October 22, 2007.

Let L be a Lie algebra.

1. A linear map N:L->L is called a Nijenhuis tensor if [Nx,Ny] - N([Nx,y]) - N([x,Ny]) + N^2[x,y] = 0. This comes I am not sure where from - differential forms, Poisson-Lie groups, etc... A simple but remarkable fact is that there is a relationship (one-to-one if a Lie algebra is finite-dimensional, Fitting decomposition plays a role) between decomposition of an algebra into the vector space direct sum of two subalgebras and Nijenhuis tensors on it (Kosmann-Schwarzbach-Magri).

2. A linear map J:L->L is called a Kaehler structure if J^2 = -1 and [Jx,Jy] - J([Jx,y]) - J([x,Jy]) - [x,y] = 0 (in reality, often more complex conditions arise, like validity of this identitiy modulo some ideal). This comes from Kaehler geometry, deformations of complex structures, etc. So, Kaehler structures are Nijenhuis tensors with N^2 = -1.

3. A famous R-matrix is a solution of the (modified?) classical Yang-Baxter equation [Rx,Ry] - R([Rx,y] - R([x,Ry]) + [x,y] = 0. This comes everybody knows where from - integrability of Hamiltonian systems, etc. There, a big role plays a decomposition of some (loop?) Lie algebras into the vector space direct sum of two subalgebras. So, in the class of operators with R^2 = 1, Nijenhuis tensors coincide with R-matrices (I am not sure that in reality such R-matrices exist).

4. A Saletan contraction is a map U:L->L such that [Ux,Uy]_N - U([Ux,y]_N) - U([x,Uy]_N) + U^2([x,y]_N) = 0, where N is a "1-part" of a Fitting decomposition of L with respect to U, and _N means projection on N. This comes from... ugh... mainly physics, I guess, in a constant quest for different types of symmetries and relations between them. Those are the "simplest" types of contractions with "linear" contracting term. Contractions, in a sense, are opposite to deformations. I guess, this equation is true for finite-dimensional Lie algebras only, as Fitting decomposition is utilized. So, Saletan contractions are Nijenhuis tensors on L with respect to multiplication [.,.]_N.

There should be a divine harmony behind all this. Hope I got the signs everywhere correctly.