So this question probably shows my inner model theoretic ignorance, but:


In "Two remarks on elementary embeddings of the universe" (<http://projecteuclid.org/download/pdf_1/euclid.pjm/1102969567>), Jech defines - for $j: V\rightarrow M$ a definable-with-parameters elementary embedding - an inner model $L(j)$, and proves that $L(j)$ is the smallest inner model which *admits* $j$ (in the sense that the inner model thinks $j$ is an elementary embedding as well).


This is a neat, short argument, and the definition of $L(j)$ is very straightforward. Naively, I would suspect that properties of the inner model $L(j)$ would tell us interesting information about the embedding $j$, and similarly for models $L(\overline{j})$ defined for sequences of embeddings $\overline{j}$. 


For instance, here's a question which seems natural to me: given an inner model $M\subset V$ and a family of embeddings $j\_\eta: V\rightarrow M$ $(\eta\in \kappa)$, it's reasonable to ask for which $I\subset \kappa$ is there an inner model $N$ admitting precisely the $j\_\eta$ with $\eta\in I$? In particular, given a left distributive algebra of embeddings, which sub-left distributive algebras can be captured this way? Naively, if I want to capture $\{j\_\eta: \eta\in I\}$, my first guess would be to look at $L(\{j\_\eta: \eta\in I\})$.


*NOTE: this isn't my actual question, I'm just including it to motivate interest in the $L(j)$s.*


However, I've never run across these models before, and I can't seem to find any recent reference to them. So, my question is: 



> 
> Are the $L(j)$s interesting at all from the point of view of modern inner model theory? If not, why not, and if so, why doesn't there seem to be any modern literature about them (maybe they are extremely hard to work with, or maybe there is such literature which I just haven't found yet)?
> 
> 
> 


(For my purposes, "modern" means "since 1990.")



Whatever happened to $L(j)$?
I recently noticed that Mitchell returned to $L[j]$ in ["Applications of the covering lemma for sequences of measures,"](https://www.ams.org/journals/tran/1987-299-01/S0002-9947-1987-0869398-2/S0002-9947-1987-0869398-2.pdf) where he shows that the model is quite a bit larger than one might expect. The thing is, $L[j]$ *isn't* $L[E\_j]$ where $E\_j$ is the short extender of $j$, it's $L[P]$ where $P = \{(x,A) : x\in j(A)\}$ is the entire *proper class long extender* of $j$. Assuming there is no inner model with a measurable $\kappa$ of order $\kappa^{++}$, Mitchell shows that if $j$ is nontrivial, $L[j]$ contains an iterate of the core model. Moreover, under the same assumption, he proves that if $j$ is the ultrapower embedding associated to a normal ultrafilter, then $L[j]$ satisfies $V = K$. So $L[j]$ contains many measurable cardinals of high order if $K$ does. As far as I know, no one has worked on the question at the level of strong/Woodin cardinals.


