TY - JOUR

T1 - Poincaré–Birkhoff–Witt isomorphisms and Kapranov dg-manifolds

AU - Laurent-Gengoux, Camille

AU - Stiénon, Mathieu

AU - Xu, Ping

N1 - Funding Information:
Research partially supported by NSF grants DMS-2001599 , DMS-1707545 , DMS-1406668 , DMS-1101827 , and NSA grant H98230-14-1-0153 .
Publisher Copyright:
© 2021

PY - 2021/8/27

Y1 - 2021/8/27

N2 - We prove that to every inclusion A↪L of Lie algebroids over the same base manifold M corresponds a Kapranov dg-manifold structure on A[1]⊕L/A, which is canonical up to isomorphism. As a consequence, Γ(Λ•A∨⊗L/A) carries a canonical L∞[1] algebra structure whose unary bracket is the Chevalley–Eilenberg differential dA∇Bott corresponding to the Bott representation of A on L/A and whose binary bracket is a cocycle representative of the Atiyah class of the Lie pair (L,A). To this end, we construct explicit isomorphisms of C∞(M)-coalgebras Γ(S(L/A))→∼[Formula Presented], which we elect to call Poincaré–Birkhoff–Witt maps. These maps admit a recursive characterization that allows for explicit computations. They generalize both the classical symmetrization map S(g)→U(g) of Lie theory and (the inverse of) the complete symbol map for differential operators. Finally, we prove that the Kapranov dg-manifold A[1]⊕L/A is linearizable if and only if the Atiyah class of the Lie pair (L,A) vanishes.

AB - We prove that to every inclusion A↪L of Lie algebroids over the same base manifold M corresponds a Kapranov dg-manifold structure on A[1]⊕L/A, which is canonical up to isomorphism. As a consequence, Γ(Λ•A∨⊗L/A) carries a canonical L∞[1] algebra structure whose unary bracket is the Chevalley–Eilenberg differential dA∇Bott corresponding to the Bott representation of A on L/A and whose binary bracket is a cocycle representative of the Atiyah class of the Lie pair (L,A). To this end, we construct explicit isomorphisms of C∞(M)-coalgebras Γ(S(L/A))→∼[Formula Presented], which we elect to call Poincaré–Birkhoff–Witt maps. These maps admit a recursive characterization that allows for explicit computations. They generalize both the classical symmetrization map S(g)→U(g) of Lie theory and (the inverse of) the complete symbol map for differential operators. Finally, we prove that the Kapranov dg-manifold A[1]⊕L/A is linearizable if and only if the Atiyah class of the Lie pair (L,A) vanishes.

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U2 - 10.1016/j.aim.2021.107792

DO - 10.1016/j.aim.2021.107792

M3 - Article

AN - SCOPUS:85111008927

VL - 387

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 107792

ER -