Macaulay2 » Documentation
Packages » GradedLieAlgebras :: minimalModel
next | previous | forward | backward | up | index | toc

minimalModel -- compute the minimal model

Synopsis

Description

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

See also

Ways to use minimalModel :

For the programmer

The object minimalModel is a method function.