check if a list of vectors defines a Lie subalgebra - Maple Help

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Query[Subalgebra] - check if a list of vectors defines a Lie subalgebra

Calling Sequences

     Query(S, "Subalgebra")

     Query(S, parm, "Subalgebra")


     S       - a list of independent vectors in a Lie algebra 𝔤

     parm    - (optional) a set of parameters appearing in the list of vectors S.  It is assumed that the set of vectors S is well-defined when the parameters vanish.



A list of vectors S defines a basis for a Lie subalgebra if  x, y  spanS for all x,y S. 


Query(S, "Subalgebra") returns true if the set S defines a subalgebra.


Query(S, parm, "Subalgebra") returns a sequence TF, Eq, Soln, SubAlgList.  Here TF is true if Maple finds parameter values for which S is a subalgebra and false otherwise; Eq is the set of equations (with the variables in parm as unknowns) which must be satisfied for S to be a subalgebra; Soln is the list of solutions to the equations Eq; and SubAlgList is the list of subalgebras obtained from the parameter values given by the different solutions in Soln.


The program calculates the defining equations Eq for S to be a subalgebra as follows.  First the list of vectors S is evaluated with the parameters set to zero to obtain a set of vectors S0.  The program ComplementaryBasis is then used to calculate a complement C to S0  The list of vectors B= S, C then gives a basis for the entire Lie algebra 𝔤.  For each x,y S  the bracket x,y is calculated and expressed as a linear combination of the vectors in the basis B. The components of [x,y[ in C must all vanish for S to be a Lie subalgebra.


We remark that the equations Eq, which the parameters must satisfy in order for S to be a subalgebra, will in general be a system of coupled quadratic equations.  Maple may not be able to solve these equations or may not solve them in full generality.


The command Query is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...).




Example 1.

First initialize a Lie algebra.






The vectors S1 = e2, e3, e4 do not determine a subalgebra while the vectors S2 = e1, e3, e4 do.

Alg > 


Alg > 



Alg > 


Alg > 





We find the values of the parameters  a1, a2 for which S3 = e2, e1+a1e3 +a2e4 determines a Lie subalgebra.

Alg > 


Alg > 





There are no values of the parameters  a1, a2  for which S4 = e2, e3+a1e2 +a2e4 determines a Lie subalgebra.

Alg > 


Alg > 




See Also

DifferentialGeometry, LieAlgebras, Query

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