On Lie superalgebroids of Poisson type

Dennise Garc\'ia-Beltr\'an; \'Oscar Guajardo

arXiv:1904.02114·math.DG·May 14, 2019

On Lie superalgebroids of Poisson type

Dennise Garc\'ia-Beltr\'an, \'Oscar Guajardo

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TL;DR

This paper investigates Lie superalgebroid structures on superdifferential forms over supermanifolds, constructed from graded Poisson structures, advancing the understanding of their algebraic and geometric properties.

Contribution

It introduces a novel construction of Lie superalgebroids from graded Poisson structures on supermanifolds' differential forms.

Findings

01

Established a correspondence between graded Poisson structures and Lie superalgebroids.

02

Provided explicit examples illustrating the construction.

03

Enhanced the theoretical framework connecting supergeometry and algebraic structures.

Abstract

The purpose of this work is to study Lie superalgebroid structures on the space of superdifferential $1$ -forms over the supermanifolds whose superfunctions are the differential forms on its underlying manifold. These superalgbroids are constructed from graded Poisson structures defined on the latter superalgebra.

Equations88

d : A a \to Ω^{1} (A) \mapsto a \otimes 1 - 1 \otimes a

d : A a \to Ω^{1} (A) \mapsto a \otimes 1 - 1 \otimes a

⟨ D, d a ⟩ = D (a)

⟨ D, d a ⟩ = D (a)

a \otimes (α \otimes β) = a α \otimes β and (α \otimes β) \otimes a = α \otimes β a

a \otimes (α \otimes β) = a α \otimes β and (α \otimes β) \otimes a = α \otimes β a

[[α, β]] = ρ (α) ┘ d β - (- 1)^{(⌊ α ⌉ + k) (⌊ β ⌉ + k)} ρ (β) ┘ d α + d (β (ρ (α)))

[[α, β]] = ρ (α) ┘ d β - (- 1)^{(⌊ α ⌉ + k) (⌊ β ⌉ + k)} ρ (β) ┘ d α + d (β (ρ (α)))

[[α, β]] = ρ (α) ┘ d β - (- 1)^{⌊ α ⌉ ⌊ β ⌉} ρ (β) ┘ d α + d (β (ρ (α)))

[[α, β]] = ρ (α) ┘ d β - (- 1)^{⌊ α ⌉ ⌊ β ⌉} ρ (β) ┘ d α + d (β (ρ (α)))

{ϕ, \cdot}

{ϕ, \cdot}

{μ, ν}

{μ, ν \land ξ}

[[α, β]] = f_{1} {h_{1}, f_{2}} d h_{2} - (- 1)^{⌊ α ⌉ ⌊ β ⌉} f_{2} {h_{2}, f_{1}} d h_{1} - (- 1)^{⌊ α ⌉ ⌊ β ⌉ + ⌊ h_{2} ⌉ ⌊ f_{1} ⌉} f_{2} f_{1} d {h_{2}, h_{1}}

[[α, β]] = f_{1} {h_{1}, f_{2}} d h_{2} - (- 1)^{⌊ α ⌉ ⌊ β ⌉} f_{2} {h_{2}, f_{1}} d h_{1} - (- 1)^{⌊ α ⌉ ⌊ β ⌉ + ⌊ h_{2} ⌉ ⌊ f_{1} ⌉} f_{2} f_{1} d {h_{2}, h_{1}}

{f, h} = (d h) (ρ (d f)) = (ρ (d f)) (h),

{f, h} = (d h) (ρ (d f)) = (ρ (d f)) (h),

{f, h h^{'}} = (ρ (d f)) (h h^{'}) = ρ (d f) (h) h^{'} + (- 1)^{⌊ f ⌉ ⌊ h ⌉} h ρ (d f) (h^{'}) = {f, h} h^{'} + (- 1)^{⌊ f ⌉ ⌊ h ⌉} h {f, h^{'}},

{f, h h^{'}} = (ρ (d f)) (h h^{'}) = ρ (d f) (h) h^{'} + (- 1)^{⌊ f ⌉ ⌊ h ⌉} h ρ (d f) (h^{'}) = {f, h} h^{'} + (- 1)^{⌊ f ⌉ ⌊ h ⌉} h {f, h^{'}},

α (ρ (β)) = - (- 1)^{⌊ α ⌉ ⌊ β ⌉} β (ρ (α))

α (ρ (β)) = - (- 1)^{⌊ α ⌉ ⌊ β ⌉} β (ρ (α))

{f, {g, h}} = {{f, g}, h} + (- 1)^{⌊ f ⌉ ⌊ g ⌉} {g, {f, h}}

{f, {g, h}} = {{f, g}, h} + (- 1)^{⌊ f ⌉ ⌊ g ⌉} {g, {f, h}}

[ρ (d f), ρ (d g)] (h) = ρ (d f) (ρ (d g) (h)) - (- 1)^{⌊ f ⌉ ⌊ g ⌉} ρ (d g) (ρ (d f) (h)) = {f, {g, h}} - (- 1)^{⌊ f ⌉ ⌊ g ⌉} {g, {f, h}} = {{f, g}, h} = ρ (d {f, g}) (h) .

[ρ (d f), ρ (d g)] (h) = ρ (d f) (ρ (d g) (h)) - (- 1)^{⌊ f ⌉ ⌊ g ⌉} ρ (d g) (ρ (d f) (h)) = {f, {g, h}} - (- 1)^{⌊ f ⌉ ⌊ g ⌉} {g, {f, h}} = {{f, g}, h} = ρ (d {f, g}) (h) .

{{f, g}, h} = ρ (d {f, g}) (h) = [ρ (d f), ρ (d g)] (h) = {f, {g, h}} - (- 1)^{⌊ f ⌉ ⌊ g ⌉} {g, {f, h}}

{{f, g}, h} = ρ (d {f, g}) (h) = [ρ (d f), ρ (d g)] (h) = {f, {g, h}} - (- 1)^{⌊ f ⌉ ⌊ g ⌉} {g, {f, h}}

π^{♯} (α) ┘ d β - (- 1)^{⌊ α ⌉ ⌊ β ⌉} π^{♯} (β) ┘ d α + d (β (ρ (α)))

π^{♯} (α) ┘ d β - (- 1)^{⌊ α ⌉ ⌊ β ⌉} π^{♯} (β) ┘ d α + d (β (ρ (α)))

[[α, β]] = ρ (α) ┘ d β - (- 1)^{⌊ α ⌉ ⌊ β ⌉} ρ (β) ┘ d α + d (β (ρ (α))) .

[[α, β]] = ρ (α) ┘ d β - (- 1)^{⌊ α ⌉ ⌊ β ⌉} ρ (β) ┘ d α + d (β (ρ (α))) .

