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GradedLieAlgebras :: extAlgLie

extAlgLie -- the matrix of dimensions of the Ext-algebra

Synopsis

Description

The columns in the output matrix are referring to the degree, indexed from 1, and the rows are referring to the homological degree, indexed from 1. In the example below S is a Koszul algebra and hence S is equal to the cohomology algebra of L, ExtUL(k,k), where k=L.field. Also, since S is a complete intersection, L disappears after degree two.

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

o1 = R

o1 : PolynomialRing
i2 : I={x^2,y^2,z^2}

       2   2   2
o2 = {x , y , z }

o2 : List
i3 : S=R/ideal I

o3 = S

o3 : QuotientRing
i4 : L=koszulDualLie(S)

o4 = L

o4 : LieAlgebra
i5 : extAlgLie 3

o5 = | 3 0 0 |
     | 0 3 0 |
     | 0 0 1 |

              3        3
o5 : Matrix ZZ  <--- ZZ
i6 : hilbertSeries(S,Order=>4)

                2    3
o6 = 1 + 3T + 3T  + T

o6 : ZZ[T]
i7 : dimsLie 3

o7 = {3, 3, 0}

o7 : List

See also

Ways to use extAlgLie :