Poisson bracket Lie n-algebra

The Lie n-algebra that generalizes the Poisson bracket from symplectic geometry to n-plectic geometry: the *Poisson bracket $L_\infty$-algebra of local observables* in higher prequantum geometry.

The detailed discussion is currently still here at *n-plectic geometry*.

**slice-automorphism ∞-groups in higher prequantum geometry**

cohesive ∞-groups: | Heisenberg ∞-group | $\hookrightarrow$ | quantomorphism ∞-group | $\hookrightarrow$ | ∞-bisections of higher Courant groupoid | $\hookrightarrow$ | ∞-bisections of higher Atiyah groupoid |
---|---|---|---|---|---|---|---|

L-∞ algebras: | Heisenberg L-∞ algebra | $\hookrightarrow$ | Poisson L-∞ algebra | $\hookrightarrow$ | Courant L-∞ algebra | $\hookrightarrow$ | twisted vector fields |

**higher and integrated Kostant-Souriau extensions**

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $\mathbb{G}$-principal ∞-connection)

$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)$

(extension are listed for sufficiently connected $X$)

The Poisson bracket $L_\infty$-algebras were proposed in

- Chris Rogers,
*$L_\infty$ algebras from multisymplectic geometry*, Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (arXiv:1005.2230, journal).

A comprehensive account is in

- Chris Rogers,
*Higher symplectic geometry*PhD thesis (2011) (arXiv:1106.4068)

An general account in higher geometry is in

Revised on March 30, 2013 15:04:09
by Urs Schreiber
(82.113.99.161)