[[α, β]] = [[f_{1} d g_{1}, f_{2} d g_{2}]] = ρ (f_{1} d g_{1}) (f_{2}) d g_{2} - (- 1)^{⌊ α ⌉ ⌊ β ⌉} f_{2} ρ (d g_{2}) (f_{1}) d g_{1} - (- 1)^{⌊ α ⌉ ⌊ β ⌉ + ⌊ g_{2} ⌉ ⌊ f_{1} ⌉} f_{2} f_{1} [d g_{2}, d g_{1}]

[[α, β]] = [[f_{1} d g_{1}, f_{2} d g_{2}]] = ρ (f_{1} d g_{1}) (f_{2}) d g_{2} - (- 1)^{⌊ α ⌉ ⌊ β ⌉} f_{2} ρ (d g_{2}) (f_{1}) d g_{1} - (- 1)^{⌊ α ⌉ ⌊ β ⌉ + ⌊ g_{2} ⌉ ⌊ f_{1} ⌉} f_{2} f_{1} [d g_{2}, d g_{1}]

= f_{1} {g_{1}, f_{2}} d g_{1} - (- 1)^{⌊ α ⌉ ⌊ β ⌉} f_{2} {g_{2}, f_{1}} d g_{1} + (- 1)^{⌊ f_{1} ⌉ ⌊ f_{2} ⌉ + ⌊ g_{1} ⌉ ⌊ f_{2} ⌉} f_{2} f_{1} d {g_{1}, g_{2}} = f_{1} {g_{1}, f_{2}} d g_{1} + (- 1)^{1 + ⌊ f_{2} ⌉ ⌊ g_{2} ⌉} f_{1} {g_{1}, g_{2}} d f_{2} - (- 1)^{⌊ α ⌉ ⌊ β ⌉} f_{2} {g_{2}, f_{1}} d g_{1}

= f_{1} {g_{1}, f_{2}} d g_{1} - (- 1)^{⌊ α ⌉ ⌊ β ⌉} f_{2} {g_{2}, f_{1}} d g_{1} + (- 1)^{⌊ f_{1} ⌉ ⌊ f_{2} ⌉ + ⌊ g_{1} ⌉ ⌊ f_{2} ⌉} f_{2} f_{1} d {g_{1}, g_{2}} = f_{1} {g_{1}, f_{2}} d g_{1} + (- 1)^{1 + ⌊ f_{2} ⌉ ⌊ g_{2} ⌉} f_{1} {g_{1}, g_{2}} d f_{2} - (- 1)^{⌊ α ⌉ ⌊ β ⌉} f_{2} {g_{2}, f_{1}} d g_{1}

+ (- 1)^{1 + ⌊ g_{1} ⌉ ⌊ f_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ f_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ g_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ g_{1} ⌉} f_{2} {g_{1}, g_{2}} d f_{1} + (- 1)^{⌊ f_{1} ⌉ ⌊ f_{2} ⌉ + ⌊ g_{1} ⌉ ⌊ f_{2} ⌉} f_{2} f_{1} d {g_{1}, g_{2}} + (- 1)^{⌊ f_{2} ⌉ ⌊ g_{2} ⌉} f_{1} {g_{1}, g_{2}} d f_{2} + (- 1)^{⌊ g_{1} ⌉ ⌊ f_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ f_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ g_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ g_{1} ⌉} f_{2} {g_{1}, g_{2}} d f_{1} = ρ (f_{1} d g_{1}) ┘ d (f_{2} d g_{2}) - (- 1)^{⌊ f_{1} d g_{1} ⌉ ⌊ f_{2} d g_{2} ⌉} ρ (f_{2} d g_{2}) ┘ d (f_{1} d g_{1}) + d (f_{2} d g_{2} (ρ (f_{1} d g_{1}))) = ρ (α) ┘ d β - (- 1)^{⌊ α ⌉ ⌊ β ⌉} ρ (β) ┘ d α + d (β (ρ (α))) \qed

+ (- 1)^{1 + ⌊ g_{1} ⌉ ⌊ f_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ f_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ g_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ g_{1} ⌉} f_{2} {g_{1}, g_{2}} d f_{1} + (- 1)^{⌊ f_{1} ⌉ ⌊ f_{2} ⌉ + ⌊ g_{1} ⌉ ⌊ f_{2} ⌉} f_{2} f_{1} d {g_{1}, g_{2}} + (- 1)^{⌊ f_{2} ⌉ ⌊ g_{2} ⌉} f_{1} {g_{1}, g_{2}} d f_{2} + (- 1)^{⌊ g_{1} ⌉ ⌊ f_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ f_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ g_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ g_{1} ⌉} f_{2} {g_{1}, g_{2}} d f_{1} = ρ (f_{1} d g_{1}) ┘ d (f_{2} d g_{2}) - (- 1)^{⌊ f_{1} d g_{1} ⌉ ⌊ f_{2} d g_{2} ⌉} ρ (f_{2} d g_{2}) ┘ d (f_{1} d g_{1}) + d (f_{2} d g_{2} (ρ (f_{1} d g_{1}))) = ρ (α) ┘ d β - (- 1)^{⌊ α ⌉ ⌊ β ⌉} ρ (β) ┘ d α + d (β (ρ (α))) \qed

[[α, β]] = ρ (α) ┘ d β - (- 1)^{⌊ α ⌉ ⌊ β ⌉} ρ (β) ┘ d α + d (β (ρ (α))),

[[α, β]] = ρ (α) ┘ d β - (- 1)^{⌊ α ⌉ ⌊ β ⌉} ρ (β) ┘ d α + d (β (ρ (α))),

d [[α, β]] = d (ρ (α) ┘ d β - (- 1)^{⌊ α ⌉ ⌊ β ⌉} ρ (β) ┘ d α + d (β (ρ (α)))) = d^{2} (β (ρ (α))) = 0,

d [[α, β]] = d (ρ (α) ┘ d β - (- 1)^{⌊ α ⌉ ⌊ β ⌉} ρ (β) ┘ d α + d (β (ρ (α)))) = d^{2} (β (ρ (α))) = 0,

C (α, β) = [[α, β]] - d (β (ρ (α)))

C (α, β) = [[α, β]] - d (β (ρ (α)))

d C (α, β) = d ([[α, β]]) - d^{2} (β (ρ (α))) = 0,

d C (α, β) = d ([[α, β]]) - d^{2} (β (ρ (α))) = 0,

C (f_{1} d g_{1}, d g_{2}) = [[f_{1} d g_{1}, d g_{2}]] - d (d g_{2} (ρ (f_{1} d g_{1}))) = - (- 1)^{⌊ f_{1} ⌉ ⌊ g_{2} ⌉ + ⌊ g_{1} ⌉ ⌊ g_{2} ⌉} ρ (d g_{2}) (f_{1}) d g_{1} - (- 1)^{⌊ g_{1} ⌉ ⌊ g_{2} ⌉} f_{1} [d g_{2}, d g_{1}] - d (f_{1} ρ (d g_{1}) (g_{2})) = - (- 1)^{⌊ f_{1} ⌉ ⌊ g_{2} ⌉ + ⌊ g_{1} ⌉ ⌊ g_{2} ⌉} ρ (d g_{2}) (f_{1}) d g_{1} - (- 1)^{⌊ g_{1} ⌉ ⌊ g_{2} ⌉} f_{1} [d g_{2}, d g_{1}] - (- 1)^{⌊ f_{1} ⌉ ⌊ g_{1} ⌉ + ⌊ f_{1} ⌉ ⌊ g_{2} ⌉} ρ (d g_{1}) (g_{2}) d f_{1} - f_{1} d (ρ (d g_{1}) (g_{2})) = f_{1} C (d g_{1}, d g_{2}) - (- 1)^{⌊ g_{1} ⌉ ⌊ g_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ g_{2} ⌉} ρ (d g_{2}) ┘ d (f_{1} d g_{1})

