A grading for a Lie algebra \(L\) decomposition L = \(\bigoplus_{i=1}^n L_i\) that respects the Lie bracket, \textit{i.e.} \([L_i,L_j]\subseteq L_{i+j}\).}. Any nilpotent Lie algebras is graded by taking \(L_i = \gamma_i(L)/\gamma_{i+1}(L)\). Let \(L\) be a nilpotent Lie algebra of maximal class, the two-step centralizers are the sets \(C_i = C_{L_1}(L_i) = \{x \in L_1 \mid [x,L_i] = 0 \}\) for all \(2\leq i \leq c\). Let \(\mathcal{C}=\{C_i\}\setminus L_1\), we say that a Lie algebra of maximal class is covered if the set \(\mathcal{C}\) consist of all one-dimensional subspaces of \(L_1\). Let \(L\) be a graded Lie algebra \(L = \bigoplus_{i=1}^n L_i\) over \(K=\F_q\) for some prime power \(q=p^n\). The field extension \(A=K[\varepsilon]/\langle \varepsilon^p \rangle\) is a vector space over \(K\) of dimension \(p\) and \(A\) is an associative commutative algebra with unit. The algebra \(L\otimes A\) over \(K\) defined by \([x\otimes a, y\otimes b] = [x,y]\otimes ab\) is a graded Lie algebra. Let \(M\) be a maximal ideal of \(L\) and consider the Lie subalgebra \(M^\uparrow = M\otimes A\). Let \(D\) be a derivation of \(L\) of degree 1. Define \(D^\uparrow\) via: \(D^\uparrow(x\otimes \varepsilon^i)=D(x)\otimes \varepsilon^i\), this is a derivation of \(M^\uparrow\) of degree 1. Define the derivation \(E\in \Der M^\uparrow\) as \(E(x\otimes \varepsilon^i) = D^\uparrow(x\otimes\varepsilon^i\cdot\varepsilon^{p-1})+1\otimes\partial_\varepsilon(\varepsilon^i)\). Let \(s\in L_1\setminus M\), take \(D=\mathrm{ad}_s\) and extend it to \(M^\uparrow\) in the natural way to \(M^\uparrow\). Denote \(E_{s^\prime}\) as previously. The \textit{inflation} \({}^{M}L\) of \(L\) at \(M\) by \(s\in L_1\setminus M\) is the graded Lie algebra obtained as an extension of \(M^\uparrow\) by an element \(s^\prime\) which is the extension of \(s\) that induces the derivation \(E_{s^\prime}\), that is, \[ [x\otimes a, s^\prime] = E_{s^\prime}(x\otimes a) = \mathrm{ad}_{s^\prime}(x\otimes\varepsilon^i)+x\otimes\partial_\varepsilon(\varepsilon^i).\]
‣ InfoInflation | ( info class ) |
Info class for the functions of the inflation of Lie algebras.
‣ LieNilpotentGrading( L ) | ( attribute ) |
Computes a grading of the nilpotent Lie algebra \(L\) using the lower central series.
‣ LieTwoStepCentralizers( L ) | ( attribute ) |
Computes the two step centralizers of the Lie algebra \(L\).
‣ IsLieCovered( L ) | ( property ) |
Returns: true or false
Returns whether the Lie algebra \(L\) is covered or not.
‣ PolynomialAlgebra( F ) | ( attribute ) |
For a finite field \(F\) computes the polynomial algebra \(F[x]\).
‣ LieCoveredInflated( n ) | ( attribute ) |
For \(n=2,3\) computes the covered Lie algebras of maximal class using the polynomial algebra of dimension \(n\).
‣ LieMinimalQuotientClassBreadth( L ) | ( attribute ) |
Given a covered Lie algebras of maximal class \(L\) computes the minimal quotient that not holds the class-breadth conjecture.
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