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Commutative algebra

Let A {\displaystyle A} be a commutative ring and M {\displaystyle M} an A-module. There are different equivalent definitions of a connection on M {\displaystyle M} .²

First definition

If k → A {\displaystyle k\to A} is a ring homomorphism, a k {\displaystyle k} -linear connection is a k {\displaystyle k} -linear morphism

∇ : M → Ω A / k 1 ⊗ A M {\displaystyle \nabla :M\to \Omega _{A/k}^{1}\otimes _{A}M}

which satisfies the identity

∇ ( a m ) = d a ⊗ m + a ∇ m {\displaystyle \nabla (am)=da\otimes m+a\nabla m}

A connection extends, for all p ≥ 0 {\displaystyle p\geq 0} to a unique map

∇ : Ω A / k p ⊗ A M → Ω A / k p + 1 ⊗ A M {\displaystyle \nabla :\Omega _{A/k}^{p}\otimes _{A}M\to \Omega _{A/k}^{p+1}\otimes _{A}M}

satisfying ∇ ( ω ⊗ f ) = d ω ⊗ f + ( − 1 ) p ω ∧ ∇ f {\displaystyle \nabla (\omega \otimes f)=d\omega \otimes f+(-1)^{p}\omega \wedge \nabla f} . A connection is said to be integrable if ∇ ∘ ∇ = 0 {\displaystyle \nabla \circ \nabla =0} , or equivalently, if the curvature ∇ 2 : M → Ω A / k 2 ⊗ M {\displaystyle \nabla ^{2}:M\to \Omega _{A/k}^{2}\otimes M} vanishes.

Second definition

Let D ( A ) {\displaystyle D(A)} be the module of derivations of a ring A {\displaystyle A} . A connection on an A-module M {\displaystyle M} is defined as an A-module morphism

∇ : D ( A ) → D i f f 1 ( M , M ) ; u ↦ ∇ u {\displaystyle \nabla :D(A)\to \mathrm {Diff} _{1}(M,M);u\mapsto \nabla _{u}}

such that the first order differential operators ∇ u {\displaystyle \nabla _{u}} on M {\displaystyle M} obey the Leibniz rule

∇ u ( a p ) = u ( a ) p + a ∇ u ( p ) , a ∈ A , p ∈ M . {\displaystyle \nabla _{u}(ap)=u(a)p+a\nabla _{u}(p),\quad a\in A,\quad p\in M.}

Connections on a module over a commutative ring always exist.

The curvature of the connection ∇ {\displaystyle \nabla } is defined as the zero-order differential operator

R ( u , u ′ ) = [ ∇ u , ∇ u ′ ] − ∇ [ u , u ′ ] {\displaystyle R(u,u')=[\nabla _{u},\nabla _{u'}]-\nabla _{[u,u']}\,}

on the module M {\displaystyle M} for all u , u ′ ∈ D ( A ) {\displaystyle u,u'\in D(A)} .

If E → X {\displaystyle E\to X} is a vector bundle, there is one-to-one correspondence between linear connections Γ {\displaystyle \Gamma } on E → X {\displaystyle E\to X} and the connections ∇ {\displaystyle \nabla } on the C ∞ ( X ) {\displaystyle C^{\infty }(X)} -module of sections of E → X {\displaystyle E\to X} . Strictly speaking, ∇ {\displaystyle \nabla } corresponds to the covariant differential of a connection on E → X {\displaystyle E\to X} .

Graded commutative algebra

The notion of a connection on modules over commutative rings is straightforwardly extended to modules over a graded commutative algebra.³ This is the case of superconnections in supergeometry of graded manifolds and supervector bundles. Superconnections always exist.

Noncommutative algebra

If A {\displaystyle A} is a noncommutative ring, connections on left and right A-modules are defined similarly to those on modules over commutative rings.⁴ However these connections need not exist.

