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 2-algebroids [2] -manifolds Poisson 2-representations Dorfman Lie 2 -algebroids of Sheffield The University M. Jotz Lean algebroids Courant and degenerate 1 / 50
Poisson Lie Overview of the vector field. graded manifolds of degree 2 with a compatible homological Lie 2-algebroids, or in other words symplectic positively found that Courant algebroids were equivalent to symplectic A few years later, Roytenberg, and independently Severa bialgebroid is a Courant algebroid. Courant algebroids and proved that the bicrossproduct of a Lie Later, in the late 90's, Liu, Weinstein and Xu defined general the late 80's. given a smooth manifold M , was discovered by Ted Courant in The standard Courant algebroid structure on TM ⊕ T ∗ M , algebroids and symplectic Lie 2-algebroids Motivation: the correspondence between Courant geometrisation algebroids 2-algebroids Lie 2 -algebroids and degenerate Courant algebroids M. Jotz Lean The University of Sheffield Dorfman Courant 2-representations Poisson [2] -manifolds Poisson Lie 2-algebroids (Degenerate) 2 / 50
Poisson Lie Courant Back to theorem 3 ⟦𝑓 1 , 𝑓 2 ⟧ + ⟦𝑓 2 , 𝑓 1 ⟧ = ρ ∗ 𝐞⟨𝑓 1 , 𝑓 2 ⟩ 2 ρ(𝑓 1 )⟨𝑓 2 , 𝑓 3 ⟩ = ⟨⟦𝑓 1 , 𝑓 2 ⟧, 𝑓 3 ⟩ + ⟨𝑓 2 , ⟦𝑓 1 , 𝑓 3 ⟧⟩ , 1 ⟦𝑓 1 , ⟦𝑓 2 , 𝑓 3 ⟧⟧ = ⟦⟦𝑓 1 , 𝑓 2 ⟧, 𝑓 3 ⟧ + ⟦𝑓 2 , ⟦𝑓 1 , 𝑓 3 ⟧⟧ , ρ: 𝖥 → TM , which satisfy the following conditions bracket ⟦⋅ , ⋅⟧ on the smooth sections Γ(𝖥) , and an anchor with a fibrewise nondegenerate symmetric bilinear form ⟨⋅ , ⋅⟩ , a A Courant algebroid over a manifold M is a vector bundle 𝖥 → M The classical definition of a Courant algebroid geometrisation Overview of the algebroids (Degenerate) 2-algebroids 2-algebroids Poisson Lie [2] -manifolds Poisson 2-representations Dorfman Lie 2 -algebroids of Sheffield The University M. Jotz Lean algebroids Courant and degenerate 3 / 50 for all 𝑓 1 , 𝑓 2 , 𝑓 3 ∈ Γ(𝖥) .
Poisson Lie Courant structure. "semidirect product'' of the Poisson structure and the dual of the Lie and the anchor of a Courant algebroid can be retrieved as a kind of The aim of this talk is to show how the Courant algebroid bracket homological vector field. algebroid 𝔽 are derived from the positively graded manifold and the Usually the Courant algebroid bracket and the anchor of a Courant algebroids and symplectic Lie 2-algebroids Motivation: the correspondence between Courant geometrisation Overview of the algebroids (Degenerate) 2-algebroids 2-algebroids Poisson Lie [2] -manifolds Poisson 2-representations Dorfman Lie 2 -algebroids of Sheffield The University M. Jotz Lean algebroids Courant and degenerate 4 / 50
Poisson Lie Courant structure. "semidirect product'' of the Poisson structure and the dual of the Lie and the anchor of a Courant algebroid can be retrieved as a kind of The aim of this talk is to show how the Courant algebroid bracket homological vector field. algebroid 𝔽 are derived from the positively graded manifold and the Usually the Courant algebroid bracket and the anchor of a Courant algebroids and symplectic Lie 2-algebroids Motivation: the correspondence between Courant geometrisation Overview of the algebroids (Degenerate) 2-algebroids 2-algebroids Poisson Lie [2] -manifolds Poisson 2-representations Dorfman Lie 2 -algebroids of Sheffield The University M. Jotz Lean algebroids Courant and degenerate 4 / 50
Poisson Lie (Degenerate) 6 Overview of the geometrisation 5 (Degenerate) Courant algebroids 4 Poisson Lie 2-algebroids 3 Poisson [2] -manifolds 2 Dorfman 2-representations 1 Lie 2 -algebroids Outline geometrisation Overview of the algebroids Courant 2-algebroids 2-algebroids Poisson Lie [2] -manifolds Poisson 2-representations Dorfman Lie 2 -algebroids of Sheffield The University M. Jotz Lean algebroids Courant and degenerate 5 / 50
