In this package we compute the breadth of Lie algebras. This package also include functions to compute covered Lie algebras of maximal class using a proces called inflation, this is described by Caranti, Mattarei, and Newman [CMN97].
Given a Lie 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 \(c \in \N\) such that \(L^{c+1} = {0}\), and the minimal such \(c\) is called the class \(\mathrm{cl}(L)\) of \(L\). For a nilpotent Lie algebra \(L\), the type of \(L\) is the vector \((d_1,\dots,d_c)\), where \(d_i = \dim L^i\). A nilpotent Lie 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 \(L\). This 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
\[ b_i \cdot b_j = \sum_{k=1}^n c_{ijk} b_k \quad 1\leq i,j \leq n. \]
For each \(k\), write \(C_k = (c_{ijk})_{1 \leq i,j \leq n}\) for the \(n \times n\) matrix over \(K\) with entries \(c_{ijk}\), 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 \(C_k \overline{x}^{tr}\), this is the adjoint matrix of \(L\) with respect to the basis \(B\).
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