Noncommutative geometry
Noncommutative geometry (NCG) is a branch of
An approach giving deep insight about noncommutative spaces is through
Motivation
The main motivation is to extend the commutative duality between spaces and functions to the noncommutative setting. In mathematics, spaces, which are geometric in nature, can be related to numerical functions on them. In general, such functions will form a commutative ring. For instance, one may take the ring C(X) of continuous complex-valued functions on a topological space X. In many cases (e.g., if X is a compact Hausdorff space), we can recover X from C(X), and therefore it makes some sense to say that X has commutative topology.
More specifically, in topology, compact
Functions on a topological space can be multiplied and added pointwise hence they form a commutative algebra; in fact these operations are local in the topology of the base space, hence the functions form a sheaf of commutative rings over the base space.
The dream of noncommutative geometry is to generalize this duality to the duality between noncommutative algebras, or sheaves of noncommutative algebras, or sheaf-like noncommutative algebraic or operator-algebraic structures, and geometric entities of certain kinds, and give an interaction between the algebraic and geometric description of those via this duality.
Regarding that the commutative rings correspond to usual affine schemes, and commutative
Applications in mathematical physics
Some applications in particle physics are described in the entries Noncommutative standard model and Noncommutative quantum field theory. The sudden rise in interest in noncommutative geometry in physics follows after the speculations of its role in M-theory made in 1997.[3]
Motivation from ergodic theory
Some of the theory developed by
Noncommutative C*-algebras, von Neumann algebras
The (formal) duals of
For the
Noncommutative differentiable manifolds
A smooth Riemannian manifold M is a topological space with a lot of extra structure. From its algebra of continuous functions C(M), we only recover M topologically. The algebraic invariant that recovers the Riemannian structure is a spectral triple. It is constructed from a smooth vector bundle E over M, e.g. the exterior algebra bundle. The Hilbert space L2(M, E) of square integrable sections of E carries a representation of C(M) by multiplication operators, and we consider an unbounded operator D in L2(M, E) with compact resolvent (e.g. the signature operator), such that the commutators [D, f] are bounded whenever f is smooth. A deep theorem[4] states that M as a Riemannian manifold can be recovered from this data.
This suggests that one might define a noncommutative Riemannian manifold as a spectral triple (A, H, D), consisting of a representation of a C*-algebra A on a Hilbert space H, together with an unbounded operator D on H, with compact resolvent, such that [D, a] is bounded for all a in some dense subalgebra of A. Research in spectral triples is very active, and many examples of noncommutative manifolds have been constructed.
Noncommutative affine and projective schemes
In analogy to the
There are also generalizations of the Cone and of the Proj of a commutative graded ring, mimicking a theorem of Serre on Proj. Namely the category of quasicoherent sheaves of O-modules on a Proj of a commutative graded algebra is equivalent to the category of graded modules over the ring localized on Serre's subcategory of graded modules of finite length; there is also analogous theorem for coherent sheaves when the algebra is Noetherian. This theorem is extended as a definition of noncommutative projective geometry by Michael Artin and J. J. Zhang,[5] who add also some general ring-theoretic conditions (e.g. Artin–Schelter regularity).
Many properties of projective schemes extend to this context. For example, there exists an analog of the celebrated Serre duality for noncommutative projective schemes of Artin and Zhang.[6]
A. L. Rosenberg has created a rather general relative concept of noncommutative quasicompact scheme (over a base category), abstracting Grothendieck's study of morphisms of schemes and covers in terms of categories of quasicoherent sheaves and flat localization functors.[7] There is also another interesting approach via localization theory, due to Fred Van Oystaeyen, Luc Willaert and Alain Verschoren, where the main concept is that of a schematic algebra.[8][9]
Invariants for noncommutative spaces
Some of the motivating questions of the theory are concerned with extending known
The theory of
Examples of noncommutative spaces
- In the deformed into a non-commutative phase space generated by the position and momentum operators.
- The standard modelof particle physics.
- The noncommutative torus, deformation of the function algebra of the ordinary torus, can be given the structure of a spectral triple. This class of examples has been studied intensively and still functions as a test case for more complicated situations.
- Snyder space[10]
- Noncommutative algebras arising from foliations.
- Examples related to Gauss shifton continued fractions, give rise to noncommutative algebras that appear to have interesting noncommutative geometries.
Connection
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In the sense of Connes
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A Connes connection is a noncommutative generalization of a connection in differential geometry. It was introduced by Alain Connes, and was later generalized by Joachim Cuntz and Daniel Quillen.
