ExteriorExtension -- A new algebra constructed from the direct sum of a Lie algebra and a module

Description

A ExteriorExtension (treated as a mutable hash table) is built by the command exteriorExtension. The menu lists the ways to interact with an ExteriorExtension. The operation takes the Lie algebra \(\mathfrak{sl}_n(\mathbb{F}\) of traceless \(n \times n\) matrices over a field \(\mathbb{F}\) and appends the non-zero graded piece of an exterior algebra \(\bigoplus_{i=1}^{n-1}\bigwedge^i \mathbb{F}^n \).

i1 : ea = exteriorExtension(4, 8);

LieAlgebra -- a placeholder for the base Lie algebra
appendage -- a placeholder for the object (exterior algebra) added on to the Lie algebra by exteriorExtension
findGrade -- Find which graded piece of the algebra an element is in
HodgeStar -- The Hodge star in an exterior algebra
bases -- gives a list of the bases of the different graded pieces of the ExteriorExtension
bracket -- Compute products in an ExteriorExtension
ad -- Constructs the standard matrix representation of the adjoint operator
powerRanks -- Compute the ranks of powers of a matrix
powerTraces -- Compute the traces of powers of a matrix
getBlock -- Extract a block of the matrix representation of an adjoint operator for ExteriorExtensions
blockPowerRanks -- Computes a list of ranks of blocks for each power of a matrix representation of an adjoint operator for ExteriorExtensions
prettyBlockPowerRanks -- same as blockPowerRanks, but prettier
structureTensor -- Constructs the structureTensor of the ExteriorExtension
KillingMatrix -- Constructs the Killing matrix of the ExteriorExtension
LieAlg2Matrix -- Converts from the LieAlgebra to Matrix form (with the standard basis)
matrix2LieAlg -- Converts from Matrix form to the LieAlgebra (with the standard basis)

Methods that use an object of class ExteriorExtension:

KillingMatrix(ExteriorExtension) (missing documentation)
structureTensor(ExteriorExtension) (missing documentation)

For the programmer

The object ExteriorExtension is a type, with ancestor classes MutableHashTable < HashTable < Thing.

The source of this document is in ExteriorExtensions.m2:321:0.

ExteriorExtension -- A new algebra constructed from the direct sum of a Lie algebra and a module

Description

Menu

Methods that use an object of class ExteriorExtension:

For the programmer