Jordan \(\varepsilon\)-homomorphisms and Jordan \(\varepsilon\)-derivations. (Q2493635)

Let \(G\) be an Abelian group with identity 1, \(R\) a \(G\)-graded algebra over a field \(F\) with \(\text{char\, }F\neq 2\), and \(H(R)\) the set of all homogeneous elements in \(R\). Assume that \(R\) is graded prime and that \(R_1\) is not commutative. Fix an anti-symmetric bicharacter \(\varepsilon\colon G\times G\to F^*\) and for \(a\in R_g\), \(b\in R_h\) set \(\varepsilon(a,b)=\varepsilon(g,h)\). For any \(a,b\in H(R)\) let \(a\circ_\varepsilon b=ab+\varepsilon(a,b)ba\). If \(T\) is a \(G\)-graded algebra over \(F\), then a linear and homogeneous \(\varphi\colon T\to R\) is an \(\varepsilon\)-homomorphism when for all \(a,b\in H(T)\), \(\varphi(ab)=\varphi(a)\varphi(b)\), \(\varphi\) is an \(\varepsilon\)-anti-homomorphism when \(\varphi(ab)=\varepsilon(a,b)\varphi(b)\varphi(a)\), and \(\varphi\) is a Jordan \(\varepsilon\)-homomorphism when \(\varphi(a\circ_\varepsilon b)=\varphi(a)\circ_\varepsilon\varphi(b)\). The first main result of the paper shows that when \(\varphi\) is a surjective Jordan \(\varepsilon\)-homomorphism, then \(\varphi\) is an \(\varepsilon\)-homomorphism or an \(\varepsilon\)-anti-homomorphism. Now for \(k\in G\), a linear \(D\colon R\to R\) satisfying \(D(R_g)\subseteq R_{kg}\) for all \(g\in G\), and with \(D(ab)=D(a)b+\varepsilon(a,b)aD(b)\) for all \(a,b\in H(R)\) is an \(\varepsilon\)-derivation of degree \(k\); \(D\) is a Jordan \(\varepsilon\)-derivation of degree \(k\) when \(D(a\circ_\varepsilon b)=D(a)\circ_\varepsilon b+\varepsilon(a,b)a\circ_\varepsilon D(b)\). Call a linear \(D\colon R\to R\) a (Jordan) \(\varepsilon\)-derivation if \(D\) is a finite sum of (Jordan) \(\varepsilon\)-derivations of different degrees. The second main result proves that any Jordan \(\varepsilon\)-derivation is an \(\varepsilon\)-derivation.