Constrained Poisson algebras and strong homotopy representations. (English) Zbl 0669.18009
Let A be a Poisson algebra, i.e., an abelian associative algebra equipped with an additional multiplication (f,g)\(\to \{f,g\}\) (the Poisson bracket \(\{\), \(\})\) which is anticommutative and acts as a derivation with respect to the abelian product: \(\{g,gh\}=\{f,g\}h+f\{g,h\}.\)
The author justifies, in terms of the algebra A, the method developed by E. S. Fradkin, I. A. Batalin, G. A. Vilkoviskij, and M. Henneaux for computing the ad I-invariant functions in A/I without going through the quotient, where I is the ideal generated by the constraints of A with respect to the abelian multiplication, supposed closed under Poisson bracket. He uses techniques of homological perturbation theory [V. K. A. M. Gugenheim and J. P. May, Mem. Am. Math. Soc. 142, 94 p. (1974; Zbl 0292.55019); V. K. A. M. Gugenheim, J. Pure Appl. Algebra 25, 197-205 (1982; Zbl 0487.55003); V. K. A. M. Gugenheim and J. D. Stasheff, Bull. Soc. Math. Belg., Ser. A 38, 237-246 1986; Zbl 0639.55008)].
The author justifies, in terms of the algebra A, the method developed by E. S. Fradkin, I. A. Batalin, G. A. Vilkoviskij, and M. Henneaux for computing the ad I-invariant functions in A/I without going through the quotient, where I is the ideal generated by the constraints of A with respect to the abelian multiplication, supposed closed under Poisson bracket. He uses techniques of homological perturbation theory [V. K. A. M. Gugenheim and J. P. May, Mem. Am. Math. Soc. 142, 94 p. (1974; Zbl 0292.55019); V. K. A. M. Gugenheim, J. Pure Appl. Algebra 25, 197-205 (1982; Zbl 0487.55003); V. K. A. M. Gugenheim and J. D. Stasheff, Bull. Soc. Math. Belg., Ser. A 38, 237-246 1986; Zbl 0639.55008)].
Reviewer: U.Cattaneo
MSC:
| 18G10 | Resolutions; derived functors (category-theoretic aspects) |
| 17B55 | Homological methods in Lie (super)algebras |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
| 81T08 | Constructive quantum field theory |
References:
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| [2] | I. A. Batalin and G. A. Vilkovisky, Existence theorem for gauge algebra, J. Math. Phys. 26 (1985), no. 1, 172 – 184. · doi:10.1063/1.526780 |
| [3] | I. A. Batalin and G. A. Vilkovisky, Quantization of gauge theories with linearly dependent generators, Phys. Rev. D (3) 28 (1983), no. 10, 2567 – 2582. , https://doi.org/10.1103/PhysRevD.28.2567 I. A. Batalin and G. A. Vilkovisky, Erratum: ”Quantization of gauge theories with linearly dependent generators”, Phys. Rev. D (3) 30 (1984), no. 2, 508. · doi:10.1103/PhysRevD.30.508 |
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| [8] | V. K. A. M. Gugenheim, On a perturbation theory for the homology of the loop-space, J. Pure Appl. Algebra 25 (1982), no. 2, 197 – 205. · Zbl 0487.55003 · doi:10.1016/0022-4049(82)90036-6 |
| [9] | V. K. A. M. Gugenheim and J. Peter May, On the theory and applications of differential torsion products, American Mathematical Society, Providence, R.I., 1974. Memoirs of the American Mathematical Society, No. 142. · Zbl 0292.55019 |
| [10] | V. K. A. M. Gugenheim and J. D. Stasheff, On perturbations and \?_{\infty }-structures, Bull. Soc. Math. Belg. Sér. A 38 (1986), 237 – 246 (1987). · Zbl 0639.55008 |
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