I have already studied the notion of Operads. As I am new in this way, I need some clarification of basic concepts related to Operas through examples.
We have the following notation:
- $\sum$: a set of symbols of algebraic operations $f$ together with their arities $ \nu(f)$
Let suppose $ \sum$ consists of one binary operation $ [.,.]$, and let $Lie$ be a variety of $\sum $ algebras. Then $ \sum ^{(2)}$ consists of two operations, say, $ [. \vdash .]$ and $[ . \dashv .]$. How is $ \sum ^ {(2)}$ described?
What are operads $Perm$ and $ComTrias$ ?
How can we construct for example Leibniz algebras via $di-Lie$, where $$ di-Lie= Perm \otimes Lie$$
Example: Defining identities of tri-Lie-algebras is as follows: An algebra from tri-Lie is a Linear space with three binary operations $[.\vdash .]$ , $ [ . \dashv .]$ and $ [. \perp .]$, $[ a \vdash b]= -[ b \dashv a]$ such that $ [. \perp .]$ is a Lie operation, $ [ . \dashv .]$ satisfies Leibniz identity and they satisfy the following axioms : $$ [ x_{1} \perp [x_{2} \dashv x_{3}]]=[[x_{1} \dashv x_{2}] \perp x_{3}] + [ x_{2} \perp [x_{1} \dashv x_{3}]]$$ , $$[x_{1} \dashv [x_{2} \perp x_{3}]]=[x_{1} \dashv [x_{2} \dashv x_{3}]]$$. I want to sort out of these example and appreciate your explanation on that.