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Poisson Lie 2-algebroids Geometry of jets and fields, Bedlewo The University of Sheffield M. Jotz Lean Courant algebroids Poisson Lie 2-algebroids and degenerate geometrisation Overview of the algebroids Courant (Degenerate) Poisson Lie

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University of Sheffield Geometry of jets and fields, Bedlewo

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Motivation: the correspondence between Courant algebroids and symplectic Lie 2-algebroids

The standard Courant algebroid structure on TM βŠ• Tβˆ—M, given a smooth manifold M, was discovered by Ted Courant in the late 80's. Later, in the late 90's, Liu, Weinstein and Xu defined general Courant algebroids and proved that the bicrossproduct of a Lie bialgebroid is a Courant algebroid. A few years later, Roytenberg, and independently Severa found that Courant algebroids were equivalent to symplectic Lie 2-algebroids, or in other words symplectic positively graded manifolds of degree 2 with a compatible homological vector field.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

The classical definition of a Courant algebroid

A Courant algebroid over a manifold M is a vector bundle π–₯ β†’ M with a fibrewise nondegenerate symmetric bilinear form βŸ¨β‹… , β‹…βŸ©, a bracket βŸ¦β‹… , β‹…βŸ§ on the smooth sections Ξ“(π–₯), and an anchor ρ: π–₯ β†’ TM, which satisfy the following conditions

1 βŸ¦π‘“1, βŸ¦π‘“2, 𝑓3⟧⟧ = βŸ¦βŸ¦π‘“1, 𝑓2⟧, 𝑓3⟧ + βŸ¦π‘“2, βŸ¦π‘“1, 𝑓3⟧⟧, 2 ρ(𝑓1)βŸ¨π‘“2, 𝑓3⟩ = βŸ¨βŸ¦π‘“1, 𝑓2⟧, 𝑓3⟩ + βŸ¨π‘“2, βŸ¦π‘“1, 𝑓3⟧⟩, 3 βŸ¦π‘“1, 𝑓2⟧ + βŸ¦π‘“2, 𝑓1⟧ = Οβˆ—πžβŸ¨π‘“1, 𝑓2⟩

for all 𝑓1, 𝑓2, 𝑓3 ∈ Ξ“(π–₯).

Back to theorem 3 / 50

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Motivation: the correspondence between Courant algebroids and symplectic Lie 2-algebroids

Usually the Courant algebroid bracket and the anchor of a Courant algebroid 𝔽 are derived from the positively graded manifold and the homological vector field. The aim of this talk is to show how the Courant algebroid bracket and the anchor of a Courant algebroid can be retrieved as a kind of "semidirect product'' of the Poisson structure and the dual of the Lie structure.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Motivation: the correspondence between Courant algebroids and symplectic Lie 2-algebroids

Usually the Courant algebroid bracket and the anchor of a Courant algebroid 𝔽 are derived from the positively graded manifold and the homological vector field. The aim of this talk is to show how the Courant algebroid bracket and the anchor of a Courant algebroid can be retrieved as a kind of "semidirect product'' of the Poisson structure and the dual of the Lie structure.

4 / 50

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Outline

1 Lie 2-algebroids 2 Dorfman 2-representations 3 Poisson [2]-manifolds 4 Poisson Lie 2-algebroids 5 (Degenerate) Courant algebroids 6 Overview of the geometrisation

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Positively graded manifolds

An N-manifold or β„•-graded manifold β„³ of degree π‘œ and dimension (π‘ž; 𝑠1, … , π‘ π‘œ) is a smooth π‘ž-dimensional manifold M endowed with a sheaf C∞(β„³) of β„•-graded commutative associative unital ℝ-algebras, whose degree 0 term is C∞(M) and which can locally be written C∞(β„³)U = C∞(U) [ΞΎ1

1, … , ξ𝑠1 1 , ΞΎ1 2, … , ξ𝑠2 2 , … , ΞΎ1 π‘œ, … , ΞΎ π‘ π‘œ π‘œ ]

with 𝑠1 + … + π‘ π‘œ graded commutative generators ΞΎπ‘˜

𝑗 of degree 𝑗 for

𝑗 ∈ {1, … , π‘œ} and π‘˜ ∈ {1, … , 𝑠𝑗}.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Positively graded manifolds are noncanonically split

For any β„•-graded manifold β„³ of degree π‘œ and dimension (π‘ž; 𝑠1, … , π‘ π‘œ), there exist smooth vector bundles Eβˆ’1, Eβˆ’2, … , Eβˆ’π‘œ

  • f ranks 𝑠1, … , π‘ π‘œ over M such that β„³ is isomorphic (in a

noncanonical manner) to the split [π‘œ]-manifold Eβˆ’1[βˆ’1] βŠ• … βŠ• Eβˆ’π‘œ[βˆ’π‘œ], which has local basis sections of Eβˆ’π‘—

βˆ— as

local generators of degree 𝑗, for 𝑗 = 1, … , π‘œ.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Degree 1 and degree 2 cases

In particular, if β„³ is an β„•-graded manifold of base M and of degree 1, then C∞(β„³) is (canonically) isomorphic to Ξ“(β‹€β€’ Eβˆ—) for a vector bundle E over M. If β„³ is an β„•-graded manifold of base M and of degree 2, then C∞(β„³) is (noncanonically) isomorphic to Ξ“(β‹€β€’ Eβˆ—

βˆ’1 βŠ— Sβ€’Eβˆ— βˆ’2) for

two vector bundles Eβˆ’1 and Eβˆ’2 over M.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Degree 1 and degree 2 cases

In particular, if β„³ is an β„•-graded manifold of base M and of degree 1, then C∞(β„³) is (canonically) isomorphic to Ξ“(β‹€β€’ Eβˆ—) for a vector bundle E over M. If β„³ is an β„•-graded manifold of base M and of degree 2, then C∞(β„³) is (noncanonically) isomorphic to Ξ“(β‹€β€’ Eβˆ—

