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Sino-Russian Mathematics Center-JLU Colloquium(2022-010)- Lie 2-algebras from geometric structures

发表于: 2022-05-06   点击: 

报告题目:Lie 2-algebras from geometric structures

报 告 人:刘张炬 (北京大学,河南大学)

报告时间:2022年5月13日 14:00-16:00

报告地点:腾讯会议 ID:347-371-151


报告摘要: The notion of Lie 2-algebras is introduced as categorification of Lie algebras, Which is one of the fundamental objects in higher Lie theory and has close connection with strongly homotopy Lie algebras. The Lie 2-algebra structure has enjoyed significant applications in both geometry and mathematical physics. Strict Lie 2-algebras are equivalent to Lie algebra crossed modules, which are classified by the third cohomology of a Lie algebra.

In this talk, we’ll review several Lie 2-algebras that come from geometric structures, namely, 2-plectic manifolds; Courant algebroids; homotopy Poisson manifolds and affine structures on Lie groupoids. A 2-plectic structure on a manifold is a nondegenerate closed 3-form. There is a Lie 2-algebra structure on functions and Hamiltonian 1-forms of a 2-plectic manifold A Courant algebroid is a vector bundle together with a bilinear form, a skew-symmetric bracket and an anchor map. The bracket satisfies the Jacobi identity up to a coboundary, which generates a Lie 2-algebra on the section space of the bundle and functions on the base manifold. Parallel to the fact that there is a one-to-one correspondence between Lie algebra structures on a vector space and linear Poisson structures on the dual space, there is a one-to-one correspondence between Lie 2-algebra structures on a 2-vector space and linear homotopy Poisson structures on the dual 2-vector space. On a Lie groupoid, vector fields that are compatible with the groupoid multiplication are called multiplicative. Multiplicative vector fields with the Schouten bracket form a Lie algebra, which is not invariant under the Morita equivalence of Lie groupoids. To define vector fields on a differentiable stack, one needs to extend the Lie algebra to a Lie 2-algebra formed by affine vector fields on a Lie groupoid, which is Morita invariant.