‣ LieClass( L ) | ( attribute ) |
Computes the class of the nilpotent Lie algebra L.
‣ LieType( L ) | ( attribute ) |
Computes the type of the nilpotent Lie algebra L.
‣ IsOfMaximalClass( L ) | ( property ) |
Returns: true or false
Returns whether the nilpotent Lie algebra L is of maximal class or not.
‣ StructureMatrices( L ) | ( attribute ) |
Computes the structure matrices of the nilpotent Lie algebra L.
‣ BasisLieCenter( L ) | ( attribute ) |
Returns the basis elements of the Lie algebra L that are contained in the center of L.
‣ BasisLieDerived( L ) | ( attribute ) |
Returns the basis elements of the Lie algebra L that are contained in the derived subalgebra of L.
‣ LieAdjointMatrix( L ) | ( attribute ) |
Computes the adjoint matrix of the Lie algebra L.
‣ PrintLiePresentation( L ) | ( attribute ) |
Prints the Lie presentation of the Lie algebra L.
‣ InfoLieBreadth | ( info class ) |
Info class for the functions of the breadth of Lie algebras.
‣ LieBreadth( L ) | ( attribute ) |
Computes the breadth of the Lie algebra L.
‣ IsTrueClassBreadth( L ) | ( property ) |
Returns: true or false
Returns whether the class-breadth conjecture holds for the Lie algebra L.
‣ TGroupBreadth( G ) | ( function ) |
Computes the breadth of the \mathcal{T}-group G.
A grading for a Lie algebra L is a decomposition L = \bigoplus_{i=1}^n L_i that respects the Lie bracket, i.e. [L_i,L_j]\subseteq L_{i+j}. All nilpotent Lie algebras can be 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 = \bigoplus_{i=1}^n L_i be a graded Lie algebra over K=\mathbb{F}_q for some prime power q=p^n. Consider the field extension A=K[\varepsilon]/\langle \varepsilon^p \rangle which is a vector space over K of dimension p. The algebra A is an associative, commutative algebra with a 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 by: D^\uparrow(x\otimes \varepsilon^i)=D(x)\otimes \varepsilon^i, which is a derivation of M^\uparrow of degree 1. Define the derivation E\in \mathrm{Der} M^\uparrow by E(x\otimes \varepsilon^i) = D^\uparrow(x\otimes\varepsilon^i\cdot\varepsilon^{p-1})+1\otimes\partial_\varepsilon(\varepsilon^i). For s\in L_1\setminus M, take D=\mathrm{ad}_s, and extend it naturally to M^\uparrow. Denote E_{s^\prime} the derivation defined above. The 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=2,3,4, computes the covered Lie algebras of maximal class using the polynomial algebra of dimension n. For n=3, one can directly load the Lie algebra by reading the file \texttt{inflation\_3.g}.
‣ LieMinimalQuotientClassBreadth( L ) | ( attribute ) |
Given a covered Lie algebras of maximal class L, computes the minimal quotient for which the class-breadth conjecture does not hold.
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