FFT-Filter frequency cutoff and negative frequencies When using a standard implementation for the FFT, for example the numpy implementation, the Fourier transformed output is in so called "standard" order. To quote the numpy implemenetation

If $A$ = fft($a$, $n$), [where $a$ is the input array and $n$ is the length of the FFT] then $A$[0] contains the zero-frequency term (the sum of the signal), which is always purely real for real inputs. Then $A$[1:n/2] contains the positive-frequency terms, and $A$[$n/2+1:$] contains the negative-frequency terms, in order of decreasingly negative frequency.
  [Odd frequencies analogous]

In short, the frequencies in the array slots are given by 
$$\begin{align}f &= [0, 1, ...,   n/2-1,     -n/2, ..., -1] / (d*n)\quad n\ \mathrm{even} \\
f &= [0, 1, ..., (n-1)/2, -(n-1)/2, ..., -1] / (d*n) \quad n \mathrm{\ odd} \end{align}$$
However, the FFT is sometimes also described using the formula (Ref.)
$$A_k = \sum_{m=0}^{n-1}a_m \exp\left\lbrace-2\pi\mathrm{i}\frac{mk}{n}\right\rbrace$$
When I want to implement a FFT-filter I have to cut of the frequencies after some cutoff frequency (or multiply all frequencies with some window function).
My question is how to treat these negative frequencies? From the equation above it seems as if I can treat them either as negative frequency components or as higher frequency components above the Nyquist frequency.
Suppose, I want to implement a simple filter that removes frequencies above some cutoff $\omega_c$. Do I have to cut of the negative frequencies below $-\omega_c$ or all negative frequencies if $\omega_c$ is smaller than the Nyquist frequency?
To be more specific, consider $n$ even, $k = \frac{n}{2}-1$ the next frequency component according to "standard" order corresponds to $-\frac{n}{2}\cdot\frac{1}{n}$ whereas the next component of the formula is simply $\frac{n}{2}\cdot\frac{1}{n}$ which is obviously not the same.
 A: 
From the equation above it seems as if I can treat them either as negative frequency components or as higher frequency components above the Nyquist frequency.

They are definitely not higher frequency components. This is just a way of ordering negative frequencies. No information is gained/loss if you've represented the negative frequencies before the positive ones. Actually MATLAB does that with its fftshift().function.

Do I have to cut of the negative frequencies below $-\omega_c$ or all negative frequencies if $\omega_c$  is smaller than the Nyquist frequency?

This is done automatically. Consider the ideal Low pass filter, namely
$$h(t) = \text{sinc}(2\omega_ct)$$
Sampling it at a rate of Nyquist, the above filter will (more or less) cut off all frequencies outside the range $[-\omega_c,+\omega_c]$.

To be more specific, consider $n$ even, $k = \frac{n}{2}-1$ the next frequency component according to "standard" order corresponds to $-\frac{n}{2}\cdot\frac{1}{n}$ whereas the next component of the formula is simply $\frac{n}{2}\cdot\frac{1}{n}$ which is obviously not the same.

The FFT according to the numPY would be (aligning with your formula)
\begin{align}FFT(x) &= [A_0, A_1, ...,  A_{ n/2-1},     A_{-n/2}, ...,A_{ -1}] / (d*n)\quad n\ \mathrm{even} \\
FFT(x) &= [A_0, A_1, ..., A_{(n-1)/2}, A_{-(n-1)/2}, ..., A_{-1}] / (d*n) \quad n \mathrm{\ odd} \end{align}
