Bipartite graphs and the structure of finite-dimensional semisimple Leibniz algebras

Given a finite connected bipartite graph, finite-dimensional indecomposable semisimple Leibniz algebras are constructed. Furthermore, any finite-dimensional indecomposable semisimple Leibniz algebra admits a similar construction.


Introduction
The finite-dimensional simple Lie algebras over an algebraically closed field of characteristic zero are classified and it is a classical result that a finite-dimensional semisimple Lie algebra is a direct sum of simple Lie subalgebras (see [6]). A finitedimensional module over a semisimple Lie algebra due to Weyl's theorem is completely reducible into a direct sum of simple submodules. Furthermore, a Lie algebra admits a Levi decomposition -a semi-direct sum of a semisimple subalgebra and a maximal solvable ideal.
In this paper a "non-commutative" generalization of Lie algebras, introduced by Bloh ( [3]) and later by Loday ([9], [10]) -the so-called Leibniz algebras are studied. Although the classical simplicity for Leibniz algebras implies that it is a Lie algebra, a modified definition of the simplicity was introduced in [5] and has been in use in the various papers on the structure theory of Leibniz algebras. Generalization of semisimplicity for Leibniz algebras draws a parallel with semisimple Lie algebras, which is the main focus of the current work. However, it is well-known that a semisimple Leibniz algebra is not in general a direct sum of simple Leibniz algebras, and the question on the structure of semisimple Leibniz algebras has been open. Recently, in [1,Theorem 3.5] the authors establish the description of the finite-dimensional indecomposable semisimple non-Lie Leibniz algebra using a graph, whose vertexes are the simple Lie subalgebras of the liezation of the Leibniz algebra.

BIPARTITE GRAPHS, FINITE-DIMENSIONAL SEMISIMPLE LEIBNIZ ALGEBRAS 123
Motivated by the results of [1], the goal of this work is to shed light on how finitedimensional semisimple Leibniz algebras are built. Turns out, the structure of a finite-dimensional semisimple Leibniz algebra is more clear if instead of a graph (cf [1]), one uses a bipartite graph. The main results are presented in the last section, consisting of the description of semisimple Leibniz algebras from [1] with different proofs and a new construction of finite-dimensional semisimple Leibniz algebras using the bipartition of an associated graph.

Preliminaries
In the following section necessary definitions and results on Lie algebra and representation theory is given. Connection of Leibniz algebra with Lie algebra and its modules is provided. One of the main ingredients in this work, an analogue of Levi's theorem for Leibniz algebra is obtained directly from the results of T.
Pirashvili [12] in Subsection 2.2.  Left and right actions induce Lie algebra structure on M ⊕ g, where M becomes an abelian ideal and g is a subalgebra. If one has the right action of g on M and sets the left action to be zero, this induces a new type of a product that generalizes the Lie bracket on g given in the following.
holds for any x, y, z ∈ L and this algebra is not a Lie algebra if the action of g on M is not trivial.
A similar construction is given in [5]. There is the following short exact sequence and the epimorphism f is universal in the sense that a Leibniz algebra homomorphism from L to any Lie algebra g factors through f : Note that, due to I annihilating the Leibniz algebra whenever multiplied from the right, I admits a structure of a right Lie algebra g L -module with the well-defined where s : g L → L is a linear section.
proper ideals. An algebra without this property is called indecomposable.
Any finite-dimensional algebra is either indecomposable or is a finite direct sum of indecomposable algebras. A Lie algebra with no non-trivial ideals is called simple.
Simple Lie algebras are indecomposable, while the converse is not necessarily true.
A Leibniz algebra with only one non-trivial ideal I, so-called simple Leibniz algebra, is indecomposable. Levi-Malcev decomposition of a finite-dimensional Lie algebra g as a semidirect sum of a semisimple subalgebra s and the solvable radical, so that g = s Rad(g).
For Leibniz algebras similar result was proved by D. Barnes [2] in 2011. Note that, the same result is implicit from [12, Proposition 2.4] given below.
Proposition 2.8. Let φ : L → g be an epimorphism from an arbitrary finitedimensional Leibniz algebra L to a semisimple Lie algebra g. Then φ admits a section.
Indeed, consider a finite-dimensional Leibniz algebra L and apply Levi-Malcev decomposition to its liezation g L . Applying Proposition 2.8 for an epimorphism g • f one obtains a section: Clearly, the kernel of the epimorphism g • f is Rad(L) and we have an analogue of Levi decomposition for Leibniz algebra L ∼ = s Rad(L).
It is remarkable, that much earlier than the authors mentioned above, A. Bloh The Malcev part of the theorem is not true in general for the case of Leibniz algebras as shown in [2]. In some cases, conjugacy of Levi subalgebras is possible (see [8] and [11]).  Since I is a g L -module, and over a semisimple Lie algebra by Weyl's semisimplicity I decomposes into a direct sum of simple g L -submodules, one obtains where g i 's are simple Lie algebras and I k 's are simple ⊕ m i=1 g i -modules. Let a semisimple Leibniz algebra L be decomposable, that is L = L 1 ⊕. . .⊕L t and L 1 , . . . , L t are indecomposable Leibniz algebras. Obviously, L 1 is also semisimple and Corollary 2.11 implies L 1 = g 1 I 1 , where g 1 as the liezation of L 1 must be a subalgebra of g L . Thus, it is a direct sum of some simple components of g L . Since L 1 L, I 1 is a g L -module and using I 1 ⊆ I it is a sum of simple g-submodules of I. This implies that not only L 1 admits the structure of decomposition (2), but  Proof. Without loss of generality let us consider [I 1 , g 1 ]. Since I 1 is a simple ⊕ n i=1 g i -module, by Theorem 2.12 I 1 = ⊗ n i=1 J i for simple g i -modules J i . Note that the action is [I 1 , g 1 ] = [J 1 ⊗· · ·⊗J n , g 1 ] = (J 1 ⊗· · ·⊗J n ).g 1 = (J 1 .g 1 )⊗J 2 ⊗· · ·⊗J n . Now this is either {0} or J 1 ⊗ · · · ⊗ J n = I 1 since J 1 is an irreducible g 1 -module.

