In the book Shun-Jen Cheng, Weiqiang Wang Dualities and Representations of Lie Superalgebrasm. One founds the following definition(Definition 1.3):

Let $\mathfrak{g}$ and $\mathfrak{g'}$ be Lie superalgebras. A homomorphism of Lie superalgebras is an even linear map $f: \mathfrak{g} \rightarrow \mathfrak{g'}$ satisfying $$f([a,b])=[f(a),f(b)],~ a, b \in \mathfrak{g}. ~~~~(*)$$

Here is my question:

Must a homomorphism of Lie superalgebras be even?

Assume $\mathfrak{g}$ is a Lie superalgebra, $A$ is a trivial $\mathfrak{g}$-supermodule. Then $A$ can be viewed as a Lie superalgebra with the zero superbracket. Let $\mathfrak{g'}=\mathfrak{g}\oplus A$ with $\mathfrak{g'}_{\bar 0}=\mathfrak{g}_{\bar 0}+A_{\bar 1}$, $\mathfrak{g'}_{\bar 1}=\mathfrak{g}_{\bar 1}+A_{\bar 0}$. Then $\mathfrak{g'}$ is a Lie superalgebra. We can define embeeding map $i: A \rightarrow \mathfrak{g'}=\mathfrak{g}\oplus A$, with the degree of $i$ is ${\bar 1}$. The map $i$ satisfies $$[i(a),i(b)]=i([a,b]).$$

Can we say this $i$ is a homomorphism of Lie superalgebras?

This post imported from StackExchange MathOverflow at 2014-11-12 16:42 (UTC), posted by SE-user Ruby