C (f_{1} d g_{1}, d g_{2}) = [[f_{1} d g_{1}, d g_{2}]] - d (d g_{2} (ρ (f_{1} d g_{1}))) = - (- 1)^{⌊ f_{1} ⌉ ⌊ g_{2} ⌉ + ⌊ g_{1} ⌉ ⌊ g_{2} ⌉} ρ (d g_{2}) (f_{1}) d g_{1} - (- 1)^{⌊ g_{1} ⌉ ⌊ g_{2} ⌉} f_{1} [d g_{2}, d g_{1}] - d (f_{1} ρ (d g_{1}) (g_{2})) = - (- 1)^{⌊ f_{1} ⌉ ⌊ g_{2} ⌉ + ⌊ g_{1} ⌉ ⌊ g_{2} ⌉} ρ (d g_{2}) (f_{1}) d g_{1} - (- 1)^{⌊ g_{1} ⌉ ⌊ g_{2} ⌉} f_{1} [d g_{2}, d g_{1}] - (- 1)^{⌊ f_{1} ⌉ ⌊ g_{1} ⌉ + ⌊ f_{1} ⌉ ⌊ g_{2} ⌉} ρ (d g_{1}) (g_{2}) d f_{1} - f_{1} d (ρ (d g_{1}) (g_{2})) = f_{1} C (d g_{1}, d g_{2}) - (- 1)^{⌊ g_{1} ⌉ ⌊ g_{2} ⌉ + ⌊ f_{1} ⌉ ⌊ g_{2} ⌉} ρ (d g_{2}) ┘ d (f_{1} d g_{1})

C (g_{1} d g_{1}, d g_{2}) = g_{1} C (d g_{1}, d g_{2}) - (- 1)^{⌊ g_{1} ⌉ ⌊ g_{2} ⌉ + ⌊ g_{1} ⌉ ⌊ g_{2} ⌉} ρ (d g_{2}) ┘ d (g_{1} d g_{1}) = g_{1} C (d g_{1}, d g_{2})

C (g_{1} d g_{1}, d g_{2}) = g_{1} C (d g_{1}, d g_{2}) - (- 1)^{⌊ g_{1} ⌉ ⌊ g_{2} ⌉ + ⌊ g_{1} ⌉ ⌊ g_{2} ⌉} ρ (d g_{2}) ┘ d (g_{1} d g_{1}) = g_{1} C (d g_{1}, d g_{2})

0 = d C (g_{1} d g_{1}, d g_{2}) = d (g_{1} C (d g_{1}, d g_{2})) = d g_{1} \land C (d g_{1}, d g_{2}) + g_{1} d C (d g_{1}, d g_{2}) = d g_{1} \land C (d g_{1}, d g_{2}) .

0 = d C (g_{1} d g_{1}, d g_{2}) = d (g_{1} C (d g_{1}, d g_{2})) = d g_{1} \land C (d g_{1}, d g_{2}) + g_{1} d C (d g_{1}, d g_{2}) = d g_{1} \land C (d g_{1}, d g_{2}) .

0 = d g_{2} \land C (d g_{2}, d g_{1}) = - (- 1)^{⌊ g_{1} ⌉ ⌊ g_{2} ⌉} d g_{2} \land C (d g_{1}, d g_{2}) .

0 = d g_{2} \land C (d g_{2}, d g_{1}) = - (- 1)^{⌊ g_{1} ⌉ ⌊ g_{2} ⌉} d g_{2} \land C (d g_{1}, d g_{2}) .

[[d g_{1}, d g_{2}]] = d (d g_{2} (ρ (d g_{1}))) = d {g_{1}, g_{2}}

[[d g_{1}, d g_{2}]] = d (d g_{2} (ρ (d g_{1}))) = d {g_{1}, g_{2}}

Ω^{r} (A) := (Ω^{1} (A))^{r} and Ω (A) := k \geq 0 ⨁ Ω^{k} (A);

Ω^{r} (A) := (Ω^{1} (A))^{r} and Ω (A) := k \geq 0 ⨁ Ω^{k} (A);

[f, g] [f, d g] [d f, d g] = 0 = [d f, g] = {f, g} = d {f, g}

[f, g] [f, d g] [d f, d g] = 0 = [d f, g] = {f, g} = d {f, g}

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TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Advanced Algebra and Geometry

Full text

On Lie superalgebroids of Poisson type

Dennise García-Beltrán

Óscar Guajardo

Abstract

The purpose of this work is to study Lie superalgebroid structures on the space of superdifferential $1$ -forms over a special kind of supermanifolds —which we call cotangent supermanifolds— whose superfunctions are the differential forms on its underlying manifold. These superalgbroids are constructed from graded Poisson structures defined on the latter superalgebra.

MSC2010: Primary: 53D17. Secondary: 17B60, 17B66.

Keywords: Superalgebroids, Poisson supergeometry, Gerstenhaber superalgebras.

Introduction

Poisson manifolds and Lie algebroids are two well-studied structures in differential geometry, both for their applications in other areas of mathematics, as well as for their usefulness in describing physical phenomena; it is also well-known the interplay between these two structures (see for instance [4], [8], and references therein).

On the other hand, given a Lie algebroid there is associated to it another algebraic structure, a Gerstenhaber algebra, whose prototype is the $\mathbb{Z}$ -graded algebra of sections of multivectors on a manifold $M$ , endowed with both the wedge product and the Schouten-Nijenhuis bracket $[\cdot,\cdot]_{\operatorname{SN}}$ ; this bracket is the only extension to this space that restricts to the usual Lie bracket of vector fields, endows the former with the structure of a Lie algebra (actually, of a $\mathbb{Z}$ -graded Lie algebra), and which is also compatible with the wedge product, in the sense that maps of the form $[P,\cdot]_{\operatorname{SN}}$ are derivations of the wedge product, for any section $P$ of $\Lambda TM$ . These relations have also been well-studied—cf. e.g. [7], [13]. Gerstenhaber algebras have also been used in theoretical physics, specifically in the study of BRST quantisation (see for example [6]). In a related direction, graded Poisson structures are also of interest, not only as a generalisation of (ungraded) Poisson structures to graded and supermanifolds, but also as algebraic structures relevant for the aforementioned studies—cf. [1], [2], for instance.

