2025/12/15 by Ilja Gogić, Gogić, Ilja, Mateo Tomašević +1
Mathematics · Computer Science · #Advanced Topics in Algebra #Matrix Theory and Algorithms #Advanced Algebra and Logic
paper · pdf · doi:10.48550/arxiv.2512.13085
Let Mn(\mathbbF) denote the algebra of n × n matrices over an algebraically closed field \mathbbF of characteristic different from 2. For n ≥ 2, we classify all maps ϕ: Mn(\mathbbF) → Mn(\mathbbF) satisfying the mixed Jordan-power identity ϕ(Ak ∘ B) = ϕ(A)k ∘ ϕ(B), for all A,B ∈ Mn(\mathbbF), where ∘ denotes the (normalized) Jordan product A ∘ B := \tfrac12(AB + BA) and k ∈ ℕ. We show that every such map is either constant, taking a fixed (k+1)-potent value, or there exist an invertible matrix T ∈ Mn(\mathbbF), a ring monomorphism ω: \mathbbF → \mathbbF, and a k-th root of unity ε ∈ \mathbbF such that ϕ takes one of the forms ϕ(X) = ε T ω(X) T-1 or ϕ(X) = ε T ω(X)t T-1, where ω(X) denotes the matrix obtained by applying ω entrywise to X, and (⋅)t denotes matrix transposition. In particular, every nonconstant solution is necessarily additive. The classification relies fundamentally on the preservation of (k+1)-potents and their intrinsic structural properties.