# extAlgebra -- compute the Ext-algebra of a Lie algebra

## Synopsis

• Usage:
E=extAlgebra(n,L)
• Inputs:
• Outputs:

## Description

The Ext-algebra of the Lie algebra $L$ is $Ext_{UL}(F,F)$, 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 $Ext_{UL}(F,F)$ 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