  For a quantum Markov semigroup $\T$ on the algebra $\B$ with a faithful
invariant state $\rho$, we can define an adjoint $\widetilde{\T}$ with respect
to the scalar product determined by $\rho$. In this paper, we solve the open
problems of characterising adjoints $\widetilde{\T}$ that are also a quantum
Markov semigroup and satisfy the detailed balance condition in terms of the
operators $H,L_k$ in the Gorini Kossakowski Sudarshan Lindblad representation
$\Ll(x)=i[H,x] - {1/2}\sum_k(L^*_kL_k x-2L^*_kxL_k + xL^*_kL_k)$ of the
generator of $\T$. We study the adjoint semigroup with respect to both scalar
products $<a,b> = \tr(\rho a^* b)$ and $<a,b> = \tr(\rho^{1/2} a^*
\rho^{1/2}b)$.
