caa a22 4500 37892091X CHVBK 20180305123605.0 cr unu---uuuuu 161128e200406 xx s 000 0 eng 10.1515/GMJ.2004.255 doi (NATIONALLICENCE)gruyter-10.1515/GMJ.2004.255 Giunashvili Z. A. Razmadze Mathematical Institute, Georgian Academy of Sciences, 1, M. Aleksidze St., Tbilisi 0193, Georgia Department of Applied Mathematics, Georgian Technical University, 77, Kostava St., Tbilisi 0175, Georgia. E-mail: zaqro@gtu.edu.ge Noncommutative Symplectic Foliation, Bott Connection and Phase Space Reduction [Elektronische Daten] [Z. Giunashvili] We investigate the geometric, algebraic and homological properties of Poisson structures on smooth manifolds and introduce noncommutative foundations of these structures for associative Poisson algebras. Noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold, symplectic foliation and symplectic leaf for associative Poisson algebras are given. These structures are considered for the case of the endomorphism algebra of a vector bundle, and a full description of the family of Poisson structures for this algebra is given. An algebraic construction of the reduction procedure for degenerate noncommutative Poisson structures is developed. A noncommutative generalization of Bott connection on foliated manifolds is introduced using the notions of a noncommutative submanifold and a quotient manifold. This definition is applied to degenerate noncommutative Poisson algebras, which allows us to consider Bott connection not only for regular but also for singular Poisson structures. © Heldermann Verlag Noncommutative geometry nationallicence Poisson structure nationallicence endomorphism algebra nationallicence Schouten-Nijenhuis bracket nationallicence Bott connection nationallicence Georgian Mathematical Journal Walter de Gruyter GmbH & Co. KG 11/2(2004-06), 255-282 1072-947X 11:2<255 2004 11 GMJ https://doi.org/10.1515/GMJ.2004.255 text/html Onlinezugriff via DOI 1 research article jats NATIONALLICENCE 856 40 https://doi.org/10.1515/GMJ.2004.255 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Giunashvili Z. A. Razmadze Mathematical Institute, Georgian Academy of Sciences, 1, M. Aleksidze St., Tbilisi 0193, Georgia Department of Applied Mathematics, Georgian Technical University, 77, Kostava St., Tbilisi 0175, Georgia. E-mail: zaqro@gtu.edu.ge NATIONALLICENCE 773 0- Georgian Mathematical Journal Walter de Gruyter GmbH & Co. KG 11/2(2004-06), 255-282 1072-947X 11:2<255 2004 11 GMJ CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-gruyter