A note on Lie and Jordan structures of Leavitt path algebras

Let $L_K(E)$ be the Leavitt path algebra of a directed graph $E$ over a field $K$. In this paper, we determine $E$ and $K$ for the Lie algebra $\mathbf{K}{L_K(E)}$ and the Jordan algebra $\mathbf{S}{L_K(E)}$ arising from $L_K(E)$ with respect to the standard involution to be solvable.

💡 Research Summary

The paper investigates the Lie and Jordan structures that arise from the standard involution on a Leavitt path algebra (L_K(E)) of a directed graph (E) over a field (K). For any associative algebra (A) equipped with an involution (\star), the set of skew‑symmetric elements (K_A={x\mid x^\star=-x}) forms a Lie subalgebra of (A^{-}) (the Lie algebra defined by the commutator), while the set of symmetric elements (S_A={x\mid x^\star=x}) forms a Jordan subalgebra of (A^\circ) (the Jordan algebra defined by the symmetrized product). The authors denote these two substructures for a Leavitt path algebra by (\mathbf K_{L_K(E)}) and (\mathbf S_{L_K(E)}), respectively, and ask for which pairs ((E,K)) these Lie and Jordan algebras are solvable.

The technical heart of the work lies in a detailed analysis of the Lie solvability of the skew‑symmetric Lie algebra (K_{M_n(A)}) when the involution on a unital algebra (A) is extended to the matrix algebra (M_n(A)) by the usual transpose‑type rule ((a_{ij})^\star=(a_{ji}^\star)). Proposition 3.1 establishes a complete picture:

• If (n\ge3) then (K_{M_n(A)}) is never Lie solvable, regardless of the characteristic of (K) or the nature of the involution.
• For (n=2) and (\operatorname{char}K=2), (K_{M_2(A)}) is Lie solvable. If the involution on (A) is trivial (the transpose), the derived series terminates after two steps; if the involution is non‑trivial, the derived series may require three steps, but solvability still holds.
• If (\operatorname{char}K\neq2) and the involution is trivial, the commutator subalgebra (