The second cohomology group of simple Leibniz algebras

Abstract

In this paper we prove some general results on Leibniz 2-cocycles for simple Leibniz algebras. Applying these results we establish the triviality of the second Leibniz cohomology for a simple Leibniz algebra with coefficients in itself, whose associated Lie algebra is isomorphic to sl 2 .

On Leibniz cohomology

“…HL n (a, M s ) ∼ = H n−2 rel (a, M ) for every integer n ≥ 3. Moreover, we have that HL 0 (a, M s ) ∼ = M a and HL 1…”

“…if n ≡ 0 (mod 3) P (1) if n ≡ 1 (mod 3) , P (0) ⊕ L(2) if n ≡ 2 (mod 3) where P (n) denotes the projective cover (and at the same time also the injective hull) of L(n). As a consequence, we have that (L(2) ⊗ L(n)) g ∼ = F if n ≡ 2 (mod 3) 0 if n ≡ 2 (mod 3) .…”

“…In [1], the authors study the cohomology of semi-simple Leibniz algebras, i.e., the cohomology of finite-dimensional Leibniz algebras L with an ideal of squares Leib(L) such that the corresponding canonical Lie algebra L Lie := L/Leib(L) is semi-simple, and conjecture that HL 2 (L, L ad ) = 0. In [16], the authors determine the deviation of the second Leibniz cohomology of a complex Lie algebra with adjoint or trivial coefficients from the corresponding Chevalley-Eilenberg cohomology.…”

Maurer-Cartan characterizations and cohomologies of compatible Lie algebras

“…Then Loday rediscovered this algebraic structure and called them Leibniz algebras [18]. A Leibniz algebra is a vector space g, endowed with a linear map [•, •] g : g ⊗ g −→ g satisfying (1) [x, [y, z]…”

“…The cohomology theory of Leibniz algebras was also developed by Loday and Pirashvili in [18]. See [1,6,7,8,9,11] for more applications of Loday-Pirashvili cohomologies of Leibniz algebras. We also develop the omni-cohomology theory for omnirepresentations introduced above, and give the relation between omni-cohomology groups and Loday-Pirashvili cohomology groups.…”