Poisson Lie algebroids 𝑗 ∈ {1, … , 𝑜} and 𝑘 ∈ {1, … , 𝑠 𝑗 } . 𝑠 𝑜 𝑜 , … , ξ 2 , … , ξ 𝑠 2 1 , … , ξ 𝑠 1 which can locally be written associative unital ℝ -algebras, whose degree 0 term is C ∞ (M) and endowed with a sheaf C ∞ (ℳ) of ℕ -graded commutative dimension (𝑞; 𝑠 1 , … , 𝑠 𝑜 ) is a smooth 𝑞 -dimensional manifold M An N-manifold or ℕ -graded manifold ℳ of degree 𝑜 and Positively graded manifolds 2-algebroids Overview of the geometrisation Courant of Sheffield and degenerate Courant algebroids M. Jotz Lean (Degenerate) The University Lie 2 -algebroids Dorfman 2-representations Poisson [2] -manifolds Poisson Lie 2-algebroids 6 / 50 C ∞ (ℳ) U = C ∞ (U) [ξ 1 1 , ξ 1 2 , … , ξ 1 𝑜 ] with 𝑠 1 + … + 𝑠 𝑜 graded commutative generators ξ 𝑘 𝑗 of degree 𝑗 for
Poisson Lie 2-algebroids local generators of degree 𝑗 , for 𝑗 = 1, … , 𝑜 . E −1 [−1] ⊕ … ⊕ E −𝑜 [−𝑜] , which has local basis sections of E −𝑗 noncanonical manner) to the split [𝑜] -manifold (𝑞; 𝑠 1 , … , 𝑠 𝑜 ) , there exist smooth vector bundles E −1 , E −2 , … , E −𝑜 For any ℕ -graded manifold ℳ of degree 𝑜 and dimension Positively graded manifolds are noncanonically split geometrisation Overview of the algebroids Courant (Degenerate) 2-algebroids Poisson Lie [2] -manifolds Poisson 2-representations Dorfman Lie 2 -algebroids of Sheffield The University M. Jotz Lean algebroids Courant and degenerate 7 / 50 of ranks 𝑠 1 , … , 𝑠 𝑜 over M such that ℳ is isomorphic (in a ∗ as
Poisson Lie 2-algebroids −2 ) for If ℳ is an ℕ -graded manifold of base M and of degree 2 , then a vector bundle E over M . In particular, if ℳ is an ℕ -graded manifold of base M and of Degree 1 and degree 2 cases geometrisation Overview of the algebroids Courant 2-algebroids (Degenerate) Poisson Lie The University and degenerate Courant algebroids [2] -manifolds M. Jotz Lean of Sheffield Lie 2 -algebroids Dorfman 2-representations Poisson 8 / 50 degree 1 , then C ∞ (ℳ) is (canonically) isomorphic to Γ(⋀ • E ∗ ) for C ∞ (ℳ) is (noncanonically) isomorphic to Γ(⋀ • E ∗ −1 ⊗ S • E ∗ two vector bundles E −1 and E −2 over M .
Poisson Lie 2-algebroids −2 ) for If ℳ is an ℕ -graded manifold of base M and of degree 2 , then a vector bundle E over M . In particular, if ℳ is an ℕ -graded manifold of base M and of Degree 1 and degree 2 cases geometrisation Overview of the algebroids Courant 2-algebroids (Degenerate) Poisson Lie The University and degenerate Courant algebroids [2] -manifolds M. Jotz Lean of Sheffield Lie 2 -algebroids Dorfman 2-representations Poisson 8 / 50 degree 1 , then C ∞ (ℳ) is (canonically) isomorphic to Γ(⋀ • E ∗ ) for C ∞ (ℳ) is (noncanonically) isomorphic to Γ(⋀ • E ∗ −1 ⊗ S • E ∗ two vector bundles E −1 and E −2 over M .
Poisson Lie (Degenerate) satisfies graded Leibniz and graded Jacobi identities. [ϕ, ψ] = ϕψ − (−1) |ϕ||ψ| ψϕ is graded skew-symmetric and The Lie bracket on graded vector fields, defined by for a homogeneous element ξ ∈ C ∞ (ℳ) . |ϕ(ξ)| = 𝑘 + |ξ| graded derivation ϕ of C ∞ (ℳ) such that Let ℳ be an [𝑜] -manifold. A vector field of degree 𝑘 on ℳ is a Graded vector fields geometrisation Overview of the algebroids Courant 2-algebroids 2-algebroids Poisson Lie [2] -manifolds Poisson 2-representations Dorfman Lie 2 -algebroids of Sheffield The University M. Jotz Lean algebroids Courant and degenerate 9 / 50
Poisson Lie (Degenerate) satisfies graded Leibniz and graded Jacobi identities. [ϕ, ψ] = ϕψ − (−1) |ϕ||ψ| ψϕ is graded skew-symmetric and The Lie bracket on graded vector fields, defined by for a homogeneous element ξ ∈ C ∞ (ℳ) . |ϕ(ξ)| = 𝑘 + |ξ| graded derivation ϕ of C ∞ (ℳ) such that Let ℳ be an [𝑜] -manifold. A vector field of degree 𝑘 on ℳ is a Graded vector fields geometrisation Overview of the algebroids Courant 2-algebroids 2-algebroids Poisson Lie [2] -manifolds Poisson 2-representations Dorfman Lie 2 -algebroids of Sheffield The University M. Jotz Lean algebroids Courant and degenerate 9 / 50
Poisson Lie 2-algebroids together with a homological vector field on it. A Lie 𝑜 -algebroid is a pair (ℳ, ) of a positively graded manifold [, ] = 2 ∘ = 0. vector field of degree 1 that commutes with itself A homological vector field on a graded manifold ℳ is a graded Homological vector fields geometrisation Overview of the algebroids Courant (Degenerate) Poisson Lie 2-algebroids [2] -manifolds Poisson 2-representations Dorfman Lie 2 -algebroids of Sheffield The University M. Jotz Lean algebroids Courant and degenerate 10 / 50
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