On a generalized Mizuhara construction (Q6610184)

An algebra \(A\) is left-symmetric (pre-Lie) if the associator \((x, y, z):= (xy)z - x(yz)\) on \(A\) is left-symmetric, i.e., symmetric in the first two elements: \((x, y, z)=(y, x, z).\) The main non-associative examples of left-symmetric algebras are Novikov and assosymmetric algebras. The paper aims to find some new examples of simple left-symmetric algebras due to (generalized) Mizuhara construction, firstly presented in [\textit{A. Mizuhara}, Sci. Math. Jpn. 57, No. 2, 325--337 (2003; Zbl 1043.17019)].\N\NLet us remember the original Mizuhara construction. Let \(A\) be a left-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\[\NH(xy, z) - H(x, yz) = H(yx, z) - H(y, xz).\N\]\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\[\N\varepsilon H(x, y) = H(Dx, y) + H(x, Dy).\N\]\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\[\Nu \cdot u = \varepsilon u,\ u \cdot x = D(x), \ x \cdot u = 0, \ x \cdot y = xy + H(x, y)u.\N\]\NDenote the so-obtained algebra by \(A(H, D).\) The algebra \(A(H, D)\) is the Mizuhara \(\varepsilon\)-extension of \(A\). As noted by Mizuhara, \(A(H, D)\) is a left-symmetric algebra. Fix \(\varepsilon \in \{0, 1\}.\) Let \(A_i(H_i, D_i)\) be some Mizuhara \(\varepsilon\)-extensions of left-symmetric algebras \(A_i\), with \(i = 1, 2.\) Consider the direct sum \(A = A_1 \oplus A_2\) of \(A_1\) and \(A_2\) (as ideals). Denote by \(D\) and \(H\) the derivation and Hessian of \(A\) defined by the rules \(D|_{A_i} = D_i,\) \ \(H|_{A_i} = H_i,\) \ \(H (A_1, A_2)=0,\) \ \(i = 1, 2.\) The algebra \(A (H, D)\) is the direct Mizuhara extension of \(A_1\) and \(A_2.\) Call an algebra \(A\) simply indecomposable if \(A\) is not the direct Mizuhara extension of two simple algebras. An algebra \(A (H, D)\) is extensible simple provided that \(A\) is either simple or its neither proper ideal coincides with \(A,\) has codimension \(1,\) and lacks the principal \(\varepsilon\)-potent.\N\NThe main results of the present paper are as follows.\N\NTheorem 2.1. Let \(A\) be the direct Mizuhara extension of some algebras \(A:= A_1(H_1, D_1)\) and \(B := A_2(H_2, D_2).\) The algebra \(A\) is simple if and only if one of the algebras \(A\) and \(B\) is simple, and the other is extensible simple.\N\NTheorem 3.1. Every Mizuhara \(0\)-extension of \(M_n(\mathbb F),\) with \(n \geq 2,\) is a local extensible simple algebra for every nonzero derivation \(D\) and every Hessian \(H = \alpha \ Tr\). The unique proper ideal of such extension is a simple nonassociative indecomposable left-symmetric algebra \N\[\NM_n(\mathbb F)_u \ = \ \langle A + tr(A)u : A \in M_n(\mathbb F) \rangle.\N\]\N\textit{D. Burde} [Manuscr. Math. 95, No. 3, 397--411 (1998; Zbl 0907.17008)] introduced a class of simple left-symmetric algebras named as \(I_n\) (\(n \geq2\)). The multiplication table of \(I_n\) is given by the formulas \N\[\Ne_n \cdot e_n = 2e_n,\quad e_n \cdot e_j = e_j ,\ e_j \cdot e_j = e_n, \ j = 1, \ldots ,n - 1.\N\]\NTheorem 4.1. Every Mizuhara \(0\)-extension of \(I_n\) is a local algebra for every nonzero derivation \(D\) such that \(-1 \notin \mathrm{Spec} D\) and for every Hessian \(H\) on \(I_n\). Under the given hypothesis on \(D\), the unique proper ideal of the extension is a simple nonassociative left-symmetric algebra \(I_u = \langle e_n + u\rangle \oplus J_n.\) Every direct Mizuhara extension that gives \(I_u\) is an extension of two algebras with the zero product. In particular, \(I_u\) is simply indecomposable.\N\NSection 5 is dedicated to a generalization of the Mizuhara construction. The author replaced the one-dimensional vector space \(\langle u \rangle\) from the original construction with another left-symmetric algebra. As it follows, the present new construction gives a new left-symmetric algebra [Theorem 5.1]. After that, as a particular case of this new construction, the author defines the following construction.\N\NLet \(A\) be an associative commutative algebra with nonzero derivation \(d\) over a field \(\mathbb F.\) Let \(B:= \overline{A}\) be an isomorphic copy of \(A\) (as a vector space) with the classical left-symmetric product \(\overline{x} \cdot \overline{ y} = \overline{xd(y)}\). Define the mapping \(D: B^{(-)} \to\mathrm{Der}(A)\) by the rule \(D(a) = ad.\) Denote the so-obtained algebra \(A \oplus \overline{A}\) with product \N\[\Nb\cdot x = D(b)(x),\quad x \cdot b = 0,\quad x \cdot y = xy + \overline{ xy}\N\]\Nby \(A_d\) and call \(A_d\) the Witt double of \(A.\) By Theorem 5.1, \(A_d\) is a left-symmetric algebra.\N\NTheorem 5.2. Let \(A\) be a unital associative commutative algebra with nonzero derivation \(d\) over a field \(\mathbb F.\) The algebra \(A_d\) is simple if and only if \(A\) is \(d\)-simple.\N\NThe paper ends by defining a generalization of the last construction, named the generalized Witt double, and proving a Theorem similar to Theorem 5.2 for the generalized Witt double.