On Mizuhara's construction for endomorphs (Q6643488)

An algebra \(A\) is right-symmetric (pre-Lie) if the associator \((x, y, z):= (xy)z - x(yz)\) on \(A\) is right-symmetric, i.e., symmetric in the right two elements: \((x, y, z)=(x, z, y).\) The main non-associative examples of right-symmetric algebras are Novikov and assosymmetric algebras. The paper aims to study simplicity questions on the Mizuhara construction of endomorphs.\N\NLet us remember the original Mizuhara construction, firstly presented in [\textit{A. Mizuhara}, Sci. Math. Jpn. 57, No. 2, 325--337 (2003; Zbl 1043.17019)]. Let \(A\) be a right-symmetric algebra over a field \(\mathbb F\). A symmetric bilinear form \(H(\cdot, \cdot)\) on \(A\) with values in \(\mathbb F\) is a Hessian provided that \N\[\NH(xy, z) - H(x, yz) = H(xz, y) - H(x, zy).\N\]\NConsider an element \(u\) such that \(u^2 = \varepsilon u,\) where \(\varepsilon \in \{0, 1\}\) and \(u \in A\). A Hessian \(H\) and a derivation \(D\) on \(A\) are \(\varepsilon\)-consistent if \N\[\N\varepsilon H(x, y) = H(D(x), y) + H(x, D(y)).\N\]\NConsider the one-dimensional extension of \(A\) by \(\mathbb F u\) and the \(\varepsilon\)-consistent pair \((H, D)\) with the product defined by the rules \N\[\Nu \cdot u = \varepsilon u,\ u \cdot x = 0, \ x \cdot u = D(x), \ x \cdot y = xy + H(x, y)u.\N\]\NDenote the so-obtained algebra by \(A(H, D).\) The algebra \(A(H, D)\) is the Mizuhara \(\varepsilon\)-extension of \(A\). The Mizuhara construction of a right-symmetric algebra gives a right-symmetric algebra.\N\NLet us introduce the endomorph construction for an algebra \(A\). Namely, we consider the direct sum \(E(A)=A \oplus \mathrm{End}(A)\) and define the new multiplication as follows: \N\[\N\mathrm{A}\cdot a = a\mathrm{A} +[\mathrm{A}, R_a],\text{ for }a\in A\text{ and }\mathrm{A}\in \mathrm{A}.\N\]\N\(A\) and \(\mathrm{End}(A)\) are subalgebras in \(E(A)\) and \(A\) is the standard right module over \(\mathrm{End}(A).\)\N\NIt is shown that the Mizuhara construction of the endomorph of a right-symmetric algebra gives almost simple algebras, which are used to construct new examples of simple right-symmetric algebras.