βˆ’1 βŠ— Sβ€’Eβˆ— βˆ’2) for

two vector bundles Eβˆ’1 and Eβˆ’2 over M.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Graded vector fields

Let β„³ be an [π‘œ]-manifold. A vector field of degree π‘˜ on β„³ is a graded derivation Ο• of C∞(β„³) such that |Ο•(ΞΎ)| = π‘˜ + |ΞΎ| for a homogeneous element ΞΎ ∈ C∞(β„³). The Lie bracket on graded vector fields, defined by [Ο•, ψ] = Ο•Οˆ βˆ’ (βˆ’1)|Ο•||ψ|ΟˆΟ• is graded skew-symmetric and satisfies graded Leibniz and graded Jacobi identities.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Graded vector fields

Let β„³ be an [π‘œ]-manifold. A vector field of degree π‘˜ on β„³ is a graded derivation Ο• of C∞(β„³) such that |Ο•(ΞΎ)| = π‘˜ + |ΞΎ| for a homogeneous element ΞΎ ∈ C∞(β„³). The Lie bracket on graded vector fields, defined by [Ο•, ψ] = Ο•Οˆ βˆ’ (βˆ’1)|Ο•||ψ|ΟˆΟ• is graded skew-symmetric and satisfies graded Leibniz and graded Jacobi identities.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Homological vector fields

A homological vector field 𝒭 on a graded manifold β„³ is a graded vector field of degree 1 that commutes with itself [𝒭, 𝒭] = 2𝒭 ∘ 𝒭 = 0. A Lie π‘œ-algebroid is a pair (β„³, 𝒭) of a positively graded manifold together with a homological vector field on it.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Homological vector fields

A homological vector field 𝒭 on a graded manifold β„³ is a graded vector field of degree 1 that commutes with itself [𝒭, 𝒭] = 2𝒭 ∘ 𝒭 = 0. A Lie π‘œ-algebroid is a pair (β„³, 𝒭) of a positively graded manifold together with a homological vector field on it.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Example: Lie 1-algebroid

Take an β„•-graded manifold of degree 1, i.e. C∞(β„³) = Ξ“(β‹€β€’ Eβˆ—) for a vector bundle E over M and take a trivialising chart U βŠ† M for

  • E. Any homological vector field 𝒭 on β„³ can locally be written as

𝒭U = βˆ‘

π‘—π‘˜

ρ(π‘“π‘˜)(𝑦𝑗)Ξ΅π‘—βˆ‚π‘¦π‘˜ βˆ’ βˆ‘

π‘—π‘˜π‘™

⟨[𝑓𝑗, π‘“π‘˜], Ξ΅π‘™βŸ©Ξ΅π‘—Ξ΅π‘˜βˆ‚Ξ΅π‘™, defining locally a Lie algebroid structure on E|U. This structure is in fact global, and Lie 1-algebroids are equivalent to Lie algebroids. (This is due to Arkady Vaintrob.)

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Example: Lie 1-algebroid

Let us describe yet another way to get the Lie algebroid structure from the homological vector field 𝒭. 𝑓 ∈ Ξ“(E) is identified with the graded vector field 𝑓 of degree βˆ’1 that sends Ξ΅ ∈ Ξ“(Eβˆ—) to βŸ¨π‘“, Ρ⟩ and 𝑔 ∈ C∞(M) to 0. Then [𝒭, 𝑓](𝑔) = ρ(𝑓)(𝑔) and [[𝒭, 𝑓], 𝑓′] = [𝑓, 𝑓′]. The Lie algebroid structure is hence derived from the homological vector field.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Example: Lie 1-algebroid

Let us describe yet another way to get the Lie algebroid structure from the homological vector field 𝒭. 𝑓 ∈ Ξ“(E) is identified with the graded vector field 𝑓 of degree βˆ’1 that sends Ξ΅ ∈ Ξ“(Eβˆ—) to βŸ¨π‘“, Ρ⟩ and 𝑔 ∈ C∞(M) to 0. Then [𝒭, 𝑓](𝑔) = ρ(𝑓)(𝑔) and [[𝒭, 𝑓], 𝑓′] = [𝑓, 𝑓′]. The Lie algebroid structure is hence derived from the homological vector field.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Example: Lie 1-algebroid

Let us describe yet another way to get the Lie algebroid structure from the homological vector field 𝒭. 𝑓 ∈ Ξ“(E) is identified with the graded vector field 𝑓 of degree βˆ’1 that sends Ξ΅ ∈ Ξ“(Eβˆ—) to βŸ¨π‘“, Ρ⟩ and 𝑔 ∈ C∞(M) to 0. Then [𝒭, 𝑓](𝑔) = ρ(𝑓)(𝑔) and [[𝒭, 𝑓], 𝑓′] = [𝑓, 𝑓′]. The Lie algebroid structure is hence derived from the homological vector field.

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SLIDE 19

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Example: Lie 2-algebroid

On a split [2]-manifold Q[βˆ’1] βŠ• Bβˆ—[βˆ’2], a homological vector field can be written 𝒭 = βˆ‘

