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