Automorphism Groups and Isomorphism Rigidity of Quantum Genetic Lotka–Volterra Algebras on M 2 (ℂ)

We study a one-parameter family of four-dimensional real commutative non-associative algebras arising as convolution algebras on the Hermitian dual of M 2 (ℂ). The family is induced by quantum genetic Lotka–Volterra operators, but the main results are structural algebra theorems. We prove time-layer rigidity: in the nondegenerate parameter range, two layers are isomorphic only when their time parameters coincide. Thus the family is a genuine algebraic deformation rather than a flow by algebra isomorphisms. For each fixed layer we classify the normalized real algebraic automorphism group and describe it by linear centralizers and semidirect products. Adding the order constraint determined by the quantum state space yields the genuine state-space automorphism group, a compact Lie group given by an orthogonal centralizer, or equivalently by a product of orthogonal groups on eigenspaces of the parameter operator. The results separate algebraic automorphisms, one-sided state-preserving monomorphisms and order-compatible automorphisms.