𝑗,π‘˜

ρQ(π‘Ÿπ‘—)(π‘¦π‘˜)Ο„π‘—βˆ‚π‘¦π‘˜ βˆ’ βˆ‘

𝑗<π‘˜ βˆ‘ 𝑙

βŸ¨βŸ¦π‘Ÿπ‘—, π‘Ÿπ‘˜βŸ§, Ο„π‘™βŸ©Ο„π‘—Ο„π‘˜βˆ‚Ο„π‘™ + βˆ‘

𝑠,𝑙

βŸ¨βˆ‚βˆ—

Bβ𝑠, Ο„π‘™βŸ©π‘π‘ βˆ‚Ο„π‘™ βˆ’ βˆ‘ 𝑗<π‘˜<𝑙 βˆ‘ π‘š

Ο‰(π‘Ÿπ‘—, π‘Ÿπ‘˜, π‘Ÿπ‘™)(π‘π‘š)Ο„π‘—Ο„π‘˜Ο„π‘™βˆ‚π‘π‘š βˆ’ βˆ‘

π‘—π‘˜π‘š

βŸ¨βˆ‡βˆ—

π‘Ÿπ‘—Ξ²π‘˜, π‘π‘šβŸ©Ο„π‘—π‘π‘˜βˆ‚π‘π‘š,

where βˆ‚B : Qβˆ— β†’ B, ρQ : Q β†’ TM, βŸ¦β‹… , β‹…βŸ§: Ξ“(Q) Γ— Ξ“(Q) β†’ Ξ“(Q), βˆ‡: Ξ“(Q) Γ— Ξ“(B) β†’ Ξ“(B) and Ο‰ ∈ Ξ©3(Q, Bβˆ—) are the components of a split Lie 2-algebroid.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Split Lie 2-algebroids

A split Lie 2-algebroid Q βŠ• Bβˆ— β†’ M is a pair of an anchored vector bundle (Q β†’ M, ρQ) and a vector bundle B β†’ M, together with

1 a vector bundle map βˆ‚βˆ— B : Bβˆ— β†’ Q, 2 a skew-symmetric dull bracket βŸ¦β‹… , β‹…βŸ§: Ξ“(Q) Γ— Ξ“(Q) β†’ Ξ“(Q), 3 a linear connection βˆ‡βˆ— : Ξ“(Q) Γ— Ξ“(Bβˆ—) β†’ Ξ“(Bβˆ—) and 4 a vector valued 3-form Ο‰ ∈ Ξ©3(Q, Bβˆ—),

such that (i) βˆ‡βˆ—

βˆ‚βˆ—

B(Ξ²1)Ξ²2 + βˆ‡βˆ—

βˆ‚βˆ—

B(Ξ²2)Ξ²1 = 0,

(ii) βŸ¦π‘Ÿ, βˆ‚βˆ—

B(Ξ²)⟧ = βˆ‚βˆ— B(βˆ‡βˆ— π‘ŸΞ²),

(iii) JacβŸ¦β‹…,β‹…βŸ§ = βˆ’βˆ‚βˆ—

B ∘ Ο‰ ∈ Ξ©3(Q, Q),

(iv) Rβˆ‡βˆ—(π‘Ÿ1, π‘Ÿ2)Ξ² = Ο‰(π‘Ÿ1, π‘Ÿ2, βˆ‚βˆ—

B(Ξ²)), and

(v) πžβˆ‡βˆ—Ο‰ = 0 for all Ξ², Ξ²1, Ξ²2 ∈ Ξ“(Bβˆ—) and π‘Ÿ, π‘Ÿ1, π‘Ÿ2 ∈ Ξ“(Q).

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

1 Lie 2-algebroids 2 Dorfman 2-representations 3 Poisson [2]-manifolds 4 Poisson Lie 2-algebroids 5 (Degenerate) Courant algebroids 6 Overview of the geometrisation

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Dull bracket

Let (Q, ρQ) be an anchored vector bundle over M. A skew-symmetric dull bracket Ξ“(Q) is a skew-symmetric ℝ-bilinear map βŸ¦β‹… , β‹…βŸ§: Ξ“(Q) Γ— Ξ“(Q) β†’ Ξ“(Q) such that ρQβŸ¦π‘Ÿ1, π‘Ÿ2⟧ = [ρQ(π‘Ÿ1), ρQ(π‘Ÿ2)] and (the Leibniz identity) βŸ¦π‘Ÿ1, 𝑔 β‹… π‘Ÿ2⟧ = 𝑔 β‹… βŸ¦π‘Ÿ1, π‘Ÿ2⟧ + ρQ(π‘Ÿ1)(𝑔) β‹… π‘Ÿ2 for all π‘Ÿ1, π‘Ÿ2 ∈ Ξ“(Q) and 𝑔 ∈ C∞(M).

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Dual of a skew-symmetric dull bracket

Dualize the bracket in the sense of derivations: Ξ”: Ξ“(Q) Γ— Ξ“(Qβˆ—) β†’ Ξ“(Qβˆ—), βŸ¨Ξ”π‘ŸΟ„, π‘Ÿβ€²βŸ© = ρQ(π‘Ÿ)βŸ¨Ο„, π‘Ÿβ€²βŸ© βˆ’ βŸ¨βŸ¦π‘Ÿ, π‘Ÿβ€²βŸ§, Ο„βŸ© for all π‘Ÿ, π‘Ÿβ€² ∈ Ξ“(Q) and Ο„ ∈ Ξ“(Qβˆ—). The dual object Ξ” is called a skew-symmetric Dorfman Q-connection on Qβˆ—.

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SLIDE 24

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Dual of a skew-symmetric dull bracket

Dualize the bracket in the sense of derivations: Ξ”: Ξ“(Q) Γ— Ξ“(Qβˆ—) β†’ Ξ“(Qβˆ—), βŸ¨Ξ”π‘ŸΟ„, π‘Ÿβ€²βŸ© = ρQ(π‘Ÿ)βŸ¨Ο„, π‘Ÿβ€²βŸ© βˆ’ βŸ¨βŸ¦π‘Ÿ, π‘Ÿβ€²βŸ§, Ο„βŸ© for all π‘Ÿ, π‘Ÿβ€² ∈ Ξ“(Q) and Ο„ ∈ Ξ“(Qβˆ—). The dual object Ξ” is called a skew-symmetric Dorfman Q-connection on Qβˆ—.

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SLIDE 25

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Properties...

