Question about the parity of the ghost number operator in BRST quantization Given a Lie algebra $[K_i,K_j]=f_{ij}^k K_k$, and ghost fields satisfying the anticommutation relations $\\{c^i,b_j\\}=\delta_j^i$, the ghost number operator is then $U=c^ib_i$ (duplicate indices are summed). The question raised in proving the BRST operator raising the ghost number by 1, given in the Example 6.1 on the page 116-118 of the book, String theory demystified. On the second and the third lines of the formula derivation of $UC^iK_i$ $UC^iK_i=...=c^iK_i-c^i\displaystyle\sum_rc^rb_rK_i=c^iK_i+c^iK_i\displaystyle\sum_rc^rb_r=...$ where we need $c^r U K_r =-c^r K_r U$ The change of sign above is not manifestly obvious to me.

Never mind. I find the second line cheated me. It should be $UC^iK_i=...=c^iK_i+c^i\displaystyle\sum_rc^rb_rK_i=c^iK_i+c^iK_i\displaystyle\sum_rc^rb_r=...$

Furthermore, in the proof followed in this example, there a few typos need to be corrected. But the final result is right. The BRST operator raises the ghost number by 1!