Derivations

A derivation \(D\) of an algebra \(A\) is a linear operator that satisfies \(D(d_1d_2)=d_1D(d_2)+D(d_1)d_2\), for every \(d_1,d_2\in A\). If a derivation is of the form \(D(d)=[d,f]=df-fd\) for some fixed \(f\in A\), we say that \(D\) is an inner derivation.

Function as.der() returns a derivation with as.der(f)(g)=fg-gf.

Usage

as.der(S)

Arguments

S

Weyl object

Value

Returns a function, a derivation

Author

Robin K. S. Hankin

Examples

(o <- rweyl(n=2,d=2))
#> A member of the Weyl algebra:
#>   x  y dx dy     val
#>   2  0  2  2  =    2
#>   0  2  1  2  =    1
(f <- as.der(o))
#> function (x) 
#> {
#>     S * x - x * S
#> }
#> <bytecode: 0x5647be3d4d00>
#> <environment: 0x5647be3d4910>

d1 <-rweyl(n=1,d=2)
d2 <-rweyl(n=2,d=2)

f(d1*d2) == d1*f(d2) + f(d1)*d2 # should be TRUE
#> [1] TRUE