# zbMATH — the first resource for mathematics

On homotopes of Novikov algebras. (Russian, English) Zbl 1035.17003
Sib. Mat. Zh. 43, No. 1, 174-182 (2002); translation in Sib. Math. J. 43, No. 1, 140-146 (2002).
Let $$\Phi$$ be a unital associative commutative ring ($$\frac{1}{2}\in\Phi$$). Let $$V$$ be an $$\Phi$$-algebra, $$L(V)$$ be its algebra of left multiplications, $$\Gamma_l(V)$$ be the centralizer of $$L(V)$$ in $$\text{End}_{\Phi}V$$, $$\Delta(V)$$ be the set of $$\frac{1}{2}$$-derivations of $$V$$, and $$C(V)=\Gamma_l(V)\cap \Delta(V^{(-)})$$. If $$\phi\in \text{End}_{\Phi}V$$ then the operation $$x\cdot y=xy\phi$$ equips $$V$$ with the structure of a $$\Phi$$-algebra $$V_{\phi}$$ which is called a homotope of the algebra $$V$$. Let $$A$$ be a Novikov $$\Phi$$-algebra. In the article the following theorems are proved:
1. If $$\phi\in C(A)$$ then $$A_{\phi}$$ is a Novikov algebra;
2. If $$\phi\in L(A)$$ then $$A_{\phi}$$ is a Novikov algebra;
3. If $$\phi\in C(V)$$ is invertible then $$V_{\phi}$$ is a Novikov algebra iff $$V$$ is a Novikov algebra;
4. If $$A$$ is prime then $$\Gamma_l(A)$$ is a commutative subalgebra of $$\text{End}_{\Phi}A$$ and $$\Gamma_l(A)=C(A)$$;
5. $$A$$ is weakly prime (i.e. $$I_1I_2=0,\, I_2I_1=0$$ implies that either $$I_1=0$$ or $$I_2=0$$, for all ideals $$I_1,\, I_2$$) iff $$A$$ is prime.
##### MSC:
 17A30 Nonassociative algebras satisfying other identities
##### Keywords:
Novikov algebra; homotope; prime algebra
Full Text: