Conservative algebras of 2-dimensional algebras

The universal conservative superalgebra

“…It follows that associative, quasi-associative, Jordan, terminal, Lie, Leibniz, and Zinbiel superalgebras are conservative (see [10]). In particular, a superalgebra M is terminal if and only if it is conservative and the multiplication in the associated superalgebra M * can be given by…”

Conservative algebras of 2-dimensional algebras, II

“…The multiplication tabel of W (2) in this basis is given in [12]. In this work we use another basis for the algebra W (2).…”

“…. , e 4 is the conservative (and, moreover, terminal) algebra S 2 of all commutative 2-dimensional algebras with trace zero multiplication [12]. We denote by p k : W (2) → F Let Ann l (W (2)) be the space of W (2) generated by the elements e 4 , e 5 − 2e 1 − 3e 8 , e 3 + e 6 , and e 3 + e 7 .…”

“…Remark 10 It follows from the proof of [12,Theorem 3] that L e 7 and L e 8 are derivations of the algebra W (2). The map L e 7 is nilpotent and it is easy to see that T a = ∞ k=0…”

“…Some properties of the product in the algebra W (n) were studied in [11]. The study of low dimensional conservative algebras was started in [12], where derivations and subalgebras of codimension 1 of the algebra W (2) and of its principal subalgebras were described.…”