‣ 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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