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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
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