Varieties satisfying semigroup identities: Algebras over a finite field and rings

We study varieties of associative algebras over a finite field and varieties of associative rings satisfying semigroup or adjoint semigroup identities. We characterize these varieties in terms of “forbidden algebras” and discuss some corollaries of the characterizations.

GENERATORS OF DETAILED BALANCE QUANTUM MARKOV SEMIGROUPS

For a quantum Markov semigroup [Formula: see text] on the algebra [Formula: see text] with a faithful invariant state ρ, we can define an adjoint [Formula: see text] with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterizing adjoints [Formula: see text] that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators H, Lk in the Gorini–Kossakowski–Sudarshan–Lindblad representation [Formula: see text] of the generator of [Formula: see text]. We study the adjoint semigroup with respect to both scalar products 〈a, b〉 = tr (ρa*b) and 〈a, b〉 = tr (ρ1/2a*ρ1/2 b).

NILPOTENT LINEAR SEMIGROUPS

We characterize the structure of linear semigroups satisfying certain global and local nilpotence conditions and deduce various Engel-type results. For example, using a form of Zel'manov's solution of the restricted Burnside problem we are able to show that a finitely generated residually finite group is nilpotent if and only if it satisfies a certain 4-generator property of semigroups we call WMN. Methods of linear semigroups then allow us to prove that a linear semigroup is Mal'cev nilpotent precisely when it satisfies WMN. As an application, we show that a finitely generated associative algebra is nilpotent when viewed as a Lie algebra if and only if its adjoint semigroup is WMN.