Abstract
Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper, we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.
Highlights
Symmetry reduction plays an important role in theoretical classical mechanics and quantum physics, and its various mathematical formulations have been studied extensively during the last half century
The quantized system corresponds to a deformation of the commutative algebra of functions such that the Poisson bracket gets deformed into the commutator of the possibly non-commutative deformed algebra. This procedure relies on a classical principle stating that deformations of mathematical objects are governed by associated differential graded Lie algebras (DGLAs)
Formal deformations of an associative algebra A in the sense of Gerstenhaber [22] are given by formal Maurer–Cartan elements of the associated Hochschild DGLA C(A ), where two such deformations are considered to be equivalent if they lie in the same orbit of the action of the canonically associated gauge group
Summary
Symmetry reduction plays an important role in theoretical classical mechanics and quantum physics, and its various mathematical formulations have been studied extensively during the last half century. Following the abovementioned classical principle, we introduce the notion of coisotropic DGLA and we study formal deformations of the corresponding Maurer–Cartan elements. This allows us to define a deformation functor and to prove that the deformation functor commutes with reduction, in the sense that at least an injective natural transformation exists, see Theorem 3.14. Applying these techniques to the case of the coisotropic Hochschild complex of a coisotropic algebra we prove that the existence and uniqueness of formal deformations of coisotropic algebras are obstructed by its associated coisotropic Hochschild cohomology, see Theorem 4.19, Theorem 4.20. Some examples of formal deformations of coisotropic algebras from geometry are given
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