change one or more components of a graded Lie algebra - Maple Programming Help

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LieAlgebras[ChangeGradedComponent] - change one or more components of a graded Lie algebra

Calling Sequences

ChangeGradedComponent(alg, newcomponent, newalg)

Parameters

alg          - a  name or string, the name of an initialized Lie algebra $\mathrm{𝔤}$

newcomponent - a list, specifying the new graded components

newalg       - a name or string, the name of a new graded Lie algebra to be created

Description

 • Let $\mathrm{𝔤}$ be a graded Lie algebra with (for example) grading . With newcomponent given by (for example) [2 = $h$], where $h$ is a list of vectors in ${𝔤}_{2}$, the command ChangeGradedComponent will return the structure equations for the new graded Lie algebra.

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

Define a 9-dimensional Lie algebra alg1 with grading where  and . Here are the structure equations:

 > StrEq := [[x1, x6] = -x1, [x1, x8] = -x2, [x2, x7] = -x1, [x2, x9] = -x2, [x3, x4] = -x1, [x3, x5] = -x2, [x3, x6] = -(1/3)*x3, [x3, x9] = -(1/3)*x3, [x4, x5] = x3, [x4, x6] = -(2/3)*x4, [x4, x8] = -x5, [x4, x9] = (1/3)*x4, [x5, x6] = (1/3)*x5, [x5, x7] = -x4, [x5, x9] = -(2/3)*x5, [x6, x7] = x7, [x6, x8] = -x8, [x7, x8] = x6-x9, [x7, x9] = x7, [x8, x9] = -x8];
 ${\mathrm{StrEq}}{:=}\left[\left[{\mathrm{x1}}{,}{\mathrm{x6}}\right]{=}{-}{\mathrm{x1}}{,}\left[{\mathrm{x1}}{,}{\mathrm{x8}}\right]{=}{-}{\mathrm{x2}}{,}\left[{\mathrm{x2}}{,}{\mathrm{x7}}\right]{=}{-}{\mathrm{x1}}{,}\left[{\mathrm{x2}}{,}{\mathrm{x9}}\right]{=}{-}{\mathrm{x2}}{,}\left[{\mathrm{x3}}{,}{\mathrm{x4}}\right]{=}{-}{\mathrm{x1}}{,}\left[{\mathrm{x3}}{,}{\mathrm{x5}}\right]{=}{-}{\mathrm{x2}}{,}\left[{\mathrm{x3}}{,}{\mathrm{x6}}\right]{=}{-}\frac{{1}}{{3}}{}{\mathrm{x3}}{,}\left[{\mathrm{x3}}{,}{\mathrm{x9}}\right]{=}{-}\frac{{1}}{{3}}{}{\mathrm{x3}}{,}\left[{\mathrm{x4}}{,}{\mathrm{x5}}\right]{=}{\mathrm{x3}}{,}\left[{\mathrm{x4}}{,}{\mathrm{x6}}\right]{=}{-}\frac{{2}}{{3}}{}{\mathrm{x4}}{,}\left[{\mathrm{x4}}{,}{\mathrm{x8}}\right]{=}{-}{\mathrm{x5}}{,}\left[{\mathrm{x4}}{,}{\mathrm{x9}}\right]{=}\frac{{1}}{{3}}{}{\mathrm{x4}}{,}\left[{\mathrm{x5}}{,}{\mathrm{x6}}\right]{=}\frac{{1}}{{3}}{}{\mathrm{x5}}{,}\left[{\mathrm{x5}}{,}{\mathrm{x7}}\right]{=}{-}{\mathrm{x4}}{,}\left[{\mathrm{x5}}{,}{\mathrm{x9}}\right]{=}{-}\frac{{2}}{{3}}{}{\mathrm{x5}}{,}\left[{\mathrm{x6}}{,}{\mathrm{x7}}\right]{=}{\mathrm{x7}}{,}\left[{\mathrm{x6}}{,}{\mathrm{x8}}\right]{=}{-}{\mathrm{x8}}{,}\left[{\mathrm{x7}}{,}{\mathrm{x8}}\right]{=}{\mathrm{x6}}{-}{\mathrm{x9}}{,}\left[{\mathrm{x7}}{,}{\mathrm{x9}}\right]{=}{\mathrm{x7}}{,}\left[{\mathrm{x8}}{,}{\mathrm{x9}}\right]{=}{-}{\mathrm{x8}}\right]$ (1)

Use the keyword grading to specify the grading of this algebra. Initialize.

 > LD1 := LieAlgebraData(StrEq, [x1, x2, x3, x4, x5, x6, x7, x8, x9], alg1, grading = [-3, -3, -2, -1, -1, 0, 0, 0, 0]);
 ${\mathrm{LD1}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}\frac{{1}}{{3}}{}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}\frac{{1}}{{3}}{}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}\frac{{2}}{{3}}{}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}\frac{{1}}{{3}}{}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}\frac{{1}}{{3}}{}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}\frac{{2}}{{3}}{}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e6}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e8}}\right]$ (2)
 > DGsetup(LD1);
 ${\mathrm{Lie algebra: alg1}}$ (3)

Note that the vectors define a 2-dimensional subalgebra of ${\mathrm{𝔤}}_{0}.$

 alg1 > LieBracket(e6, e7);
 ${\mathrm{e7}}$ (4)

Therefore we can replace all of ${\mathrm{𝔤}}_{0}$ with just . The result is a 7-dimensional graded Lie algebra which is a sub-algebra of the one we started with.

 alg1 > newLD1a := ChangeGradedComponent(alg1, [0 = [e6, e7]], newalg1);
 ${\mathrm{newLD1a}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}\frac{{1}}{{3}}{}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}\frac{{2}}{{3}}{}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}\frac{{1}}{{3}}{}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e7}}\right]$ (5)
 alg1 > DGsetup(newLD1a);
 ${\mathrm{Lie algebra: newalg1}}$ (6)
 alg1 > Tools:-DGinfo("Grading");
 $\left[{-}{3}{,}{-}{3}{,}{-}{2}{,}{-}{1}{,}{-}{1}{,}{0}{,}{0}\right]$ (7)
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