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GradedLieAlgebras :: koszulDualLie

koszulDualLie -- gives 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 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

Caveat

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.

See also

Ways to use koszulDualLie :