In this paper we consider Poisson structures on the graded-commutative algebra of differential forms $\mathcal{A}=\Omega(M)$ of a smooth manifold $M$ . We endow the pair $(M,\mathcal{A})$ with the structure of a smooth supermanifold and consider superalgebroids defined on the space $\Omega^{1}(\mathcal{A})$ of graded (or super-) differential $1$ -forms on this supermanifold. More specifically, we consider superalgebroid structures defined using the given Poisson bracket on $\mathcal{A}$ .

The structure of the paper is as follows: in Section 1 we define the concepts relevant to this work; to wit: supermanifolds, superbundles and superalgebroids. In Section 2 we introduce graded Poisson structures on a superalgebra $\mathcal{A}$ and study some of their properties; in particular we construct Poisson-type superalgebroids on the space $\Omega^{1}(\mathcal{A})$ , which is defined as a certain subspace of $\mathcal{A}\otimes\mathcal{A}$ ; we also characterise these superalgebroids. In Section 3 we prove the equivalence between a Lie superalgebroid structure on $\Omega^{1}(\mathcal{A})$ and a Gerstenhaber superalgebra (suitably defined in this section) on graded forms $\Omega(\mathcal{A})$ ; this result is well-known in the classical case (see, for instance, [9]), but it seems to be lacking in the supergeometric setting, which is the reason why we write down a proof. This last section also establishes the equivalence between differential Gerstenhaber superalgebras and certain Poisson superstructures.

1 Preliminaries

In this work, the category of (real) supermanifolds will be understood in the sense of [5]; to wit, a supermanifold of dimension $(m|n)$ is a pair $\left({M}\middle|\mathcal{R}{M}\right)$ where $M$ is a smooth manifold of dimension $m$ and $\mathcal{R}M$ is a superalgebra bundle over $M$ such that each fibre $\mathcal{R}_{p}M$ (for any $p$ in $M$ ) is isomorphic to the exterior algebra $\Lambda{S}^{*}$ for some (real) vector space $S$ of dimension $n$ ; the smooth superfunctions are the sections of the bundle $\mathcal{R}M$ , i.e., they are the elements of the supercommutative unital algebra $\Gamma\left({\mathcal{R}M}\right)$ .

By a supervector bundle over $\left({M}\middle|\mathcal{R}{M}\right)$ we mean simply a vector bundle $E$ over $M$ such that

$E=E_{0}\oplus{E_{1}}$ , and 2. 2.

each fibre $E_{p}$ carries the structure of a (left) $\mathcal{R}_{p}M$ -supermodule.

We refer the reader to [12] for a comprehensive study of supervector bundles. Do note that the objects studied there are quite more general than what we have called a supervector bundle; for our purposes, we need only consider the supermodule structure on the space of sections of $E$ —and not a supermanifold structure on its total space—, so our definition is general enough, but the reader should bear in mind the scope of the usual definition of a supervector bundle. The space of sections over $M$ of $E$ is then a $\Gamma\left({\mathcal{R}M}\right)$ -supermodule. We will consider a special kind of these:

1.1 Definition.

Let $\mathcal{A}$ be a supercommutative algebra. A standard (left) supermodule is a bilateral supermodule $N$ such that $an=(-1)^{\lfloor{a}\rceil\lfloor{n}\rceil}na$ for any homogeneous $a\in\mathcal{A}$ and $n\in{N}$ .

There is of course an analogous definition of a right standard supermodule. Throughout this work, the symbol $\lfloor{\cdot}\rceil$ denotes the parity of a homogeneous element of a superalgebra. It will be tacitly assumed that all supervector bundles arise from standard supermodules. In particular we adopt the convention that the module of supervector fields is a left standard module (over the superalgebra $\mathcal{A}=\Gamma\left({\mathcal{R}M}\right)$ ), whereas the space of superforms $\Omega\left({M}\middle|\mathcal{R}{M}\right)$ is a bigraded superalgebra (with bigrade over $\mathbb{Z}_{2}\times\mathbb{Z}$ ) and a right standard module; we will refer to the $\mathbb{Z}_{2}$ -grading as the parity of a bihomogeneous superform, and to its $\mathbb{Z}$ -grading as the degree.

We are interested mostly on the supermanifolds whose smooth superfunctions are the differential forms on ordinary manifolds, that is where $\mathcal{R}M=\Lambda{T}^{*}M$ .

1.2 Definition.

The supermanifold $\left({M}\middle|{\Lambda{T^{*}}M}\right)$ is the cotangent supermanifold on $M$ . We will denote it by $\left({M}\middle|{TM}\right)$ (cf. [5, Section 1.4] for this notation and its meaning).

In what follows, in order to save space, we shall denote a supermanifold by $\mathcal{M}=\left({M}\middle|\mathcal{R}{M}\right)$ , its tangent superbundle by $T\mathcal{M}$ , and a supervector bundle simply by $E$ .

1.3 Definition.

Let $\mathcal{M}$ be a supermanifold and $E$ a supervector bundle over $\mathcal{M}$ . A superalgebroid structure on $E$ is an even bundle map $\rho:\Gamma(E)\rightarrow\Gamma(T\mathcal{M})\simeq\operatorname{Der}(\Gamma(\mathcal{R}M))$ (the anchor) and a bilinear operation $\llbracket\cdot,\cdot\rrbracket:\Gamma(E)\times\Gamma(E)\rightarrow\Gamma(E)$ (the bracket) such that

$\llbracket\cdot,\cdot\rrbracket$ gives $\Gamma(E)$ the structure of a Lie superalgebra, and 2. 2.

$\llbracket X,fY\rrbracket=\rho(X)(f)Y+(-1)^{\lfloor{X}\rceil\lfloor{f}\rceil}f\llbracket X,Y\rrbracket$ for any $X,Y\in\Gamma(E)$ , $f\in\Gamma\left({\mathcal{R}M}\right)$ (graded Leibniz identity).

1.4 Examples.

The supertangent bundle $T\mathcal{M}$ is a Lie superalgebroid with $\rho=\operatorname{Id}$ and whose brackets is the usual supercommutator of supervector fields. 2. 2.

If $L\colon{T}\mathcal{M}\to{T^{*}}\mathcal{M}$ is a bundle isomorphism, then $T^{*}\mathcal{M}$ becomes a Lie superalgebroid by defining $\rho={L}^{\operatorname{-}\!1}$ and $\llbracket\alpha,\beta\rrbracket=L([{L}^{\operatorname{-}\!1}(\alpha),{L}^{\operatorname{-}\!1}(\beta)])$ , where $[\cdot,\cdot]$ is the same bracket of the previous example.

2 Poisson-type superalgebroids

We now turn our attention to special sorts of $\mathbb{R}$ -algebras $\mathcal{A}$ .

2.1 Definition.

A graded Poisson algebra of degree $k$ $(\mathcal{A},\{\cdot,\cdot\})$ is a $\mathbb{Z}$ -graded associative and graded-commutative algebra $(\mathcal{A}^{\bullet},\cdot)$ together with an $\mathbb{R}$ -bilinear operation, the Poisson bracket, $\{\cdot,\cdot\}:\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$ such that

$\{\mathcal{A}^{a},\mathcal{A}^{b}\}\subseteq\mathcal{A}^{a+b+k}$ (the bracket has degree $k$ ); 2. 2.

