Binomial formula for powers of a derivation
Contents
Statement
Suppose is a Non-associative ring (?) (i.e., a not necessarily associative ring -- this includes the associative ring, Lie ring, and other cases) and is a derivation of . For any nonnegative integer , if we denote by the composition of with itself times, with defined as the identity map.
We then have:
Note that this in particular applies to the case of a Derivation of a Lie ring (?) and a Derivation of an associative ring (?).
Related facts
Applications
Proof
Proof idea
The idea is to use the Leibniz rule (which establishes the case ; is vacuous) and prove by induction. The proof is almost exactly like the proof of the binomial formula for for a commutative associative algebra.
Proof details: case
In the case , both sides are , so the equality holds vacuously.
Proof details: case
In this case, the left side is . The right side is:
By the Leibniz rule, the left side equals the right side.
Proof details: inductive step
Inductive hypothesis:
To prove:
Proof: We have by definition:
By the inductive hypothesis, we can expand the inside and we get:
Since is additive, we can pull it inside the summation on the right side and get:
Now using the Leibniz rule on the inside, we get:
We now rearrange the summation and obtain:
We use Pascal's identity on the sum of binomial coefficients and obtain what we want:
References
- Limits of abelian subgroups of finite p-groups by Jonathan Lazare Alperin and George Isaac Glauberman, Journal of Algebra, ISSN 00218693, Volume 203, Page 533 - 566(Year 1998): ^{Weblink for Elsevier copy}^{More info}, Proposition 2.5(a)