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
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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
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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
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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
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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
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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
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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
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The object koszulDual is a method function.