In this elementary tutorial, we give a brief introduction on how to use the package GradedLieAlgebras.
The most common way to construct a Lie algebra is by means of the constructor lieAlgebra.
i1 : L1 = lieAlgebra({a,b}, {}) o1 = L1 o1 : LieAlgebra |
i2 : computeLie 5 o2 = {2, 1, 2, 3, 6} o2 : List |
The above list is the dimensions of the free Lie algebra on two generators in degrees 1 to 5. To get an explicit basis in a certain degree, use basisLie.
i3 : basisLie 2 o3 = {[b, a]} o3 : List |
i4 : basisLie 3 o4 = {[a, b, a], [b, b, a]} o4 : List |
The basis elements in degree 3 above should be interpreted as [a,[b,a]] and [b,[b,a]]. To multiply two elements, use multLie.
i5 : prod=multLie([a,b],[a,a,b]) o5 = {{-1, 1}, {[a, b, a, b, a], [b, a, a, b, a]}} o5 : List |
The output above should be interpreted as -[a,[b,[a,[b,a]]]] + [b,[a,[a,[b,a]]]], that is, we use the right associative convention for Lie monomials, see monomialLie. The output from multLie is a linear combination of the basis elements of degree 5.
i6 : basisLie 5 o6 = {[a, a, a, b, a], [b, a, a, b, a], [a, b, a, b, a], [b, b, a, b, a], [a, ------------------------------------------------------------------------ b, b, b, a], [b, b, b, b, a]} o6 : List |
An expression like prod above may be used as a relation in a Lie algebra.
i7 : L2=lieAlgebra({a,b},{prod}) o7 = L2 o7 : LieAlgebra |
i8 : computeLie 5 o8 = {2, 1, 2, 3, 5} o8 : List |
As expected, the dimension in degree 5 of L2 is one less than that of L1. Each relation in the second argument to the constructor lieAlgebra is a general Lie expression, generalExpressionLie, see How to write Lie elements.