The Ext-algebra of the Lie algebra L is ExtUL(F,F), where F is the field of L and UL is the enveloping algebra of L. It is computed using the minimal model of L, see ExtAlgebra. If R is a quadratic (skew)commutative Koszul algebra, and L is the value of koszulDual(R) then extAlgebra(n,L) represents the ring R up to degree n. A basis for ExtUL(F,F) as a vector space is represented by generators(ExtAlgebra). The symbol SPACE is used as multiplication of elements in the Ext-algebra and for multiplication by scalars.
i1 : F = lieAlgebra({a,b,c},Weights=>{{1,0},{1,0},{2,1}}, Signs=>{1,1,1},LastWeightHomological=>true) o1 = F o1 : LieAlgebra |
i2 : D = differentialLieAlgebra{0_F,0_F,a a + b b} o2 = D o2 : LieAlgebra |
i3 : L=D/{a b,a c} o3 = L o3 : LieAlgebra |
i4 : E=extAlgebra(3,L) o4 = E o4 : ExtAlgebra |
i5 : describe E o5 = generators => {ext_0, ext_1, ext_2, ext_3, ext_4} Weights => {{1, 1}, {1, 1}, {2, 2}, {2, 2}, {3, 3}} Signs => {0, 0, 0, 0, 0} lieAlgebra => L Field => QQ computedDegree => 3 |
i6 : (ext_0 - 2 ext_1) ext_2 o6 = - 4ext_4 o6 : E |
i7 : R=QQ[a,b,c]/{a*a,b*b,c*c} o7 = R o7 : QuotientRing |
i8 : L=koszulDual(R) o8 = L o8 : LieAlgebra |
i9 : E=extAlgebra(4,L) o9 = E o9 : ExtAlgebra |
i10 : describe E o10 = generators => {ext_0, ext_1, ext_2, ext_3, ext_4, ext_5, ext_6} Weights => {{1, 1}, {1, 1}, {1, 1}, {2, 2}, {2, 2}, {2, 2}, {3, 3}} Signs => {0, 0, 0, 0, 0, 0, 0} lieAlgebra => L Field => QQ computedDegree => 4 |
i11 : ext_0 ext_1 ext_2 o11 = ext_6 o11 : E |