Ξ”: Ξ“(Q) Γ— Ξ“(Qβˆ—) β†’ Ξ“(Qβˆ—) has the following properties

1 Ξ”π‘Ÿ(𝑔τ) = 𝑔 β‹… Ξ”π‘ŸΟ„ + ρQ(π‘Ÿ)(𝑔) β‹… Ο„, 2 Ξ”π‘”π‘ŸΟ„ = 𝑔 β‹… Ξ”π‘ŸΟ„ + βŸ¨π‘Ÿ, Ο„βŸ© β‹… Οβˆ— Qπžπ‘” 3 βŸ¨Ξ”π‘Ÿ1Ο„, π‘Ÿ2⟩ + βŸ¨Ξ”π‘Ÿ2Ο„, π‘Ÿ1⟩ = ρQ(π‘Ÿ1)βŸ¨Ο„, π‘Ÿ2⟩ + ρQ(π‘Ÿ2)βŸ¨Ο„, π‘Ÿ1⟩ 4 Ξ”π‘Ÿ(Οβˆ— Qπžπ‘”) = Οβˆ— Q𝐞(ρQ(π‘Ÿ)(𝑔)).

for all 𝑔 ∈ C∞(M), π‘Ÿ, π‘Ÿ1, π‘Ÿ2 ∈ Ξ“(Q) and Ο„ ∈ Ξ“(Qβˆ—).

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Curvature of a Dorfman connection

The Jacobiator in Leibniz form JacβŸ¦β‹… ,β‹…βŸ§(π‘Ÿ1, π‘Ÿ2, π‘Ÿ3) =βŸ¦βŸ¦π‘Ÿ1, π‘Ÿ2⟧, π‘Ÿ3⟧ + βŸ¦π‘Ÿ2, βŸ¦π‘Ÿ1, π‘Ÿ3⟧⟧ βˆ’ βŸ¦π‘Ÿ1, βŸ¦π‘Ÿ2, π‘Ÿ3⟧⟧ is equivalent to the curvature of the Dorfman connection: JacβŸ¦β‹… ,β‹…βŸ§(π‘Ÿ1, π‘Ÿ2, π‘Ÿ3) = RΞ”(π‘Ÿ1, π‘Ÿ2)βˆ—π‘Ÿ3 for all π‘Ÿ1, π‘Ÿ2, π‘Ÿ3.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Dorfman connections

Lie algebroid brackets are dual to flat, skew-symmetric Dorfman connections, which we call Dorfman representations. Split Lie 2-algebroid brackets are dual to Dorfman 2-representations.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Dorfman connections

Lie algebroid brackets are dual to flat, skew-symmetric Dorfman connections, which we call Dorfman representations. Split Lie 2-algebroid brackets are dual to Dorfman 2-representations.

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SLIDE 29

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Dorfman 2-representations

Let (Q β†’ M, ρQ) be an anchored vector bundle and B a vector bundle over M. A (Q, ρQ)-Dorfman 2-representation on Qβˆ— βŠ• B is

1 a vector bundle morphism βˆ‚B : Qβˆ— β†’ B, 2 a linear connection βˆ‡: Ξ“(Q) Γ— Ξ“(B) β†’ Ξ“(B) such that

βˆ‡βˆ—

βˆ‚βˆ—

BΞ²1Ξ²2 + βˆ‡βˆ—

βˆ‚βˆ—

BΞ²2Ξ²1 = 0,

3 a skew-symmetric Dorfman connection

Ξ”: Ξ“(Q) Γ— Ξ“(Qβˆ—) β†’ Ξ“(Qβˆ—) such that βˆ‚B ∘ Ξ”π‘Ÿ = βˆ‡π‘Ÿ ∘ βˆ‚B and

4 a vector-valued 2-form R ∈ Ξ©2(Q, Hom(B, Qβˆ—)) such that 1 βˆ‚B ∘ R(π‘Ÿ1, π‘Ÿ2) = Rβˆ‡(π‘Ÿ1, π‘Ÿ2) and R(π‘Ÿ1, π‘Ÿ2) ∘ βˆ‚B = RΞ”(π‘Ÿ1, π‘Ÿ2), 2 R(π‘Ÿ1, π‘Ÿ2)βˆ—π‘Ÿ3 = βˆ’R(π‘Ÿ1, π‘Ÿ3)βˆ—π‘Ÿ2 and 3 πžβ™¦R(π‘Ÿ1, π‘Ÿ2, π‘Ÿ3) = βˆ‡βˆ— β‹… (R(π‘Ÿ1, π‘Ÿ2)βˆ—π‘Ÿ3)

for all ΞΎ1, ΞΎ2 ∈ Ξ“(Bβˆ—) and π‘Ÿ, π‘Ÿ1, π‘Ÿ2, π‘Ÿ3 ∈ Ξ“(Q) and 𝑔 ∈ C∞(M).

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SLIDE 30

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

2-term representations up to homotopy

Let A be a Lie algebroid and E0 βŠ• E1 a 2-term graded vector

  • bundle. A 2-representation of A on E0 βŠ• E1is

1 a map βˆ‚: E0 β†’ E1, 2 two A-connections, βˆ‡0 and βˆ‡1 on E0 and E1, respectively,

such that βˆ‚ ∘ βˆ‡0 = βˆ‡1 ∘ βˆ‚,

3 an element R ∈ Ξ©2(A, Hom(E1, E0)) such that Rβˆ‡0 = R ∘ βˆ‚,

Rβˆ‡1 = βˆ‚ ∘ R and πžβˆ‡HomR = 0, where βˆ‡Hom is the connection induced on Hom(E1, E0) by βˆ‡0 and βˆ‡1.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Example

Let (π–₯ β†’ M, ρπ–₯, βŸ¦β‹… , β‹…βŸ§π–₯, βŸ¨β‹… , β‹…βŸ©) be a Courant algebroid. Choose a linear connection βˆ‡: π”œ(M) Γ— Ξ“(π–₯) β†’ Ξ“(π–₯) that preserves the pairing.

1 βˆ‚TM = ρπ–₯ : π–₯ β†’ TM. 2 Ξ”: Ξ“(π–₯) Γ— Ξ“(π–₯) β†’ Ξ“(π–₯), Δ𝑓𝑓′ = βŸ¦π‘“, π‘“β€²βŸ§π–₯ + βˆ‡Οπ–₯(𝑓′)𝑓 is a

Dorfman connection.