In contrast with connections on left and right modules, there is a problem how to define a connection on an R-S-bimodule over noncommutative rings R and S. There are different definitions of such a connection.⁵ Let us mention one of them. A connection on an R-S-bimodule P {\displaystyle P} is defined as a bimodule morphism

∇ : D ( A ) ∋ u → ∇ u ∈ D i f f 1 ( P , P ) {\displaystyle \nabla :D(A)\ni u\to \nabla _{u}\in \mathrm {Diff} _{1}(P,P)}

which obeys the Leibniz rule

∇ u ( a p b ) = u ( a ) p b + a ∇ u ( p ) b + a p u ( b ) , a ∈ R , b ∈ S , p ∈ P . {\displaystyle \nabla _{u}(apb)=u(a)pb+a\nabla _{u}(p)b+apu(b),\quad a\in R,\quad b\in S,\quad p\in P.}

See also

• Connection (vector bundle)
• Connection (mathematics)
• Noncommutative geometry
• Supergeometry
• Differential calculus over commutative algebras

Notes

• Koszul, Jean-Louis (1950). "Homologie et cohomologie des algèbres de Lie" (PDF). Bulletin de la Société Mathématique de France. 78: 65–127. doi:10.24033/bsmf.1410.
• Koszul, J. L. (1986). Lectures on Fibre Bundles and Differential Geometry (Tata University, Bombay, 1960). doi:10.1007/978-3-662-02503-1 (inactive 1 November 2024). ISBN 978-3-540-12876-2. S2CID 51020097. Zbl 0244.53026.{{cite book}}: CS1 maint: DOI inactive as of November 2024 (link)
• Bartocci, Claudio; Bruzzo, Ugo; Hernández-Ruipérez, Daniel (1991). The Geometry of Supermanifolds. doi:10.1007/978-94-011-3504-7. ISBN 978-94-010-5550-5.
• Dubois-Violette, Michel; Michor, Peter W. (1996). "Connections on central bimodules in noncommutative differential geometry". Journal of Geometry and Physics. 20 (2–3): 218–232. arXiv:q-alg/9503020. doi:10.1016/0393-0440(95)00057-7. S2CID 15994413.
• Landi, Giovanni (1997). An Introduction to Noncommutative Spaces and their Geometries. Lecture Notes in Physics. Vol. 51. arXiv:hep-th/9701078. doi:10.1007/3-540-14949-X. ISBN 978-3-540-63509-3. S2CID 14986502.
• Mangiarotti, L.; Sardanashvily, G. (2000). Connections in Classical and Quantum Field Theory. doi:10.1142/2524. ISBN 978-981-02-2013-6.

External links

• Sardanashvily, G. (2009). "Lectures on Differential Geometry of Modules and Rings". arXiv:0910.1515 [math-ph].

References

1. (Koszul 1950) - Koszul, Jean-Louis (1950). "Homologie et cohomologie des algèbres de Lie" (PDF). Bulletin de la Société Mathématique de France. 78: 65–127. doi:10.24033/bsmf.1410. http://www.numdam.org/item/10.24033/bsmf.1410.pdf ↩

2. (Koszul 1950),(Mangiarotti & Sardanashvily 2000) - Koszul, Jean-Louis (1950). "Homologie et cohomologie des algèbres de Lie" (PDF). Bulletin de la Société Mathématique de France. 78: 65–127. doi:10.24033/bsmf.1410. http://www.numdam.org/item/10.24033/bsmf.1410.pdf ↩

3. (Bartocci, Bruzzo & Hernández-Ruipérez 1991), (Mangiarotti & Sardanashvily 2000) - Bartocci, Claudio; Bruzzo, Ugo; Hernández-Ruipérez, Daniel (1991). The Geometry of Supermanifolds. doi:10.1007/978-94-011-3504-7. ISBN 978-94-010-5550-5. https://doi.org/10.1007%2F978-94-011-3504-7 ↩

4. (Landi 1997) - Landi, Giovanni (1997). An Introduction to Noncommutative Spaces and their Geometries. Lecture Notes in Physics. Vol. 51. arXiv:hep-th/9701078. doi:10.1007/3-540-14949-X. ISBN 978-3-540-63509-3. S2CID 14986502. https://arxiv.org/abs/hep-th/9701078 ↩

5. (Dubois-Violette & Michor 1996),(Landi 1997) - Dubois-Violette, Michel; Michor, Peter W. (1996). "Connections on central bimodules in noncommutative differential geometry". Journal of Geometry and Physics. 20 (2–3): 218–232. arXiv:q-alg/9503020. doi:10.1016/0393-0440(95)00057-7. S2CID 15994413. https://arxiv.org/abs/q-alg/9503020 ↩