$\{f,g\}=-(-1)^{(\left|{f}\right|+k)(\left|{g}\right|+k)}\{g,f\}$ (graded skew-symmetry); 3. 3.

$\{f,\{g,h\}\}=\{\{f,g\},h\}+(-1)^{(\left|{f}\right|+k)(\left|{g}\right|+k)}\{g,\{f,h\}\}$ (graded Jacobi identity), and 4. 4.

$\{f,gh\}=\{f,g\}h+(-1)^{(\left|{f}\right|+k)(\left|{g}\right|)}g\{f,h\}$ (graded Leibniz identity),

for all $f,g,h\in\mathcal{A}$ . Here $\left|{\cdot}\right|$ denotes the degree of an element of $\mathcal{A}$ .

Note that a graded-commutative algebra furnishes a supercommutative algebra by defining the parity of a $\mathbb{Z}$ -homogeneous element as the parity of its degree; in this case, parity and degree are indistinguishable.

For any graded algebra $\mathcal{A}$ , the space $\Omega^{1}(\mathcal{A})$ is defined as the kernel of the multiplication map $\mathbf{m}\colon\mathcal{A}\otimes\mathcal{A}\to\mathcal{A}$ ; let us define

[TABLE]

and observe that $\Omega^{1}(\mathcal{A})$ is generated by the image of this map; also, note that $\mathbf{d}$ is a derivation with values in the latter module, which is the dual module to the one of derivations $\operatorname{Der}\mathcal{A}$ with the pairing

[TABLE]

(all this is in accordance with [4, p.3]).

2.2 Remark.

The module $\mathcal{A}\otimes\mathcal{A}$ is technically a bimodule with left and right multiplications

[TABLE]

respectively. Since the definition of $\Omega^{1}(\mathcal{A})$ renders it as a submodule of the one above, we can consider it to be a bilateral module on its own, but a right module with respect to the pairing (2.2). However, since our definition of a superalgebroid implicitly assumes such an object to be a left module, we shall treat $\Omega^{1}(\mathcal{A})$ as a left module whenever necessary, considering it as a submodule of $\mathcal{A}\otimes\mathcal{A}$ ; this is done in order to avoid clumsy computations below.

Let us note, however, that since we shall be working with $\mathcal{A}=\Gamma\left({\mathcal{R}M}\right)$ , the space $\Omega^{1}(\mathcal{A})$ is in fact the space of $1$ -superforms on $\mathcal{M}$ and the map $\mathbf{d}$ is simply the exterior superderivative, which is even and has degree $1$ ; its definition agrees with the above.

2.3 Examples.

Any Poisson algebra in the classical sense is a graded Poisson algebra whose bracket has degree [math]. 2. 2.

If $(V,g)$ is a pseudoeuclidean vector space then defining on the exterior algebra $\Lambda V$ the bracket $\{\cdot,\cdot\}$ by the relations

(a)

$\{1,\cdot\}=0$ ; 2. (b)

$\{x,y\}=g(x,y)$ for $x$ and $y$ in $V$ , and 3. (c)

$\{x,y\wedge z\}=\{x,y\}z-\{x,z\}y$ for all $x,y$ and $z$ in $V$ ,

we obtain a graded Poisson algebra of degree $-2$ . 3. 3.

If $(M,(\cdot,\cdot))$ is a Poisson manifold, then by defining on $\mathcal{A}=\Omega(M)$ the bracket $\{\cdot,\cdot\}$ by the relations

(a)

$\{a,b\}=0$ for $a,b\in\Omega^{0}(M)$ ; 2. (b)

$\{{d}a,b\}=\{a,{d}b\}=(a,b)$ for $a,b\in\Omega^{0}(M)$ ; 3. (c)

$\{{d}a,{d}b\}={d}(a,b)$ , and 4. (d)

$\{\alpha,\beta\wedge\gamma\}=\{\alpha,\beta\}\wedge\gamma+(-1)^{(\left|{\alpha}\right|-1)\left|{\beta}\right|}\beta\wedge\{\alpha,\gamma\}$ for all $\alpha,\beta,\gamma$ in $\Omega^{1}(M)$ ,

we obtain a graded Poisson algebra on $\mathcal{A}$ of degree $-1$ . This bracket was completely characterised in [1].

If $(\mathcal{A},\{\cdot,\cdot\})$ is graded Poisson algebra of degree $k$ , we can define an $\mathcal{A}$ -module morphism $\Omega^{1}(\mathcal{A})\rightarrow\operatorname{Der}(\mathcal{A})$ by $\rho(\mathbf{d}{f})=\{f,\cdot\}$ for $f\in\mathcal{A}$ (where $\mathbf{d}$ is defined by (2.1)), and extending by $\mathcal{A}$ -linearity. This map furnishes a derivation of degree $k$ on $\mathcal{A}$ .

We can also define a bracket $\llbracket\cdot,\cdot\rrbracket:\Omega^{1}(\mathcal{A})\times\Omega^{1}(\mathcal{A})\rightarrow\Omega^{1}(\mathcal{A})$ by

[TABLE]

where $\lrcorner$ denotes interior product and which satisfies

$\llbracket\alpha,\beta\rrbracket=-(-1)^{(\lfloor{\alpha}\rceil+k)(\lfloor{\beta}\rceil+k)}\llbracket\beta,\alpha\rrbracket$ 2. 2.

$\llbracket\alpha,\llbracket\beta,\gamma\rrbracket\rrbracket=\llbracket\llbracket\alpha,\beta\rrbracket,\gamma\rrbracket+(-1)^{(\lfloor{\alpha}\rceil+k)(\lfloor{\beta}\rceil+k)}\llbracket\beta,\llbracket\alpha,\gamma\rrbracket\rrbracket$ 3. 3.

$\llbracket\alpha,f\beta\rrbracket=\rho(\alpha)(f)\beta+(-1)^{(\lfloor{\alpha}\rceil+k)\lfloor{f}\rceil}f\llbracket\alpha,\beta\rrbracket$

for all $\alpha,\beta\in\Omega^{1}(\mathcal{A})$ and $f\in\mathcal{A}$ , which can be verified with direct computations.

2.4 Definition.

When $(\mathcal{A},\{\cdot,\cdot\})$ is a graded Poisson algebra of even degree, then $(\mathcal{A},\{\cdot,\cdot\})$ is called an Poisson superalgebra.

Note that if $k$ is even then $\lfloor{\lfloor{a}\rceil+k}\rceil=\lfloor{a}\rceil$ for all $\alpha$ in $\Omega^{1}(\mathcal{A})$ , so it suffices to consider the case $k=0$ .

2.5 Theorem.

If $(\mathcal{A},\{\cdot,\cdot\})$ is a Poisson superalgebra, then $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ is a Lie superalgebroid with anchor map $\rho$ defined by $\rho(\mathbf{d}{f})=\{f,\cdot\}$ for $f\in\mathcal{A}$ and bracket defined by

[TABLE]

for every $\alpha,\beta\in\Omega^{1}(\mathcal{A})$ .