3 The map βˆ‡π‘π‘π‘‘ : Ξ“(π–₯) Γ— π”œ(M) β†’ π”œ(M),

βˆ‡π‘π‘π‘‘

𝑓 X = [ρπ–₯(𝑓), X] + ρπ–₯(βˆ‡X𝑓) is a linear connection. 4 The basic curvature R𝑐𝑏𝑑 Ξ”

∈ Ξ©2(π–₯, Hom(TM, π–₯)) is R𝑐𝑏𝑑

Ξ” (𝑓1, 𝑓2)X = βˆ’ βˆ‡XβŸ¦π‘“1, 𝑓2⟧π–₯ + βŸ¦βˆ‡X𝑓1, 𝑓2⟧π–₯ + βŸ¦π‘“1, βˆ‡X𝑓2⟧π–₯

+ βˆ‡βˆ‡π‘π‘π‘‘

𝑓2 X𝑓1 βˆ’ βˆ‡βˆ‡π‘π‘π‘‘ 𝑓1 X𝑓2 βˆ’ Ξ²βˆ’1βŸ¨βˆ‡βˆ‡π‘π‘π‘‘ β‹…

X𝑓1, 𝑓2⟩

for all 𝑓1, 𝑓2 ∈ Ξ“(π–₯) and X ∈ π”œ(M).

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SLIDE 32

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

1 Lie 2-algebroids 2 Dorfman 2-representations 3 Poisson [2]-manifolds 4 Poisson Lie 2-algebroids 5 (Degenerate) Courant algebroids 6 Overview of the geometrisation

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Degree βˆ’2 Poisson bracket on a [2]-manifold

A Poisson [2]-manifold is a [2]-manifold endowed with a Poisson bracket of degree βˆ’2. A Poisson bracket of degree βˆ’2 is graded skew-symmetric and satisfies |{ΞΎ, Ξ·}| = |ΞΎ| + |Ξ·| βˆ’ 2 and graded Leibniz and Jacobi identities.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Degree βˆ’2 Poisson bracket on a [2]-manifold

A Poisson [2]-manifold is a [2]-manifold endowed with a Poisson bracket of degree βˆ’2. A Poisson bracket of degree βˆ’2 is graded skew-symmetric and satisfies |{ΞΎ, Ξ·}| = |ΞΎ| + |Ξ·| βˆ’ 2 and graded Leibniz and Jacobi identities.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Split Poisson [2]-manifolds are equivalent to self-dual 2-term representations up to homotopy

Theorem (J.L. 2015) A split Poisson [2]-manifold (β„³ = Q[βˆ’1] βŠ• Bβˆ—[βˆ’2], {β‹… , β‹…}) defines as follows a Lie algebroid structure on B, a VB-morphism βˆ‚Q : Qβˆ— β†’ Q and a 2-term representation up to homotopy (βˆ‡, βˆ‡βˆ—, R) of B on βˆ‚Q : Qβˆ— β†’ Q:

1 {𝑔1, 𝑔2} = {𝑔, Ο„} = 0, 2 {Ο„1, Ο„2} = βŸ¨Ο„2, βˆ‚Q(Ο„1)⟩, 3 {𝑐, 𝑔} = ρB(𝑐)(𝑔) with an anchor ρB : B β†’ TM, 4 {𝑐, Ο„} = βˆ‡π‘Ο„ with a linear B-connection βˆ‡ on Qβˆ—, 5 {𝑐1, 𝑐2} = [𝑐1, 𝑐2] βˆ’ R(𝑐1, 𝑐2) with [β‹…, β‹…] a Lie algebroid

bracket on B and R ∈ Ξ©2(B, Hom(Q; Qβˆ—)). The 2-term representation up to homotopy is self-dual: βˆ‚βˆ—

Q = βˆ‚Q

and Rβˆ— = βˆ’R.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Split Poisson [2]-manifolds are equivalent to self-dual 2-term representations up to homotopy

Theorem (J.L. 2015) A split Poisson [2]-manifold (β„³ = Q[βˆ’1] βŠ• Bβˆ—[βˆ’2], {β‹… , β‹…}) defines as follows a Lie algebroid structure on B, a VB-morphism βˆ‚Q : Qβˆ— β†’ Q and a 2-term representation up to homotopy (βˆ‡, βˆ‡βˆ—, R) of B on βˆ‚Q : Qβˆ— β†’ Q:

1 {𝑔1, 𝑔2} = {𝑔, Ο„} = 0, 2 {Ο„1, Ο„2} = βŸ¨Ο„2, βˆ‚Q(Ο„1)⟩, 3 {𝑐, 𝑔} = ρB(𝑐)(𝑔) with an anchor ρB : B β†’ TM, 4 {𝑐, Ο„} = βˆ‡π‘Ο„ with a linear B-connection βˆ‡ on Qβˆ—, 5 {𝑐1, 𝑐2} = [𝑐1, 𝑐2] βˆ’ R(𝑐1, 𝑐2) with [β‹…, β‹…] a Lie algebroid

bracket on B and R ∈ Ξ©2(B, Hom(Q; Qβˆ—)). The 2-term representation up to homotopy is self-dual: βˆ‚βˆ—

Q = βˆ‚Q

and Rβˆ— = βˆ’R.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Split Poisson [2]-manifolds are equivalent to self-dual 2-term representations up to homotopy

Conversely, any self-dual 2-representation defines a split Poisson [2]-manifold.