Main results
Moreover, it is well-known from representation theory of Lie algebras that [g 1 , I 1 ] = {0} if and only if J 1 = C (a one-dimensional representation is trivial). Proof. If for some 1 ≤ k ≤ n one has [I k , g i ] = {0} for all 1 ≤ i ≤ m, then I k is a direct summand of L (in fact, it is a contradiction with I k ⊆ I being generated by the squares). Thus, from Proposition 3.1 it follows that [I k , g i ] = I k for some 1 ≤ i ≤ m which implies the first part of the statement. If graph Γ is disconnected, let {g i } i∈A be some connected component of Γ. Using Proposition 3.1 build a set of indexes B, where for p ∈ B there is some q ∈ A such that [I p , g q ] = I p . Then (⊕ i∈A g i ) (⊕ k∈B I k ) is a direct summand of the Leibniz algebra L, which is a contradiction.
Using decomposition (2)   Proof. Assume that BΓ is not connected. Let ({g i } i∈A , {I k } k∈B ) be one of the connected components of BΓ. Then ( i∈A g i ) ( k∈B I k ) is a direct summand of the Leibniz algebra, thus the algebra is decomposable.
Conversely, if the Leibniz algebra is decomposable, then the corresponding bipartite graph is disconnected.
The following corollary establishes that the converse of Theorem 3.3 is also valid. The graphs Γ and BΓ corresponding to that decomposition are: Although Γ is connected, but since [I 1 , ⊕ 4 i=1 g i ] = {0}, the Leibniz algebra cannot be indecomposable. In fact, the graph BΓ shows that such Leibniz algebra is not only indecomposable, but does not exist since I 1 is not being generated by any elements of the Leibniz algebra.
The next statement shows the construction of an indecomposable semisimple Leibniz algebra from any finite connected bipartite graph. Indeed, pick any simple Lie algebra g 1 , . . . , g m and simple g i -modules M ji for all where J ki = M ki if a ki = 1 and J ki = C otherwise. By Theorem 2.12 the (⊕ m i=1 g i )-module I k is simple for all 1 ≤ k ≤ n. Then L = (⊕ m i=1 g i ) (⊕ n k=1 I k ) is the Leibniz algebra with corresponding bipartite graph BΓ. Note that by Theorem 3.5 it follows that L is indecomposable.

BIPARTITE GRAPHS, FINITE-DIMENSIONAL SEMISIMPLE LEIBNIZ ALGEBRAS 129
Example 3.9. Up to an isomorphism there are exactly two connected bipartite graphs with essential submatrix A being a 2 × 2 matrix: Then by the construction given in the Theorem 3.8 the corresponding indecomposable semisimple Leibniz algebras are the following: where J pq is a simple g q -module (and for L 2 , g-module J 21 ∼ = J 21 ⊗ C). Note that for both L 1 and L 2 the graph Γ is the same simple connected graph on two vertexes.
In conclusion, the study of the structure of semisimple finite-dimensional Leibniz algebras is complete. Indecomposable such algebras are build by construction from the proof of Theorem 3.8 if one chooses finite-dimensional simple Lie algebras g 1 , . . . , g m , well-known finite-dimensional irreducible g q -modules J pq for all 1 ≤ q ≤ m, 1 ≤ p ≤ n and a connected bipartite graph with bipartition of vertexes into m and n vertexes.