Faithful representations of Leibniz algebras (Q2845447)

The paper is devoted to Leibniz algebras which present a non anti-symmetric generalization of Lie algebras. Equivalently, a (left) Leibniz algebra can be defined as a linear algebra \(L\) whose left multiplication operators \(d_a:L\rightarrow L\), defined by \(d_a(x)=ax\) for all \(a,x\in L\), are derivations.NEWLINENEWLINEA module for the Leibniz algebra \(L\) is a vector space \(V\) with two bilinear compositions \(xv\), \(vx\) for \(x\in L\) and \(v\in V\) such that NEWLINE\[NEWLINEx(yv)=(xy)v+y(xv),\quad x(vy)=(xv)y+v(xy),\quad v(xy)=(vx)y+x(vy)NEWLINE\]NEWLINE for all \(x, y\in L\) and \(v\in V\). The main result of the present paper asserts that if \(L\) is an \(n\)-dimensional Leibniz algebra then there exists a faithful \(L\)-module of dimension less than or equal to \(n+1\) with some additional properties.