Definition
Given a right A-module E, a Connes connection on E is a linear map
that satisfies the
See also
- Commutativity
- Fuzzy sphere
- Koszul connection
- Moyal product
- Noncommutative algebraic geometry
- Noncommutative topology
- Phase space formulation
- Quasi-free algebra
Citations
- ^ Khalkhali & Marcolli 2008, p. 171.
- ^ Khalkhali & Marcolli 2008, p. 21.
- S2CID 7562354.
- S2CID 17287100.
- ISSN 0001-8708.
- ISSN 0002-9939.
- MSRI lecture Noncommutative schemes and spaces (Feb 2000): video
- ISBN 0-8247-0424-X- New York: Dekker, 2000.- 287 p. - (Monographs and textbooks in pure and applied mathematics, 232)
- ISSN 0022-4049.
- ISSN 0031-899X.
- ^ Vale 2009, Definition 8.1.
References
- ISBN 978-0-12-185860-5
- MR 2408150
- MR 2371808
- Gracia-Bondia, Jose M; Figueroa, Hector; Varilly, Joseph C (2000), Elements of Non-commutative geometry, Birkhauser, ISBN 978-0-8176-4124-5
- Khalkhali, Masoud; Marcolli, Matilde (2008). Khalkhali, Masoud; Marcolli, Matilde (eds.). An Invitation to non-Commutative Geometry. World Scientific. ISBN 978-981-270-616-4.
- Landi, Giovanni (1997), An introduction to noncommutative spaces and their geometries, Lecture Notes in Physics. New Series m: Monographs, vol. 51, Berlin, New York: MR 1482228
- Van Oystaeyen, Fred; Verschoren, Alain (1981), Non-commutative algebraic geometry, Lecture Notes in Mathematics, vol. 887, ISBN 978-3-540-11153-5
References for Connes connection
- Connes, Alain (1980). "C* algèbres et géométrie différentielle". C. R. Acad. Sci. Paris Sér. A (in French). 290 (13): 599–604.
- Connes, Alain (2001). "C* algebras and differential geometry". arXiv:hep-th/0101093.
- Connes, Alain (2001). "C* algebras and differential geometry".
- Connes, Alain (1985). "Non-commutative differential geometry". S2CID 122740195.
- Connes, Alain (1995). Noncommutative Geometry. Academic Press. ISBN 978-0-08-057175-1.
- Cuntz, Joachim; Quillen, Daniel (1995). "Algebra Extensions and Nonsingularity". JSTOR 2152819.
- García-Beltrán, Dennise; a-Beltrán, Dennise; Vallejo, José A.; Vorobjev, Yuriĭ (2012). "On Lie Algebroids and Poisson Algebras". Symmetry, Integrability and Geometry: Methods and Applications. 8: 006. S2CID 5946411.
- * Vale, R. (2009). "notes on quasi-free algebras" (PDF).
- "Connections". Topics in Cyclic Theory. 2020. pp. 201–228. ISBN 9781108855846.
Further reading
- Zbl 1245.00040
- Grensing, Gerhard (2013). Structural aspects of quantum field theory and noncommutative geometry. Hackensack New Jersey: World Scientific. ISBN 978-981-4472-69-2.
External links
- Introduction to Quantum Geometry by Micho Đurđevich
- Ginzburg, Victor (2005). "Lectures on Noncommutative Geometry". arXiv:math/0506603.
- Khalkhali, Masoud (2004). "Very Basic Noncommutative Geometry". arXiv:math/0408416.
- Marcolli, Matilde (2004). "Lectures on Arithmetic Noncommutative Geometry". arXiv:math/0409520.
- Madore, J. (2000). "Noncommutative Geometry for Pedestrians". Classical and Quantum Nonlocality: 111. S2CID 15595586.
- Masson, Thierry (2006). "An informal introduction to the ideas and concepts of noncommutative geometry". arXiv:math-ph/0612012. (An easier introduction that is still rather technical)
- Noncommutative geometry on arxiv.org
- MathOverflow, Theories of Noncommutative Geometry
- Mahanta, Snigdhayan (2005). "On some approaches towards non-commutative algebraic geometry". arXiv:math/0501166.
- Sardanashvily, G. (2009). "Lectures on Differential Geometry of Modules and Rings". ].
- Noncommutative geometry and particle physics
- connection in noncommutative geometry in nLab