The input Q is a quotient of a polynomial algebra by a quadratic ideal (which might be zero). Some of the variables may be declared as SkewCommutative and the variables may have multidegrees where the first degree is equal to one. The quadratic ideal must be homogeneous with respect to the multidegree 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=>{}] o1 = R1 o1 : PolynomialRing |
i2 : I1={x^2,y^2,z^2} 2 2 2 o2 = {x , y , z } o2 : List |
i3 : L1=koszulDualLie(R1/ideal I1) o3 = L1 o3 : LieAlgebra |
i4 : L1.relsLie o4 = {[ko , ko ], [ko , ko ], [ko , ko ]} 0 1 0 2 1 2 o4 : List |
i5 : R2=QQ[x,y,z, SkewCommutative=>{x,z},Degrees=>{{1,1},{1,2},{1,3}}] o5 = R2 o5 : PolynomialRing |
i6 : I2=ideal{y^2+x*z,x*y} 2 o6 = ideal (y + x*z, x*y) o6 : Ideal of R2 |
i7 : L2=koszulDualLie(R2/I2) o7 = L2 o7 : LieAlgebra |
i8 : peek L2 o8 = LieAlgebra{cache => CacheTable{...9...} } compdeg => 0 deglength => 3 field => QQ genDiffs => {[], [], []} genSigns => {0, 1, 0} gensLie => {ko , ko , ko } 0 1 2 genWeights => {{1, 1, 0}, {1, 2, 0}, {1, 3, 0}} numGen => 3 1 relsLie => {{{-1, -}, {[ko , ko ], [ko , ko ]}}, [ko , ko ]} 2 0 2 1 1 2 1 |
Generators in the polynomial ring used in input should not be used also as generators of a Lie algebra, since in that case the generators will not be of class Symbol.