# koszulDual -- compute the Lie algebra whose enveloping algebra is the Koszul dual of a quadratic algebra

## Synopsis

• Usage:
L = koszulDual(Q)
• Inputs:
• Outputs:

## Description

The input $Q$ is a quotient of a polynomial algebra by a quadratic ideal (which might be 0). Some of the variables may be declared as SkewCommutative. Moreover, the variables may have multi-degrees where the first degree is equal to $1$. The quadratic ideal must be homogeneous with respect to the multi-degree and the "skew-degree". The output is the Lie algebra whose enveloping algebra is the Koszul dual of $Q$.

 i1 : R1=QQ[x,y,z, SkewCommutative=>{y,z}] o1 = R1 o1 : PolynomialRing, 2 skew commutative variables i2 : I1={x^2,y*z} 2 o2 = {x , y*z} o2 : List i3 : L1=koszulDual(R1/ideal I1) o3 = L1 o3 : LieAlgebra i4 : describe L1 o4 = generators => {ko , ko , ko } 0 1 2 Weights => {{1, 0}, {1, 0}, {1, 0}} Signs => {1, 0, 0} ideal => { - (ko_1 ko_0), - (ko_2 ko_0)} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 i5 : E1=extAlgebra(3,L1) o5 = E1 o5 : ExtAlgebra i6 : dims(3,E1) o6 = | 3 0 0 | | 0 2 0 | | 0 0 0 | 3 3 o6 : Matrix ZZ <--- ZZ

Here is an example of a non-Koszul algebra. The table for the Ext-algebra has a non-zero occurrence off the diagonal.

 i7 : R2=QQ[x,y,z, SkewCommutative=>{},Degrees=>{{1,1},{1,2},{1,3}}] o7 = R2 o7 : PolynomialRing i8 : I2=ideal{y^2+x*z,x*y,z^2} 2 2 o8 = ideal (y + x*z, x*y, z ) o8 : Ideal of R2 i9 : L2=koszulDual(R2/I2) o9 = L2 o9 : LieAlgebra i10 : describe L2 o10 = generators => {ko , ko , ko } 0 1 2 Weights => {{1, 1, 0}, {1, 2, 0}, {1, 3, 0}} Signs => {1, 1, 1} ideal => { - (1/2)(ko_0 ko_0), (1/2)(ko_1 ko_1) - (ko_2 ko_0), - (ko_2 ko_1)} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 i11 : E2=extAlgebra(4,L2) o11 = E2 o11 : ExtAlgebra i12 : dims(4,E2) o12 = | 3 0 0 0 | | 0 3 0 0 | | 0 0 1 1 | | 0 0 0 1 | 4 4 o12 : Matrix ZZ <--- ZZ