That $M$ is a minimal model of a Lie algebra $L$ up to degree $d$ means that there exists a differential Lie algebra homomorphism $f: M \ \to\ L$ such that $H(f)$ is an isomorphism up to degree $d$, $M$ is free as a Lie algebra, and the linear part of the differential on $M$ is zero. The homomorphism $f$ may be obtained using map(LieAlgebra) applied to $M$.
The generators of $M$ yield a basis for the cohomology of $L$, i.e., $Ext_{UL}(k,k)$, where $k$ is the coefficient field of $L$. This skewcommutative algebra may be obtained using extAlgebra. Multiplication of elements in $Ext_{UL}(k,k)$ is obtained using ExtElement ExtElement.
Observe that the homological weight in the cohomology algebra is one higher than the homological weight in the minimal model.
Since $R$ in the following example is a Koszul algebra it follows that the cohomology algebra of $L$ is equal to $R$. This means that the minimal model of $L$ has generators in each degree $(d,d-1)$.
i1 : R=QQ[x] o1 = R o1 : PolynomialRing |
i2 : L=koszulDual R o2 = L o2 : LieAlgebra |
i3 : describe L
o3 = generators => {ko }
0
Weights => {{1, 0}}
Signs => {1}
ideal => { - (1/2)(ko_0 ko_0)}
ambient => LieAlgebra{...10...}
diff => {}
Field => QQ
computedDegree => 0
|
i4 : E=extAlgebra(5,L) o4 = E o4 : ExtAlgebra |
i5 : dims(5,E)
o5 = | 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 1 0 0 |
| 0 0 0 1 0 |
| 0 0 0 0 1 |
5 5
o5 : Matrix ZZ <--- ZZ
|
i6 : describe minimalModel(5,L)
o6 = generators => {fr , fr , fr , fr , fr }
0 1 2 3 4
Weights => {{1, 0}, {2, 1}, {3, 2}, {4, 3}, {5, 4}}
Signs => {1, 1, 1, 1, 1}
ideal => {}
ambient => LieAlgebra{...10...}
diff => {0, (fr_0 fr_0), (fr_0 fr_1), (fr_1 fr_1) + 4 (fr_0 fr_2), 2 (fr_1 fr_2) + (fr_0 fr_3)}
Field => QQ
computedDegree => 5
map => fr => ko_0
0
fr => 0
1
fr => 0
2
fr => 0
3
fr => 0
4
source => LieAlgebra{...10...}
target => L
|
In the following example the enveloping algebra of $L1$ has global dimension $2$, which means that the computed minimal model is in fact the full minimal model of $L1$.
i7 : L1=lieAlgebra{a,b,c}/{a b,a b c}
o7 = L1
o7 : LieAlgebra
|
i8 : M1= minimalModel(3,L1) o8 = M1 o8 : LieAlgebra |
i9 : describe M1
o9 = generators => {fr , fr , fr , fr , fr }
0 1 2 3 4
Weights => {{1, 0}, {1, 0}, {1, 0}, {2, 1}, {3, 1}}
Signs => {0, 0, 0, 1, 1}
ideal => {}
ambient => LieAlgebra{...10...}
diff => {0, 0, 0, (fr_1 fr_0), (fr_1 fr_2 fr_0)}
Field => QQ
computedDegree => 3
map => fr => a
0
fr => b
1
fr => c
2
fr => 0
3
fr => 0
4
source => M1
target => L1
|
i10 : H=lieHomology M1 o10 = H o10 : VectorSpace |
i11 : dims(6,L1)===dims(6,H) o11 = true |
The object minimalModel is a method function.