A Gröbner-Bases Algorithm for the Computation of the Cohomology of Lie (Super) Algebras

Abstract

We present an effective algorithm for computing the standard cohomology spaces of finitely generated Lie (super) algebras over a commutative field K of characteristic zero. In order to reach explicit representatives of some generators of the quotient space Z k B k of cocycles Z k modulo coboundaries B k , we apply Gröbner bases techniques (in the appropriate linear setting) and take advantage of their strength. Moreover, when the considered Lie (super) algebras enjoy a grading -a case which often happens both in representation theory and in differential geometry -, all cohomology spaces Z k B k naturally split up as direct sums of smaller subspaces, and this enables us, for higher dimensional Lie (super) algebras, to improve the computer speed of calculations. Lastly, we implement our algorithm in the MAPLE software and evaluate its performances via some examples, most of which have several applications in the theory of Cartan-Tanaka connections.

An affirmative answer to a conjecture on the Metoki class

“…§2.1(I-3)). In [4], (1) For the weight w ≤ 0, the structure of H • c (ham 2n , sp(2n, R)) w is completely determined, and (2) When n = 1 and w > 0, the next holds true: H • c (ham 2 , sp(2, R)) w = 0 (w = 1, 2, . .…”

“…[1]). The quotient space ker(f : Y → Z)/Im(g : X → Y ) is equipped with the basis GB k/e = Basis([NF(ϕ, GB e , Ord y ) | ϕ ∈ GB k ], Ord y )…”

Euler number and Betti numbers of homology groups of pre Lie superalgebra

“…(1.3) Then we call g a pre (or Z-graded) Lie superalgebra. A Lie superalgebra g is graded by Z 2 as g = g [0] ⊕ g [1] and the condition (1.1) is regarded as [g [1] , g [1] ] ⊂ g [0] in modulo 2 sense. Remark 1.1 Super Jacobi identity (1.3) above is equivalent to the one of the following.…”

“…Suppose g = j∈Z g j is a pre Lie superalgebra. Let g [0] = i is even g i and g [1] = i is odd g i . Then g = g [0] ⊕ g [1] holds and this is a Lie superalgebra.…”

Effective Cartan–Tanaka connections for C6-smooth strongly pseudoconvex hypersurfaces M3⊂C2

“…Especially for k = 2, the cohomology spaces H k := ker(∂ k )/im(∂ k−1 ) encode deformations of Lie algebras and are useful for Cartan connections ( [13,4,6]), cf. [1] for an algorithm using Gröbner bases. [4,6,2]) For any three x , x , x ∈ g, one has at every point p ∈ P:…”