Cohomology of the Poisson Superalgebra on Spaces of Superdimension (2, n −)

Abstract

Under certain assumptions about the continuity of cochains, we study the cohomology spaces of a Poisson superalgebra realized on the space of smooth Grassmann-valued functions with compact support in R 2 . We find the zeroth, first, and second cohomology spaces in the adjoint representation in the case of a constant nondegenerate Poisson superbracket.

General form of the deformation of the Poisson superbracket on a (2, n)-dimensional superspace

“…Then Eq. (A4.4) reduces to The expression for d ad 2T 3 (x|f, g, h) was calculated in [4]: d ad 2T 3 (x|f, g, h) = {h(x), V 2 (x)Θ(x|∂ 2 f g − f ∂ 2 g)} + E x h(x)a({f, g}) + +∂ 2 h(x)Θ(x|V 1 {f, g}) + {h(x),ν(x|{f, g})}.…”

“…where I 1 (p, q, r) = ([p, q + r] 3 + [p + r, q] 3 − [p, q] 3 )r 4 r 3 + [p, q] 3 (p + q) 4 (p + q) 3 + cycle(p, q, r), I 2 (p, q, r) = −6r 4 r 3 [p, q] 2 [p, q] x + cycle(p, q, r), I 3 (p, q, r) = 2r 4 r 3 {6[p, q](p 2 q 1 q 2 + p 1 p 2 q 2 )r 1 − 3[p, q](p 2 q 2 1 + p 2 1 q 2 )r 2 + +(p 3 2 q 1 − p 1 q 3 2 + 3p 2 q 1 q 2 2 − 3p 1 p 2 2 q 2 )r 2 1 + 3(p 2 1 p 2 q 2 − p 2 q 2 1 q 2 )r 1 r 2 + +(p 2 q 3 1 − p 3 1 q 2 )r 2 2 } + cycle(p, q, r), I 4 (p, q, r) = (p 4 + q 4 )(p 3 + q 3 )[(q 3 1 q 2 2 − p 3 1 p 2 2 + 3p 1 q 2 1 q 2 2 − 3p 2 1 p 2 2 q 1 + +p 3 1 p 2 q 2 − p 2 q 3 1 q 2 + 3p 2 1 q 1 q 2 2 − 3p 1 p 2 2 q 2 1 + 3p 2 1 p 2 q 1 q 2 − 3p 1 p 2 q 2 1 q 2 + +p 2 2 q 3 1 − p 3 1 q 2 2 )r 2 + (p 1 + q 1 ) 2 (p 3 2 − q 3 2 )r 1 ] + cycle(p, q, r), I 5 (p, q, r) = (p 4 + q 4 )(p 3 + q 3 )[(p 1 + q 1 ) 2 (q 1 − p 1 )r 3 2 + +3(p 1 + q 1 ) 2 (p 2 − q 2 )r 1 r 2 2 + 3(p 1 + q 1 )(q 2 2 − p 2 2 )r 2 1 r 2 + +(p 2 + q 2 )(p 2 2 − q 2 2 )r 3 1 ] + cycle(p, q, r), I 6 (p, q, r) = −(p 4 + q 4 + r 4 )(p 3 + q 3 + r 3 )[p, q] 3 + cycle(p, q, r), I 7 (p, q, r) = (p 4 + q 4 + r 4 )(p 3 + q 3 + r 3 )(p 1 + q 1 + r 1 ) 2 {[p, q](3(p 2 + q 2 )r 2 − −(p 2 + q 2 + r 2 ) 2 )} + cycle(p, q, r), I 8 (p, q, r) = (p 4 + q 4 + r 4 )(p 3 + q 3 + r 3 ){(4p 2 q 2 r 2 1 − 2(p 2 + q 2 )r 2 1 r 2 + +4(p 1 + q 1 )r 1 r 2 2 + 2r 2 1 r 2 2 )[p, q] + +(2(p 2 2 + q 2 2 )r 2 1 + 4p 1 q 1 r 2 2 − (2p 1 p 2 + 4p 1 q 2 + +4p 2 q 1 + 2q 1 q 2 )r 1 r 2 )[p, q] ξ } + cycle(p, q, r).…”

“…The following Theorem is proved in [4]: Theorem 1.1. Let the bilinear mappings m 1 , m 3 , m 0 5 , m 1 5 , m 2 5 , and m 3 5 from (D…”

“…We solve this problem for the case n + = n − = 2 and show that in this case the Poisson superalgebra has an additional deformation comparing with other cases considered in [1] and in [2]. The proposed analysis is essentially based also on the results of the papers [3] by the authors, where the first and the second cohomology spaces with coefficients in the adjoint representation of the Poisson superalgebra are found.…”

“…It occurs that the case n − = 2, where Poisson superalgebra has an additional deformation, needs separate investigation, which will be provided in [5]. The proposed analysis is essentially based on the results of the papers [4] by the authors, where the second cohomology space with coefficients in the adjoint representation of the Poisson superalgebra was found, and [3] where the general form of the * -commutator in the case of a Poisson superalgebra of smooth compactly supported functions on R n + taking values in a Grassmann algebra was found for n + ≥ 4.…”