Zur Interpretation der komplexen Elemente in der Geometrie

A Quantum Deformation of Invariants of Higher Binary Forms

“…where π ∈ S k is a permutation, as PBW basis elements instead of (20). It is a consequence of Proposition 5.3 that we obtain no further invariants by this modification.…”

“…For example for f = q 3 (01)(02)(03) we have ∆ f = −q The quadratic form ∆ has the discriminant I ∆ := [2]MK − q 2 LL + q 2 [2]KM with I ∆ = −q 20 (q 2 ADAD + qADDA + DADA + qDAAD) −q 21 [2] 2 (q 6 ACCC + q 2 CACC + q 4 CCAC + CCCA) + q 24 [2](q 4 ACBD + ACDB) +(q 20 + q 22 − q 26 )(qADBC + DABC + qBCAD + BCDA) +q 20 [2](q 6 BDAC + q 2 BDCA + q 4 CABD + CADB) +(−q 19 + q 23 + q 25 )(qCBAD + CBDA + qADCB + DACB) + q 18 [2](q 4 DBAC + DBCA) −q 21 [2] 2 (q 6 BBBD + q 2 BBDB + q 4 BDBB + DBBB) + q 24 [2] 3 (q 2 BBCC + CCBB) +(−q 20 − 2q 22 − q 24 + 2q 26 + 2q 28 − q 32 )BCBC +(−q 18 + 2q 22 + 2q 24 − q 26 − 2q 28 − q 30 )CBCB +(q 19 + q 21 − q 23 − 3q 25 − q 27 + q 29 + q 31 )CBBC. Remark 6.…”

“…The form ∆ f is related to a projective construction of complex elements due to F. Klein (cf. [20]). In the classical case the two complex conjugated zeros are those two unique complex points, from which one sees the intervals, determined by the three real zeros of f under an angle of π 3 .…”