On the Notion of Noncommutative Submanifold

D’Andrea, Francesco

doi:10.3842/sigma.2020.050

Cited by 5 publications

(7 citation statements)

References 30 publications

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“…It is important to notice that this notion recovers other examples coming from Poisson geometry, e.g. [17] and non commutative geometry, as [28] and [13].…”

Section: Introductionmentioning

confidence: 63%

“…Example 2. 15 Our notion of a coisotropic algebra generalizes and unifies previous notions used in noncommutative geometry referring to features of the derivations: i) A submanifold algebra in the sense of [28] and [13] can equivalently be described as a coisotropic algebra A with A tot = A N such that the canonical module morphism (2.22) is an isomorphism. ii) A quotient manifold algebra in the sense of [28] can equivalently be described as a coisotropic algebra A with A N ⊆ A tot a subalgebra and…”

Section: Proposition 213 Letmentioning

confidence: 81%

“…ii.) We have 13 The center of a coisotropic algebra could now be defined as Z (A ) := HH 0 (A ). Similarly, one can define the outer derivations of a coisotropic algebra by OutDer(A ) := HH 1 (A ).…”

Section: Corollary 47 Letmentioning

confidence: 99%

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Deformation and Hochschild cohomology of coisotropic algebras

Dippell

Esposito

Waldmann

2021

Annali di Matematica

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Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper, we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.

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“…It is important to notice that this notion recovers other examples coming from Poisson geometry, e.g. [17] and non commutative geometry, as [28] and [13].…”

Section: Introductionmentioning

confidence: 63%

Section: Proposition 213 Letmentioning

confidence: 81%

See 1 more Smart Citation

Deformation and Hochschild cohomology of coisotropic algebras

Dippell

Esposito

Waldmann

2021

Annali di Matematica

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show abstract

“…Example 2.15 Our notion of a coisotropic algebra generalizes and unifies previous notions used in noncommutative geometry referring to features of the derivations: i.) A submanifold algebra in the sense of [23] and [11] can equivalently be described as a coisotropic algebra A with A tot = A N such that the canonical module morphism (2.22) is an isomorphism.…”

Section: Coisotropic Algebras and Derivationsmentioning

confidence: 81%

Deformation and Hochschild Cohomology of Coisotropic Algebras

Dippell¹,

Esposito²,

Waldmann³

2020

Preprint

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Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.

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“…We note that since the L i j = f i ∂ j − f j ∂ i ∈ g involved in the twist vanish on E f , the deformation automatically disappears on it, and the twisted algebraic variety is well defined as the undeformed. For other examples of submanifold algebras that are not algebras of functions on smooth manifolds, we refer the reader to the recent paper [18]. We devote a subsection to each family and a proposition to each twist deformation; propositions are proved in [38], where twist deformations also of the other classes of quadrics are discussed in detail.…”

Section: MCmentioning

confidence: 99%

Twisted submanifolds of $${\mathbb {R}}^n$$

Fiore

Weber

2021

Lett Math Phys

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We propose a general procedure to construct noncommutative deformations of an embedded submanifold M of $${\mathbb {R}}^n$$ R n determined by a set of smooth equations $$f^a(x)=0$$ f a ( x ) = 0 . We use the framework of Drinfel’d twist deformation of differential geometry of Aschieri et al. (Class Quantum Gravity 23:1883, 2006); the commutative pointwise product is replaced by a (generally noncommutative) $$\star $$ ⋆ -product determined by a Drinfel’d twist. The twists we employ are based on the Lie algebra $$\Xi _t$$ Ξ t of vector fields that are tangent to all the submanifolds that are level sets of the $$f^a$$ f a (tangent infinitesimal diffeomorphisms); the twisted Cartan calculus is automatically equivariant under twisted $$\Xi _t$$ Ξ t . We can consistently project a connection from the twisted $${\mathbb {R}}^n$$ R n to the twisted M if the twist is based on a suitable Lie subalgebra $${\mathfrak {e}}\subset \Xi _t$$ e ⊂ Ξ t . If we endow $${\mathbb {R}}^n$$ R n with a metric, then twisting and projecting to the normal and tangent vector fields commute, and we can project the Levi–Civita connection consistently to the twisted M, provided the twist is based on the Lie subalgebra $${\mathfrak {k}}\subset {\mathfrak {e}}$$ k ⊂ e of the Killing vector fields of the metric; a twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can be characterized in terms of generators and $$\star $$ ⋆ -polynomial relations. We present in some detail twisted cylinders embedded in twisted Euclidean $${\mathbb {R}}^3$$ R 3 and twisted hyperboloids embedded in twisted Minkowski $${\mathbb {R}}^3$$ R 3 [these are twisted (anti-)de Sitter spaces $$dS_2,AdS_2$$ d S 2 , A d S 2 ].

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On the Notion of Noncommutative Submanifold

Cited by 5 publications

References 30 publications

Deformation and Hochschild cohomology of coisotropic algebras

Deformation and Hochschild cohomology of coisotropic algebras

Deformation and Hochschild Cohomology of Coisotropic Algebras

Twisted submanifolds of $${\mathbb {R}}^n$$

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