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

Synopsis

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 variable(s)
 
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

See also

Ways to use koszulDual :

For the programmer

The object koszulDual is a method function.