Proof.

All the required properties can be checked directly from the definition of the bracket (2.4). To verify the Jacobi identity, it is easier to consider a system $\left\{{\mathbf{d}{f_{i}}}\right\}_{i\in I}$ of generators of $\Omega^{1}(\mathcal{A})$ , and to express the elements $\alpha\in\Omega^{1}(\mathcal{A})$ as $\alpha=\sum_{i}{g_{i}}(\mathbf{d}{f_{i}})$ for $g_{i}\in\mathcal{A}$ . ∎

2.6 Example.

If $(M,g)$ is a Riemannian manifold, we can define

[TABLE]

for $\phi\in\mathcal{C}^{\infty}(M)$ , $\mu,\nu,\xi\in\Omega^{1}(M)$ ; here, $\alpha^{\sharp}$ is the vector field associated to a $1$ -form $\alpha$ via the musical isomorphism induced by $g$ . It is easy to see that $\{\cdot,\cdot\}$ defines a structure of graded Poisson algebra of degree $-2$ on $\mathcal{A}=\Omega(M)$ (in fact, 2 of 2.3 is a particular case of this example, when $M=\{*\}$ and $(M,g)$ is a pseudoriemannian manifold). Then, by Theorem 2.5, $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ is a Lie superalgebroid with anchor map $\rho$ defined by $\rho(\mathbf{d}{f})=\{f,\cdot\}$ for $f\in\mathcal{A}$ and bracket defined by (2.4) for $\alpha,\beta\in\Omega^{1}(\mathcal{A})$ .

More explicitly, if $\alpha=f_{1}\mathbf{d}{h_{1}}$ and $\beta=f_{2}\mathbf{d}{h_{2}}$ with $f_{1},f_{2},h_{1},h_{2}\in\mathcal{A}$ , then

[TABLE]

Theorem 2.5 tells us that given a Poisson superalgebra we have a Lie superalgebroid naturally associated to it. We are interested now in the reciprocal question: When does a Lie superalgebroid $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ determine a Poisson superalgebra on $\mathcal{A}$ ?

In order to answer this question, let us consider a Lie superalgebroid $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ ; for any $f,h\in\mathcal{A}$ we can define

[TABLE]

which is clearly $\mathbb{R}$ -bilinear and satisfies the Leibniz identity

[TABLE]

for all $f,h,h^{\prime}\in\mathcal{A}$ homogeneous.

Thus, in order to get a Poisson superalgebra on $\mathcal{A}$ we only need to take care of the skew-supersymmetry and the graded Jacobi identity.

2.7 Definition.

The anchor map $\rho$ of a Lie superalgebroid $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ is said to be skew-supersymmetric if

[TABLE]

for all $\alpha,\beta\in\Omega^{1}(\mathcal{A})$ .

Observe that if the anchor map is skew-supersymmetric, then the new operation defined by (2.9) is also skew-supersymmetric, so henceforth we will assume $\rho$ to satisfy (2.10). Let us now turn our attention to the graded Jacobi identity.

2.8 Theorem.

Let $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ be a Lie superalgebroid, and define $Q\in\Lambda^{2}(\operatorname{Der}(\mathcal{A}))$ by $Q(\mathbf{d}{f},\mathbf{d}{g})=\mathbf{d}{g}(\rho(\mathbf{d}{f}))$ for $f,g\in\mathcal{A}$ . The following conditions are equivalent:

the operation defined by $\{f,g\}=\rho(\mathbf{d}{f})(g)$ satisfies the Jacobi identity, 2. 2.

$[Q,Q]_{\operatorname{SN}}=0$ , 3. 3.

$[\rho(\mathbf{d}{f}),\rho(\mathbf{d}{g})]=\rho(\mathbf{d}\{f,g\})$ , where $\{\cdot,\cdot\}$ is defined in 1.

Here, $[\cdot,\cdot]_{SN}$ denotes the generalisation of the Schouten–Nijenhuis bracket to elements of the bigraded algebra $\Lambda\operatorname{Der}\mathcal{A}$ .

Proof.

For the equivalence of 1 and 2 see [3, Equation (3.8)]).

Suppose that the bracket $\{\cdot,\cdot\}$ satisfy the graded Jacobi identity, that is,

[TABLE]

for all $f,g\in\mathcal{A}$ . With this,

[TABLE]

Now suppose that 3 is true, then

[TABLE]

∎

This result motivates the following:

2.9 Definition.

A Lie superalgebroid $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ is of Poisson type if:

the anchor map $\rho$ is skew-supersymmetric, in the sense of (2.10); and 2. 2.

both the bracket and anchor map satisfy any of the conditions of Theorem 2.8.

In other words, a Lie superalgebroid $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ is of Poisson type if it determines a Poisson superalgebra on $\mathcal{A}$ .

Another important issue for us is to determine the form of the bracket of a Lie superalgebroid of Poisson type. We know [10, Example 2.4.5] that the classical example of Poisson supermanifolds leads to brackets of the type

[TABLE]

where $\pi$ is a Poisson structure on a supermanifold $\mathcal{M}$ and $\pi^{\sharp}\colon{T\mathcal{M}}\to{T^{*}\mathcal{M}}$ is the musical morphism associated to it. The following result characterises the class of such algebroids.

2.10 Proposition.

Let $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ be a Lie superalgebroid with skew-supersymmetric anchor $\rho$ , and $\llbracket\mathbf{d}f,\mathbf{d}g\rrbracket=\mathbf{d}\{f,g\}$ ; then the bracket of the Lie superalgebroid is of the form

[TABLE]

Proof.

Since every element of $\Omega^{1}(\mathcal{A})$ is an $\mathcal{A}$ -linear combination of elements of the form $\mathbf{d}{f_{i}}$ for some $f_{i}\in\mathcal{A}$ , it is enough to prove the statement for $\alpha=f_{1}\mathbf{d}{g_{1}}$ and $\beta=f_{2}\mathbf{d}{g_{2}}$ :

[TABLE]

An equivalent statement for the hypothesis of Proposition 2.10 is:

2.11 Proposition.

If $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ is a Lie superalgebroid with skew-supersymmetric anchor $\rho$ , then the following assertions are equivalent:

$\llbracket\mathbf{d}{f},\mathbf{d}{g}\rrbracket=\mathbf{d}(\mathbf{d}{g}(\rho(\mathbf{d}{f})))=\mathbf{d}\{f,g\}$ * for all $f,g\in\mathcal{A},$ * 2. 2.

$\mathbf{d}\alpha=0=\mathbf{d}\beta$ * implies $\llbracket\alpha,\beta\rrbracket=0$ for $\alpha,\beta\in\Omega^{1}(\mathcal{A}).$ *

Proof.

First, let us assume that 1 holds. By Proposition 2.10 we know that the bracket is of the form

[TABLE]

and hence condition 2 holds,

[TABLE]

whenever $\mathbf{d}\alpha=0=\mathbf{d}\beta$ .

