An $L_\infty$ algebra structure on polyvector fields

Details An $L_\infty$ algebra structure on polyvector fields An $L_\infty$ algebra structure on polyvector fields

2008-05-21

arxiv.org/abs/0805.3363v5

Mathematics

Quantum Algebra

21 pages, many eps figures, v5 completely rewritten and corrected

Scientific paper

In this paper we construct an $L_\infty$ structure on polyvector fields on a vector space $V$ over $\mathbb{C}$ where $V$ may be infinite-dimensional. We prove that the constructed $L_\infty$ algebra of polyvector fields is $L_\infty$ equivalent to the Hochschild complex of polynomial functions on $V$, even in the infinite-dimensional case. For a finite-dimensional space $V$, our $L_\infty$ algebra is equivalent to the classical Schouten-Nijenhuis Lie algebra of polyvector fields. For an infinite-dimensional $V$, it is essentially different. In particular, we get the higher obstructions for deformation quantization in infinite-dimensional case.

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