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Example

Let (π–₯ β†’ M, ρπ–₯, βŸ¦β‹… , β‹…βŸ§π–₯, βŸ¨β‹… , β‹…βŸ©) be a Courant algebroid. Choose a linear connection βˆ‡: π”œ(M) Γ— Ξ“(π–₯) β†’ Ξ“(π–₯) that preserves the pairing. The 2-representation (Idπ–₯ : π–₯ β†’ π–₯, βˆ‡, βˆ‡, Rβˆ‡) is then self-dual. We get so for each metric connection βˆ‡ a split Poisson manifold (π–₯[βˆ’1] βŠ• Tβˆ—M[βˆ’2], {β‹… , β‹…}βˆ‡).

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SLIDE 39

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Example

Let (π–₯ β†’ M, ρπ–₯, βŸ¦β‹… , β‹…βŸ§π–₯, βŸ¨β‹… , β‹…βŸ©) be a Courant algebroid. Choose a linear connection βˆ‡: π”œ(M) Γ— Ξ“(π–₯) β†’ Ξ“(π–₯) that preserves the pairing. The 2-representation (Idπ–₯ : π–₯ β†’ π–₯, βˆ‡, βˆ‡, Rβˆ‡) is then self-dual. We get so for each metric connection βˆ‡ a split Poisson manifold (π–₯[βˆ’1] βŠ• Tβˆ—M[βˆ’2], {β‹… , β‹…}βˆ‡).

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Example

Let (π–₯ β†’ M, ρπ–₯, βŸ¦β‹… , β‹…βŸ§π–₯, βŸ¨β‹… , β‹…βŸ©) be a Courant algebroid. Choose a linear connection βˆ‡: π”œ(M) Γ— Ξ“(π–₯) β†’ Ξ“(π–₯) that preserves the pairing. The 2-representation (Idπ–₯ : π–₯ β†’ π–₯, βˆ‡, βˆ‡, Rβˆ‡) is then self-dual. We get so for each metric connection βˆ‡ a split Poisson manifold (π–₯[βˆ’1] βŠ• Tβˆ—M[βˆ’2], {β‹… , β‹…}βˆ‡).

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SLIDE 41

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

1 Lie 2-algebroids 2 Dorfman 2-representations 3 Poisson [2]-manifolds 4 Poisson Lie 2-algebroids 5 (Degenerate) Courant algebroids 6 Overview of the geometrisation

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Poisson Lie 2-algebroids

Let (β„³, {β‹… , β‹…}) be a Poisson [2]-manifold. Assume that β„³ has in addition a Lie 2-algebroid structure 𝒭. Then (β„³, 𝒭, {β‹… , β‹…}) is a Poisson Lie 2-algebroid if the homological vector field preserves the Poisson structure: 𝒭{ΞΎ1, ΞΎ2} = {𝒭(ΞΎ1), ΞΎ2} + (βˆ’1)|ΞΎ1|{ΞΎ1, 𝒭(ΞΎ2)} for all ΞΎ1, ΞΎ2 ∈ C∞(β„³).

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

A characterisation of split Poisson Lie 2-algebroids

Theorem (J.L. 2015) (β„³ = Q[βˆ’1] βŠ• Bβˆ—[βˆ’2], 𝒭, {β‹… , β‹…}) is a Poisson Lie 2-algebroid if and only if

1 βˆ‚Q(Ξ”π‘ŸΟ„) = βˆ‡βˆ‚BΟ„π‘Ÿ + βŸ¦π‘Ÿ, βˆ‚QΟ„βŸ§ + βˆ‚βˆ— BβŸ¨Ο„, βˆ‡β‹…π‘ŸβŸ©, 2 βˆ‚B(βˆ‡π‘Ο„) = [𝑐, βˆ‚BΟ„] + βˆ‡βˆ‚Qτ𝑐, 3 βˆ‚BR(𝑐1, 𝑐2)π‘Ÿ =

βˆ’βˆ‡π‘Ÿ[𝑐1, 𝑐2] + [βˆ‡π‘Ÿπ‘1, 𝑐2] + [𝑐1, βˆ‡π‘Ÿπ‘2] + βˆ‡βˆ‡π‘2π‘Ÿπ‘1 βˆ’ βˆ‡βˆ‡π‘1π‘Ÿπ‘2,

4 βˆ‚QR(π‘Ÿ1, π‘Ÿ2)𝑐 = βˆ’βˆ‡π‘βŸ¦π‘Ÿ1, π‘Ÿ2⟧ + βŸ¦π‘Ÿ1, βˆ‡π‘π‘Ÿ2⟧ + βŸ¦βˆ‡π‘π‘Ÿ1, π‘Ÿ2⟧ +

βˆ‡βˆ‡π‘Ÿ2π‘π‘Ÿ1 βˆ’ βˆ‡βˆ‡π‘Ÿ1π‘π‘Ÿ2 + βˆ‚βˆ—

B⟨R(β‹…, 𝑐)π‘Ÿ1, π‘Ÿ2⟩, and 5 πžβˆ‡BΟ‰R = πžβˆ‡QΟ‰B ∈ Ξ©2(B, β‹€3 Qβˆ—) = Ξ©3(Q, β‹€2 Bβˆ—).

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Example

Let (π–₯ β†’ M, ρπ–₯, βŸ¦β‹… , β‹…βŸ§π–₯, βŸ¨β‹… , β‹…βŸ©) be a Courant algebroid. Choose a linear connection βˆ‡: π”œ(M) Γ— Ξ“(π–₯) β†’ Ξ“(π–₯) that preserves the pairing. The Dorfman 2-representation and the self-dual 2-representation found earlier are compatible and define a Poisson Lie 2-algebroid structure on π–₯[βˆ’1] βŠ• Tβˆ—M[βˆ’2].

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SLIDE 45

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Example

Let (π–₯ β†’ M, ρπ–₯, βŸ¦β‹… , β‹…βŸ§π–₯, βŸ¨β‹… , β‹…βŸ©) be a Courant algebroid. Choose a linear connection βˆ‡: π”œ(M) Γ— Ξ“(π–₯) β†’ Ξ“(π–₯) that preserves the pairing. The Dorfman 2-representation and the self-dual 2-representation found earlier are compatible and define a Poisson Lie 2-algebroid structure on π–₯[βˆ’1] βŠ• Tβˆ—M[βˆ’2].