Now let us assume that condition 2 holds. Define

[TABLE]

for $\alpha,\beta\in\Omega^{1}(\mathcal{A})$ . Notice that $C$ is skew-supersymmetric and

[TABLE]

for any closed $\alpha$ and $\beta$ .

Note that

[TABLE]

If $f_{1}=g_{1}$ , then

[TABLE]

Now, applying the operator $\mathbf{d}$ and taking into account that $g_{1}\mathbf{d}{g_{1}}$ is closed, we get

[TABLE]

Interchanging the roles of $g_{1}$ and $g_{2}$ , we have

[TABLE]

Relations (2.13) and (2.14) imply that $\mathbf{d}{g_{1}}$ , $\mathbf{d}{g_{2}}$ and $C(\mathbf{d}{g_{1}},\mathbf{d}{g_{2}})$ are linearly dependent for arbitrary $g_{1}$ and $g_{2}$ . In particular, if $\mathbf{d}{g_{1}}$ and $\mathbf{d}g_{2}$ are linearly independent, then $C(\mathbf{d}{g_{1}},\mathbf{d}{g_{2}})=0$ , and hence

[TABLE]

which proves the Proposition. ∎

As an immediate consequence of all the work made above, we obtain:

2.12 Theorem.

Let $\mathcal{M}$ be the cotangent supermanifold on the smooth manifold $M$ , and suppose $(\Omega(M),\{\cdot,\cdot\})$ is a Poisson superalgebra. Under these conditions, the cotangent superbundle $T^{*}\mathcal{M}$ inherits a Lie superalgebroid structure whose anchor map is given by $\rho(\mathbf{d}{f})=\{f,\cdot\}$ and whose bracket is given by (2.12).

3 Further structures on superforms

Recall we defined on (2.1) the map $\mathbf{d}\colon\mathcal{A}\to\ker{\mathbf{m}}$ that associates to any $a$ a generator of the latter space, which we have denoted as $\Omega^{1}(\mathcal{A})$ ; now we set $\Omega^{0}(\mathcal{A})=\mathcal{A}$ and define

[TABLE]

so defined, the space $\Omega(\mathcal{A})$ is a bigraded-commutative algebra; if $\mathcal{A}=\Omega(M)$ then $\Omega(\mathcal{A})$ is naturally isomorphic to the superalgebra of graded forms on the supermanifold $\left({M}\middle|{TM}\right)$ —see, for instance, [11, Section 2]—.

Let now $(\mathcal{A},\{\cdot,\cdot\})$ be a Poisson superalgebra. We define a bracket $[\cdot,\cdot]$ for elements of $\Omega^{1}(\mathcal{A})$ by the following relations:

[TABLE]

for all $f,g,h\in\mathcal{A}$ , and we extend to all of $\Omega(\mathcal{A})$ by

[TABLE]

to obtain the following structure

3.1 Definition.

Let $(\mathcal{B},\wedge)$ be a $\mathbb{Z}$ -graded superalgebra; if there is a bracket $[\cdot,\cdot]\colon\mathcal{B}\otimes\mathcal{B}\to\mathcal{B}$ such that

$[\alpha,\beta]=-(-1)^{(|\alpha|-1)(|\beta|-1)+\lfloor{\alpha}\rceil\lfloor{\beta}\rceil}[\beta,\alpha]$ 2. 2.

$[\alpha,[\beta,\gamma]]=[[\alpha,\beta],\gamma]+(-1)^{(|\alpha|-1)(|\beta|-1)+\lfloor{\alpha}\rceil\lfloor{\beta}\rceil}[\beta,[\alpha,\gamma]]$ 3. 3.

$[\alpha,\beta\wedge\gamma]=[\alpha,\beta]\wedge\gamma+(-1)^{(|\alpha|-1)|\beta|+\lfloor{\alpha}\rceil\lfloor{\beta}\rceil}\beta\wedge[\alpha,\gamma]$

for all $\alpha,\beta,\gamma\in\mathcal{B}$ , then $(\mathcal{B},\wedge,[\cdot,\cdot])$ is a Gerstenhaber superalgebra.

The following is a generalisation of a well-known result (see [13] and references therein):

3.2 Theorem.

$\Omega(\mathcal{A})$ * is a Gerstenhaber superalgebra if and only if $\Omega^{1}(\mathcal{A})$ is a Lie superalgebroid.*

Proof.

Suppose that there is Lie superbracket $[\cdot,\cdot]$ of degree $-1$ that makes $\Omega(\mathcal{A})$ into a Gerstenhaber superalgebra. It is clear that $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket)$ is a Lie superalgebra with $\llbracket\alpha,\beta\rrbracket=[\alpha,\beta]$ for $\alpha$ and $\beta\in\Omega^{1}(\mathcal{A})$ . Now, for any $\alpha\in\Omega^{1}(\mathcal{A})$ and $f,g\in\mathcal{A}$ , it follows from the properties of Gerstenhaber superalgebra that

[TABLE]

Hence, $[\alpha,\cdot]$ is a derivation on $\mathcal{A}$ . If we define $\rho:\Omega^{1}(\mathcal{A})\rightarrow\operatorname{Der}(\mathcal{A})$ by $\rho(\alpha)=[\alpha,\cdot]$ , then

[TABLE]

for $\alpha,\beta\in\Omega^{1}(\mathcal{A})$ and $f\in\mathcal{A}$ . This shows that $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ is indeed a Lie superalgebroid.

Conversely, given a Lie superalgebroid $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ ,we define

[TABLE]

for $f,g\in\mathcal{A}$ and $\alpha,\beta\in\Omega^{1}(\mathcal{A})$ ; for decomposable forms $\mathbf{A}={\alpha}_{1}{\wedge}\cdots{\wedge}{\alpha}_{k}$ and $\mathbf{B}={\beta}_{1}{\wedge}\cdots{\wedge}{\beta}_{l}$ we define

[TABLE]

where we have defined: $\mathbf{A}_{i}:={\alpha}_{1}{\wedge}\cdots{\wedge}\widehat{{\alpha}_{i}}{\wedge}\cdots{\wedge}{{\alpha}_{k}}$ , $\mathbf{B}_{j}:={\beta}_{1}{\wedge}\cdots{\wedge}\widehat{{\beta}_{j}}{\wedge}\cdots{\wedge}{{\beta}_{l}}$ , $a_{i}:=\lfloor{\alpha_{i}}\rceil\sum_{p=i+1}^{k}\lfloor{\alpha_{p}}\rceil+i$ and $b_{j}:=\lfloor{\beta_{j}}\rceil\sum_{q=1}^{j-1}\lfloor{\beta_{q}}\rceil+j$ ; the notation ${\alpha}_{1}{\wedge}\cdots{\wedge}\widehat{{\alpha}_{i}}{\wedge}\cdots{\wedge}{{\alpha}_{k}}$ means the $i$ th factor is omitted. A long computation shows that $\Omega(\mathcal{A})$ with $[\cdot,\cdot]$ is indeed a Gerstenhaber superalgebra. ∎

3.3 Definition.

Let $[\cdot,\cdot]$ be a bracket on $\Omega(\mathcal{A})$ such that $(\Omega(\mathcal{A}),\wedge,[\cdot,\cdot])$ is a Gerstenhaber superalgebra. If the exterior superderivative is a derivation of degree 1 for $[\cdot,\cdot]$ , that is if

[TABLE]

for all $\alpha,\beta\in\Omega(\mathcal{A})$ , we call it a differential Gerstenhaber superalgebra.

