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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