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Conversely, how do we recover the Courant algebroid structure from the Dorfman 2-representation and the self-dual 2-representation?

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Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

1 Lie 2-algebroids 2 Dorfman 2-representations 3 Poisson [2]-manifolds 4 Poisson Lie 2-algebroids 5 (Degenerate) Courant algebroids 6 Overview of the geometrisation

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SLIDE 48

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

The degenerate Courant algebroid on the core

Theorem (J.L. 2015) Let (β„³ = Q[βˆ’1] βŠ• Bβˆ—[βˆ’2], 𝒭, {β‹… , β‹…}) be a split Poisson Lie 2-algebroid. Then Qβˆ— inherits the structure of a degenerate Courant algebroid over M, with the anchor ρQβˆ‚Q = ρBβˆ‚B, the map 𝒠 = Οβˆ—

Q𝐞: C∞(M) β†’ Ξ“(Qβˆ—), the pairing defined by

βŸ¨Ο„1, Ο„2⟩Qβˆ— = βŸ¨Ο„1, βˆ‚QΟ„2⟩ and the bracket defined by βŸ¦Ο„1, Ο„2⟧Qβˆ— = Ξ”βˆ‚QΟ„1Ο„2 βˆ’ βˆ‡βˆ‚BΟ„2Ο„1 for all Ο„1, Ο„2 ∈ Ξ“(Qβˆ—).

Back to the definition

Given a Poisson Lie 2-algebroid, this structure does not depend on the choice of a splitting, and the map βˆ‚B : Qβˆ— β†’ B preserves the brackets and the anchors.

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SLIDE 49

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

The degenerate Courant algebroid on the core

Theorem (J.L. 2015) Let (β„³ = Q[βˆ’1] βŠ• Bβˆ—[βˆ’2], 𝒭, {β‹… , β‹…}) be a split Poisson Lie 2-algebroid. Then Qβˆ— inherits the structure of a degenerate Courant algebroid over M, with the anchor ρQβˆ‚Q = ρBβˆ‚B, the map 𝒠 = Οβˆ—

Q𝐞: C∞(M) β†’ Ξ“(Qβˆ—), the pairing defined by

βŸ¨Ο„1, Ο„2⟩Qβˆ— = βŸ¨Ο„1, βˆ‚QΟ„2⟩ and the bracket defined by βŸ¦Ο„1, Ο„2⟧Qβˆ— = Ξ”βˆ‚QΟ„1Ο„2 βˆ’ βˆ‡βˆ‚BΟ„2Ο„1 for all Ο„1, Ο„2 ∈ Ξ“(Qβˆ—).

Back to the definition

Given a Poisson Lie 2-algebroid, this structure does not depend on the choice of a splitting, and the map βˆ‚B : Qβˆ— β†’ B preserves the brackets and the anchors.

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SLIDE 50

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Example

Let (π–₯ β†’ M, ρπ–₯, βŸ¦β‹… , β‹…βŸ§π–₯, βŸ¨β‹… , β‹…βŸ©) be a Courant algebroid. Choose a linear connection βˆ‡: π”œ(M) Γ— Ξ“(π–₯) β†’ Ξ“(π–₯) that preserves the pairing. Then Δ𝑓1𝑓2 βˆ’ βˆ‡Ο(𝑓2)𝑓1 = βŸ¦π‘“1, 𝑓2⟧π–₯.

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SLIDE 51

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Symplectic Lie 2-algebroids and Courant algebroids

Theorem (J.L. 2015) Let β„³ be a symplectic Lie 2-algebroid over a base manifold M. Choose any splitting β„³ ≃ Q[βˆ’1] βŠ• Tβˆ—M[βˆ’2] of the underlying symplectic [2]-manifold. Then

1 Q ≃ Qβˆ— via βˆ‚Q and βŸ¨Ο„1, βˆ‚QΟ„2⟩ is nondegenerate. 2 The map ρQ ∘ βˆ‚Q = βˆ‚TM defines an anchor on Qβˆ—. 3 The bracket βŸ¦β‹… , β‹…βŸ§Qβˆ— defined on Ξ“(Qβˆ—) by

βŸ¦Ο„1, Ο„2⟧Qβˆ— = Ξ”βˆ‚QΟ„1Ο„2 βˆ’ {βˆ‚TMΟ„2, Ο„1} does not depend on the choice of the splitting. This anchor, pairing and bracket define a Courant algebroid structure on Qβˆ—.

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SLIDE 52

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Symplectic Lie 2-algebroids and Courant algebroids

Theorem (J.L. 2015) Let β„³ be a symplectic Lie 2-algebroid over a base manifold M. Choose any splitting β„³ ≃ Q[βˆ’1] βŠ• Tβˆ—M[βˆ’2] of the underlying symplectic [2]-manifold. Then

1 Q ≃ Qβˆ— via βˆ‚Q and βŸ¨Ο„1, βˆ‚QΟ„2⟩ is nondegenerate. 2 The map ρQ ∘ βˆ‚Q = βˆ‚TM defines an anchor on Qβˆ—. 3 The bracket βŸ¦β‹… , β‹…βŸ§Qβˆ— defined on Ξ“(Qβˆ—) by

βŸ¦Ο„1, Ο„2⟧Qβˆ— = Ξ”βˆ‚QΟ„1Ο„2 βˆ’ {βˆ‚TMΟ„2, Ο„1} does not depend on the choice of the splitting. This anchor, pairing and bracket define a Courant algebroid structure on Qβˆ—.

