Algebras of length one (Q2078413)

The paper is about the study of the length of non-associative algebras. Roughly speaking, the notion of the length of finite-dimensional algebras can be considered as an analog of the nilpotency for non-nilpotent algebras or an analog of the Gelfand-Kirillov dimension for finite-dimensional algebras. The authors describe all unital finite-dimensional algebras with the length of \(1\). For more understanding of the results of the paper, we give the following definitions. \textbf{Definition}. The length of the word equals the number of letters in the corresponding product. The set \(L_i(S) = S^i\) is the linear span of the set \(S^i\) (the set of all finite linear combinations with coefficients belonging to the basic field \({F}\)). It should be noted that \(L_0(S) = \langle 1\rangle = {F}\) for any \(S.\) The set \(L(S)\) stands for \(\bigcup\limits_{i=1}^{\infty} L_i(S).\) \(S\) is a generating set of an algebra \(A\) if and only if \(A = L(S).\) The length of a generating set \(S\) of a finite-dimensional algebra \(A\) is defined as follows: \(l(S) = \min\{k \in Z_+ : L_k(S) = A\}.\) The length of an algebra \(A\) is \(l(A) = sup \{l(S) : L(S) = A\}.\) The study of lengths of non-associative algebras becomes from the associative case: the problem of the associative algebra length computation was first discussed for the algebra of \(3 \times 3\) matrices in the context of the mechanics of isotropic continua in 1959--60. The problem of computing the length of the full matrix algebra \(M_n\) as a function of the matrix size \(n\) was stated in 1984 and is still an open problem. Recently, some authors are beginning the study of the length of non-associative algebras [\textit{A. E. Guterman} and \textit{D. K. Kudryavtsev}, J. Algebra 544, 483--497 (2020; Zbl 1462.17003)]. One of the most important results of the cited paper is the following: let \({A}\) be a non-associative algebra, then \(l({A}) \leq 2^{\dim { A} -2}.\) The study of unital associative algebras with length \(1\) is given in [\textit{O. V. Markova}, J. Math. Sci., New York 185, No. 3, 458--472 (2012; Zbl 1259.16037); translation from Fundam. Prikl. Mat. 17, No. 1, 169--188 (2012); Mosc. Univ. Math. Bull. 68, No. 1, 74--76 (2013; Zbl 1273.16023); translation from Vestn. Mosk. Univ., Ser. I 68, No. 1, 54--56 (2013)]. The present paper is about the study of a more general case. Let us summarize two principal results of the paper: \textbf{Theorem 3.8.} Let \(A\) be a unital finite-dimensional \(F\)-algebra (char \(F\) \(\neq\) 2). Then \(l(A) = 1\) if and only if there is a special basis \(B = \{1_A, a_2, \ldots , a_n\}\) of \(A\) (i.e. \(a_ia_j= \alpha_{ij} 1_A+\beta_j a_i-\beta_i a_j \)). \textbf{Theorem 4.6.} Let \(F\) be a field of characteristic \(2\) and \(A\) an unital \(F\)-algebra with \(\dim \ A \geq 4.\) Then \(l(A) = 1\) if and only if there is a basis \(B^* = \{a^*_1 = 1_A, a^*_2, \ldots , a^*_n\}\) of \(A\) whose multiplication table satisfies one of the following two conditions: (I) \((a^*_i)^2 \equiv 0,\) for \(i = 2, \ldots , n,\) \(a^*_i a^*_j \equiv a^*_j a^*_i\equiv \beta_{ij} a^*_i + \beta^{ji}a^*_j,\) (II) \((a^*)^2 \equiv a^*_i,\) for \(i = 2, \ldots, n,\) \(a^*_i a^*_j \equiv \beta_{ij} a^*_i + (1 + \beta_{ji})a^*_j.\) Notice that in the last case \(a^*_i a^*_j + a^*_j a^*_i \equiv a^*_i + a^*_j.\) Also, the authors describe \(3\)-dimensional algebras with the length of \(1\) (Corollary 3.5. and Corollary 4.5.). Another interesting corollary talks about unital Jordan algebras with the length of 1: it should be isomorphic to a Jordan algebra of a symmetric bilinear form (Corollary 3.11.).