization in algebra and topology. Vol. 330. London Math. Soc. Lecture Note Ser. Cambridge: Cambridge Univ. Press, 2006, pp. 220–313. arXiv: math/0403276. url: http://dx.doi.org/10.1017/CBO9780511526381.015.

[Var]

Joseph C. Varilly. An Introduction to Noncommutative Geometry. arXiv: physics/9709045.