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- Title
Nambu–Poisson bracket on superspace.
- Authors
Abramov, Viktor
- Abstract
We propose an extension of n -ary Nambu–Poisson bracket to superspace ℝ n | m and construct by means of superdeterminant a family of Nambu–Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace ℝ n | m . We prove in the case of the superspaces ℝ n | 1 and ℝ n | 2 that our n -ary bracket, defined with the help of superdeterminant, satisfies the conditions for n -ary Nambu–Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov–Jacobi identity (fundamental identity). We study the structure of n -ary bracket defined with the help of superdeterminant in the case of superspace ℝ n | 2 and show that it is the sum of usual n -ary Nambu–Poisson bracket and a new n -ary bracket, which we call χ -bracket, where χ is the product of two odd degree smooth functions.
- Subjects
POISSON algebras; GRASSMANN manifolds; JACOBI'S condition; MATHEMATICAL transformations; ASSOCIATIVE algebras
- Publication
International Journal of Geometric Methods in Modern Physics, 2018, Vol 15, Issue 11, pN.PAG
- ISSN
0219-8878
- Publication type
Article
- DOI
10.1142/S0219887818501906