37 / 50

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SLIDE 53

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

1 Lie 2-algebroids 2 Dorfman 2-representations 3 Poisson [2]-manifolds 4 Poisson Lie 2-algebroids 5 (Degenerate) Courant algebroids 6 Overview of the geometrisation

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SLIDE 54

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Quick review of double vector bundles

A double vector bundle is a commutative square D

Ο€B

  • Ο€Q
  • B

π‘ŸB

  • Q

π‘ŸQ

M

  • f vector bundles such that the structure maps of the vertical

bundles define morphisms of the horizontal bundles. Take a triple Q, B, C of vector bundles over a smooth manifold M. Then the fibre-product Q Γ—M B Γ—M C has a vector bundle structure

  • ver Q given by (π‘Ÿ, 𝑐, 𝑑) +Q (π‘Ÿ, 𝑐′, 𝑑′) = (π‘Ÿ, 𝑐 + 𝑐′, 𝑑 + 𝑑′), and

similarly a vector bundle structure over B.

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SLIDE 55

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Quick review of double vector bundles

Consider such a decomposed double vector bundle D = Q Γ—M B Γ—M C. For each 𝑐 ∈ Ξ“(B), we have a linear section Μƒ 𝑐 ∈ Ξ“

Q(D),

Μƒ 𝑐(π‘Ÿπ‘›) = (π‘Ÿπ‘›, 𝑐(𝑛), 0C

𝑛)

and for each 𝑑 ∈ Ξ“(C), we have a core section 𝑑† ∈ Ξ“

Q(D),

𝑑†(π‘Ÿπ‘›) = (π‘Ÿπ‘›, 0B

𝑛, 𝑑(𝑛)).

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SLIDE 56

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Theorem (J.L.2015) The category of positively graded manifolds of degree 2 is equivalent to the category of metric double vector bundles. Splittings of a positively graded manifold of degree 2 correspond to maximally isotropic decompositions of the corresponding metric double vector bundle. 𝔽

  • B
  • Q

M

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SLIDE 57

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Theorem (J.L.2015) The category of positively graded Poisson manifolds of degree 2 is equivalent to the category of self-dual VB-algebroids. Splittings of a positively graded Poisson manifold of degree 2 correspond to self-dual 2-representations, which correspond to maximally isotropic decompositions of the corresponding self-dual VB-algebroid. 𝔽

  • B
  • Q

M

(The equivalence of decomposed VB-algebroids with 2-representations is due to Gracia-Saz and Mehta.)

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SLIDE 58

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Theorem (Li-Bland 2012, J.L.2015) The category of Lie 2-algebroids is equivalent to the category of VB-Courant algebroids. Splittings of a Lie 2-algebroid correspond to Dorfman 2-representations, which correspond to maximally isotropic decompositions of the corresponding VB-Courant algebroid. 𝔽

  • B
  • Q

M

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SLIDE 59

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Theorem (Li-Bland 2012, J.L.2015) The category of Poisson Lie 2-algebroids is equivalent to the category of LA-Courant algebroids. 𝔽

  • B
  • Q

M

44 / 50

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SLIDE 60

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Symplectic Lie 2-algebroids correspond to tangent doubles of Courant algebroids. Tπ–₯

  • TM
  • π–₯

M

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SLIDE 61

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

One further application

Consider a double Lie algebroid D

  • A
  • B

M

The direct sum Dβˆ—

A βŠ• Dβˆ— B

  • Cβˆ—
  • A βŠ• B

M

is a VB-Courant algebroid.

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SLIDE 62

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

One further application

A decomposition of D gives rise to two 2-representations which form a matched pair (Gracia-Saz, J.L., Mackenzie, Mehta). A decomposition of D naturally induces a maximally isotropic decomposition of Dβˆ—

A βŠ• Dβˆ— B.

The corresponding split Lie 2-algebroid is the bicrossproduct (A βŠ• B)[βˆ’1] βŠ• Cβˆ—[βˆ’2] of a matched pair of 2-representations!

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SLIDE 63

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

One further application

A decomposition of D gives rise to two 2-representations which form a matched pair (Gracia-Saz, J.L., Mackenzie, Mehta). A decomposition of D naturally induces a maximally isotropic decomposition of Dβˆ—

A βŠ• Dβˆ— B.

The corresponding split Lie 2-algebroid is the bicrossproduct (A βŠ• B)[βˆ’1] βŠ• Cβˆ—[βˆ’2] of a matched pair of 2-representations!

47 / 50

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SLIDE 64

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

One further application

A decomposition of D gives rise to two 2-representations which form a matched pair (Gracia-Saz, J.L., Mackenzie, Mehta). A decomposition of D naturally induces a maximally isotropic decomposition of Dβˆ—

A βŠ• Dβˆ— B.

The corresponding split Lie 2-algebroid is the bicrossproduct (A βŠ• B)[βˆ’1] βŠ• Cβˆ—[βˆ’2] of a matched pair of 2-representations!

47 / 50

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SLIDE 65

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Diagrammatic table of the supergeometric objects in this talk.

[2]-manifolds Symplectic [2]-manifolds Lie 2- algebroids Poisson [2]-manifolds Poisson Lie 2-algebroids Degenerate Courant algebroids Symplectic Lie 2- algebroids Courant algebroids Cotangent functor β€œCore” β€œCore” Tangent functor 48 / 50

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SLIDE 66

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Diagrammatic table of the (classical, double) geometric objects in this talk.

Metric double vector bundles Tangent prolongations

  • f metric

vector bundles VB Courant algebroids Metric VB- algebroids Metric vector bundles LA-Courant algebroids Degenerate Courant algebroids Tangent prolongations

  • f Courant

algebroids Courant algebroids Tangent functor Core Core Tangent functor 49 / 50

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SLIDE 67

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Thank you for your attention! Happy birthday, Janusz!

50 / 50

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SLIDE 68

Poisson Lie 2-algebroids and degenerate Courant algebroids

  • M. Jotz Lean

The University

  • f Sheffield

Lie 2-algebroids Dorfman 2-representations Poisson [2]-manifolds Poisson Lie 2-algebroids (Degenerate) Courant algebroids Overview of the geometrisation

Thank you for your attention! Happy birthday, Janusz!

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