In this package we compute the breadth of Lie algebras. The details of the algorithms are available in [EF26]. This package also includes functions to compute covered Lie algebras of maximal class using a process called inflation, which is described by Caranti, Mattarei, and Newman [CMN97].
Given a finite-dimensional algebra \(L\), we define its lower central series as \(L = L^1 > L^2 > \dots\), where \(L^{i+1} = L^i L\). The algebra \(L\) is nilpotent if there exists \(c \in \mathbb{N}\) such that \(L^{c+1} = \{0\}\), and the minimal such \(c\) is called the class \(\mathrm{cl}(L)\) of \(L\). For a nilpotent algebra \(L\), the type of \(L\) is the vector \((d_1,\dots,d_c)\), where \(d_i = \dim L^i\). A nilpotent algebra \(L\) is said to be of maximal class if the type of \(L\) is \((2,1,\dots,1)\). The centralizer of \(x \in L\) is the subspace of \(L\) defined by \(C_L(x) = \{a \in L \mid ax = 0\}\). For an algebra \(L\), we define
\[ \mathrm{br}(L) = \max \{ \mathrm{br}(x) \mid x \in L\}, \;\; \mbox{ where } \;\; \mathrm{br}(x) = \dim(L) - \dim(C_L(x)). \]
The class-breadth conjecture asserts that \(\mathrm{cl}(L) \leq \mathrm{br}(L)+1\) for an algebra \(L\). The conjecture holds for nilpotent Lie algebras over infinite fields and for nilpotent associative algebras over arbitrary fields.
Let \(L\) be a Lie algebra over a field \(K\), and let \(B = \{ b_1, \ldots, b_n \}\) be a basis of \(L\). The multiplication in \(L\) is described by structure constants defined by
\[ b_i \cdot b_j = \sum_{k=1}^n c_{ijk} b_k \quad\mbox{ for } 1\leq i,j \leq n. \]
For each \(k\), write \(C_k = (c_{ijk})_{1 \leq i,j \leq n}\) for the \(n \times n\) matrix over \(K\) with entries \(c_{ijk}\), and these matrices are the structure matrices of \(L\). For \(x = x_1 b_1 + \ldots + x_n b_n \in L\), let \(\overline{x} = (x_1, \ldots, x_n) \in K^n\) denote the coefficient vector of \(x\). Then \(C_k \overline{x}^{tr}\) is a column vector over \(K\). We write \(M_B(x)\) for the \(n \times n\) matrix over \(K\) whose \(k\)-th column is \(C_k \overline{x}^{\mathrm{tr}}\), which is the adjoint matrix of \(L\) with respect to the basis \(B\).
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