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# ExtAlgebra -- the class of all Ext-algebras

## Description

This type represents the Ext-algebra of a graded differential Lie algebra $L$, $E=Ext_{UL}(F,F)$, where $F$ is the field of $L$ and $UL$ is the enveloping algebra of $L$. Each object of type ExtAlgebra is itself a type E, and homogeneous elements in E belong also to the type ExtElement, which is the parent of E. The generators of E, see generators(ExtAlgebra), represents a basis for $E$ as a vector space and correspond to the Lie algebra generators for the minimal model $M$ of $L$; however, the homological degree of a generator in $E$ is 1 more than the homological degree for the corresponding generator in $M$ (and also the sign is switched).

    i1 : L = lieAlgebra{a,b}/{a a b,b b b a}

    o1 = L

    o1 : LieAlgebra

    i2 : E = extAlgebra(5,L)

    o2 = E

    o2 : ExtAlgebra

    i3 : describe E

    o3 = generators => {ext_0, ext_1, ext_2, ext_3}
         Weights => {{1, 1}, {1, 1}, {3, 2}, {4, 2}}
         Signs => {1, 1, 0, 0}
         lieAlgebra => L
         Field => QQ
         computedDegree => 5

    i4 : parent E

    o4 = ExtElement

    o4 : Type

    i5 : ext_0 ext_1

    o5 = 0

    o5 : E

    i6 : M = minimalModel(5,L)

    o6 = M

    o6 : LieAlgebra

    i7 : describe M

    o7 = generators => {fr , fr , fr , fr }
                          0    1    2    3
         Weights => {{1, 0}, {1, 0}, {3, 1}, {4, 1}}
         Signs => {0, 0, 1, 1}
         ideal => {}
         ambient => LieAlgebra{...10...}
         diff => {0, 0, (fr_0 fr_1 fr_0), (fr_1 fr_1 fr_1 fr_0)}
         Field => QQ
         computedDegree => 5
         map => fr  => a
                  0
                fr  => b
                  1
                fr  => 0
                  2
                fr  => 0
                  3
                source => M
                target => L

    i8 : gens E

    o8 = {ext_0, ext_1, ext_2, ext_3}

    o8 : List

## See also

- ExtElement -- the class of all Ext-algebra elements
- extAlgebra -- compute the Ext-algebra of a Lie algebra
- minimalModel -- compute the minimal model

## Methods that use an object of class ExtAlgebra:

- basis(List,ExtAlgebra) -- compute a basis
- basis(ZZ,ExtAlgebra) -- compute a basis
- basis(ZZ,ZZ,ExtAlgebra) -- compute a basis
- `computedDegree(ExtAlgebra)` -- see computedDegree -- get the degree to which the computations have been performed
- `describe(ExtAlgebra)` -- see describe(LieAlgebra) -- real description
- dim(List,ExtAlgebra) -- compute the dimension
- dim(ZZ,ExtAlgebra) -- compute the dimension
- dim(ZZ,ZZ,ExtAlgebra) -- compute the dimension
- `dims(ZZ,ExtAlgebra)` -- see dims -- compute the dimensions of a Lie algebra, Ext-algebra or vector space
- `dims(ZZ,ZZ,ExtAlgebra)` -- see dims -- compute the dimensions of a Lie algebra, Ext-algebra or vector space
- generators(ExtAlgebra) -- get the generators
- ZZ _ ExtAlgebra -- get the zero element

## For the programmer

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GradedLieAlgebras/doc.m2:181:0.