3.4 Theorem.

A Gerstenhaber superbracket on $\Omega(\mathcal{A})$ is differential if and only if it is the bracket associated to a Poisson superstructure on $\mathcal{A}$ in the sense of (3.1).

Proof.

Let $[\cdot,\cdot]$ be a differential Gerstenhaber superbracket on $\Omega(\mathcal{A})$ . We define the map $\{\cdot,\cdot\}:\mathcal{A}\times\mathcal{A}\to\mathcal{A}$ by $(f,g)\mapsto\{f,g\}:=[f,\mathbf{d}g]$ which is a Poisson superbracket; indeed:

$\{f,g\}=-(-1)^{\lfloor{f}\rceil\lfloor{g}\rceil}\{g,f\}$ because

[TABLE] 2. 2.

the graded Jacobi identity holds:

[TABLE] 3. 3.

the graded Leibniz identity holds:

[TABLE]

So, $\{\cdot,\cdot\}$ is a Poisson superbracket on $\Omega(M)$ ; moreover $\{\cdot,\cdot\}$ induces $[\cdot,\cdot]$ by the relations (3.1).

Now, let $[\cdot,\cdot]:\Omega(\mathcal{A})\times\Omega(\mathcal{A})\rightarrow\Omega(\mathcal{A})$ be the bracket associated to a Poisson superalgebra $(\mathcal{A},\{\cdot,\cdot\})$ . It is enough show that it is differential for $\alpha=\mathbf{d}f$ and $\beta=g\mathbf{d}h$ for some $f,g,h\in\mathcal{A}$ :

[TABLE]

Let $(\Omega(\mathcal{A}),\wedge,[\cdot,\cdot],\mathbf{d})$ be a differential Gerstenhaber superalgebra; from the Theorem above, $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ is a Lie superalgebroid. Moreover, $\rho$ is skew-supersymmetric since

[TABLE]

for all $f_{1},f_{2},g_{2},g_{2}\in\mathcal{A}$ , where $\langle\cdot,\cdot\rangle$ denotes the pairing of supervectorfields and $1$ -superforms. Also, due to the fact that the bracket is differential we have

[TABLE]

That is, the hypotheses of Proposition 2.10 are satisfied, therefore the bracket of the Lie superalgebroid is of the form (2.4) and therefore the superalgebroid is of Poisson type. Now, we can prove the following

3.5 Proposition.

If $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ is a Lie algebroid with

$\rho$ * skew-supersymmetric* 2. 2.

$\llbracket\mathbf{d}f,\mathbf{d}g\rrbracket=\mathbf{d}(\rho(\mathbf{d}f)(g))$ * for all $f,g\in\mathcal{A}$ *

then $(\Omega(\mathcal{A}),\wedge,[\cdot,\cdot],\mathbf{d})$ is a differential Gerstenhaber superalgebra.

Proof.

We only need to prove that the Gerstenhaber superalgebra induced by the superalgebroid is differential. For this, first note that

[TABLE]

for all $f,g\in\mathcal{A}$ and remember that it is enough show that the bracket induced by $\llbracket\cdot,\cdot\rrbracket$ in $\Omega(\mathcal{A})$ it is differential for $\alpha=\mathbf{d}f$ and $\beta=g\mathbf{d}h$ for some $f,g,h\in\mathcal{A}$ :

[TABLE]

This establishes the proposition. ∎

Observe that if $(\mathcal{A},\{\cdot,\cdot\})$ is a Poisson superalgebra, we know from the previous section that $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ is a Lie superalgebroid of Poisson type with bracket defined by $\eqref{def:corchete}$ . All of the above implies that $\llbracket\cdot,\cdot\rrbracket$ coincides with the restriction on $1$ -superforms of the bracket defined by $\eqref{eq:KS}$ . We have now proved:

3.6 Theorem.

$(\mathcal{A},\{\cdot,\cdot\})$ * is a Poisson superalgebra if and only if $(\Omega^{1}(\mathcal{A}),\llbracket\cdot,\cdot\rrbracket,\rho)$ is a Lie superalgebroid of Poisson type with bracket defined by $\eqref{eq:formacorchete}$ .*

Acknowledgements: the authors wish to thank the Mexican National Council for Science and Technology (CONACYT) for the funding of their Postdoctoral Fellowships under programme 2018-000005-01-NACV —Programa de Estancias Posdoctorales Nacionales 2018-1— with reference numbers (CVU) 289063 and 344289, during which this paper was written.

Bibliography13

The reference list from the paper itself. Each links out to its DOI / PubMed record.

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2[2] F Cantrijn and LA Ibort. Introduction to poisson supermanifolds. Differential Geometry and its Applications , 1(2):133–152, 1991.
3[3] JA De Azcarraga, JM Izquierdo, AM Perelomov, and JC Pérez-Bueno. The ℤ 2 subscript ℤ 2 \mathbb{Z}_{2} -graded schouten–nijenhuis bracket and generalized super-poisson structures. Journal of Mathematical Physics , 38(7):3735–3749, 1997.
4[4] Dennise García-Beltrán, José A Vallejo, and Yurii Vorobjev. On lie algebroids and poisson algebras. Symmetry, Integrability and Geometry: Methods and Applications , 8, 2011. ar Xiv:1106.1512 .
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On Lie superalgebroids of Poisson type

Abstract

Introduction

1 Preliminaries

1.1 Definition**.**

1.2 Definition**.**

1.3 Definition**.**

1.4 Examples**.**

2 Poisson-type superalgebroids

2.1 Definition**.**

2.2 Remark**.**

2.3 Examples**.**

2.4 Definition**.**

2.5 Theorem**.**

Proof.

2.6 Example**.**

2.7 Definition**.**

2.8 Theorem**.**

Proof.

2.9 Definition**.**

2.10 Proposition**.**

Proof.

2.11 Proposition**.**

Proof.

2.12 Theorem**.**

3 Further structures on superforms

3.1 Definition**.**

3.2 Theorem**.**

Proof.

3.3 Definition**.**

3.4 Theorem**.**

Proof.

3.5 Proposition**.**

Proof.

3.6 Theorem**.**

1.1 Definition.

1.2 Definition.

1.3 Definition.

1.4 Examples.

2.1 Definition.

2.2 Remark.

2.3 Examples.

2.4 Definition.

2.5 Theorem.

2.6 Example.

2.7 Definition.

2.8 Theorem.

2.9 Definition.

2.10 Proposition.

2.11 Proposition.

2.12 Theorem.

3.1 Definition.

3.2 Theorem.

3.3 Definition.

3.4 Theorem.

3.5 Proposition.

3.6 Theorem.