Seminar talk, 12 April 2017: Difference between revisions
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| abstract = For an arbitrary manifold <math>M</math>, we consider supermanifolds <math>\Pi TM</math> and <math>\Pi T^*M</math>, where <math>\Pi</math> is the parity reversion functor. The space <math>\Pi T^*M</math> possesses canonical odd Schouten bracket and space <math>\Pi TM</math> possess canonical de Rham differential <math>d</math>. An arbitrary even function <math>P</math> on <math>\Pi T^*M</math> such that <math>[P,P]=0</math> induces a homotopy Poisson bracket on <math>M</math>, a differential, <math>d_P</math> on <math>\Pi T^*M</math>, and higher Koszul brackets on <math>\Pi TM</math>. (If <math>P</math> is fiberwise quadratic, then we arrive at standard structures of Poisson geometry.) Using the language of <math>Q</math>-manifolds and in particular of Lie algebroids, we study the interplay between canonical structures and structures depending on <math>P</math>. Then using just recently invented theory of thick morphisms we construct a non-linear map between the <math>L_{\infty}</math> algebra of functions on <math>\Pi TM</math> with higher Koszul brackets and the Lie algebra of functions on <math>\Pi T^*M</math> with the canonical odd Schouten bracket. | | abstract = For an arbitrary manifold <math>M</math>, we consider supermanifolds <math>\Pi TM</math> and <math>\Pi T^*M</math>, where <math>\Pi</math> is the parity reversion functor. The space <math>\Pi T^*M</math> possesses canonical odd Schouten bracket and space <math>\Pi TM</math> possess canonical de Rham differential <math>d</math>. An arbitrary even function <math>P</math> on <math>\Pi T^*M</math> such that <math>[P,P]=0</math> induces a homotopy Poisson bracket on <math>M</math>, a differential, <math>d_P</math> on <math>\Pi T^*M</math>, and higher Koszul brackets on <math>\Pi TM</math>. (If <math>P</math> is fiberwise quadratic, then we arrive at standard structures of Poisson geometry.) Using the language of <math>Q</math>-manifolds and in particular of Lie algebroids, we study the interplay between canonical structures and structures depending on <math>P</math>. Then using just recently invented theory of thick morphisms we construct a non-linear map between the <math>L_{\infty}</math> algebra of functions on <math>\Pi TM</math> with higher Koszul brackets and the Lie algebra of functions on <math>\Pi T^*M</math> with the canonical odd Schouten bracket. | ||
This is the joint work with T.Voronov. | This is the joint work with T. Voronov. | ||
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Latest revision as of 14:46, 3 April 2017
Speaker: Hovhannes Khudaverdian
Title: Thick morphisms and higher Koszul brackets
Abstract:
For an arbitrary manifold , we consider supermanifolds and , where is the parity reversion functor. The space possesses canonical odd Schouten bracket and space possess canonical de Rham differential . An arbitrary even function on such that induces a homotopy Poisson bracket on , a differential, on , and higher Koszul brackets on . (If is fiberwise quadratic, then we arrive at standard structures of Poisson geometry.) Using the language of -manifolds and in particular of Lie algebroids, we study the interplay between canonical structures and structures depending on . Then using just recently invented theory of thick morphisms we construct a non-linear map between the algebra of functions on with higher Koszul brackets and the Lie algebra of functions on with the canonical odd Schouten bracket.
This is the joint work with T. Voronov.