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GradedLieAlgebras :: extMultLie

extMultLie -- the (skew commutative) product in the Ext-algebra

Synopsis

Description

In the example below, ExtUL(QQ,QQ) is equal to R and a basis as a vector space up to the given degree is given by extBasisLie. Observe that the multiplication in ExtUL(QQ,QQ) has nothing to do with the polynomial multiplication in L.cache.extRepRing. The elements in ExtUL(QQ,QQ) are represented as linear polynomials in L.cache.extRepRing and multiplication in ExtUL(QQ,QQ) is determined by the differential in the minimal model of L, see minmodelLie. Instead of the prefix notation extMultLie for the multiplication one may use the infix notation SPACE, see RingElement RingElement and the example below.

i1 : R=QQ[x,y,z, SkewCommutative=>{x,y,z}]

o1 = R

o1 : PolynomialRing
i2 : L=koszulDualLie(R)

o2 = L

o2 : LieAlgebra
i3 : extTableLie 3

o3 = | 3 0 0 |
     | 0 3 0 |
     | 0 0 1 |

              3        3
o3 : Matrix ZZ  <--- ZZ
i4 : extBasisLie 3

o4 = {ext , ext , ext , ext , ext , ext , ext }
         0     1     2     3     4     5     6

o4 : List
i5 : m=extMultLie(ext_1,ext_2)

o5 = -ext
         3

o5 : QQ[ext , ext , ext , ext , ext , ext , ext ]
           0     1     2     3     4     5     6
i6 : ext_1 ext_2

o6 = -ext
         3

o6 : QQ[ext , ext , ext , ext , ext , ext , ext ]
           0     1     2     3     4     5     6
i7 : ext_0 m

o7 = ext
        6

o7 : QQ[ext , ext , ext , ext , ext , ext , ext ]
           0     1     2     3     4     5     6
i8 : peekLie minmodelLie 3

o8 = gensLie => {fr , fr , fr , fr , fr , fr , fr }
                   0    1    2    3    4    5    6
     genWeights => {{1, 0}, {1, 0}, {1, 0}, {2, 1}, {2, 1}, {2, 1}, {3, 2}}
     genSigns => {0, 0, 0, 1, 1, 1, 0}
     relsLie => {}
     genDiffs => {0, 0, 0, (fr_2 fr_1), (fr_2 fr_0), (fr_1 fr_0), (fr_0 fr_3) - (fr_1 fr_4) + (fr_2 fr_5)}
     field => QQ
     diffl => true
     compdeg => 3
     modelmap => MapLie{fr_0 => ko_0                     }
                        fr_1 => ko_1
                        fr_2 => ko_2
                        fr_3 => 0
                        fr_4 => 0
                        fr_5 => 0
                        fr_6 => 0
                        sourceLie => LieAlgebra{...14...}
                        targetLie => L

See also

Ways to use extMultLie :