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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