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GradedLieAlgebras :: minimalModel

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  → 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., ExtUL(k,k), where k is the coefficient field of L. This skewcommutative algebra may be obtained using extAlgebra. Multiplication of elements in ExtUL(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 :