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For a quantum Markov semigroup T on the algebra B(h) with a faithful invariant state ρ, we can define an adjoint T with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterising adjoints T that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators H,We study the adjoint semigroup with respect to both scalar products a, b = tr(ρa * b) and a, b = tr(ρ 1/2 a * ρ 1/2 b).

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We find the structure of generators of norm continuous quantum Markov semigroups on B(h) that are symmetric with respect to the scalar product tr(ρ 1/2 x * ρ 1/2 y) induced by a faithful normal invariant state invariant state ρ and satisfy two quantum generalisations of the classical detailed balance condition related with this non-commutative notion of symmetry: the socalled standard detailed balance condition and the standard detailed balance condition with an antiunitary time reversal.