報告題目：Lie 2-algebras from geometric structures
報 告 人：劉張炬 (北京大學，河南大學)
報告時間：2022年5月13日 14:00-16:00
報告地點：騰訊會議 ID：347-371-151
加入會議：https://meeting.tencent.com/dm/rx9v5zMU7xnV
報告摘要: 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.
報告人簡介：劉張炬，北京大學/河南大學教授，1982年1月本科畢業于吉林大學，1986年6月在北京大學獲得博士學位。1999年獲得國家傑出青年基金。主要從事數學物理、Poisson幾何等領域的研究。曾入選教育部跨世紀優秀人才支持計劃，2004年獲得教育部自然科學一等獎。
