TITLE: The cooperations algebras Johnson-Wilson theory and truncated BP-theory
QUESTION [5 upvotes]: Given a prime $p$, there is a well known homology theory $BP$, known as Brown-Peterson homology. Has several related theories, namely the Johnson-Wilson theories $E(n)$ and the truncated Brown-Peterson theories $BP\langle n\rangle$. Their homotopy groups are given by 
$$
E(n)_* = \mathbb{Z}_{(p)}[v_1, ..., v_{n-1}, v_n^{\pm 1}]
$$
and $BP\langle n \rangle$ is a connective version of $E(n)$. In the case $n=1$, $p$-local $KU$ splits as a wedge of suspensions of $E(1)$, and $ku$ splits as a wedge of $BP\langle 1 \rangle$, and in this case the algebras $E(1)_*BP\langle 1\rangle$ and $E(1)_*E(1)$ are understood and the elements have interpretations as numerical polynomials. 
Does anyone know of methods to compute these algebras for $n>1$, or even $n=2$? I think I can compute $E(1)_*E(2)$ and $E(1)_*BP\langle 2\rangle$, so I would also be interested to know if there is a method of producing $E(2)_*E(2)$ from $E(1)_*E(2)$ and $K(2)_*E(2)$.

REPLY [5 votes]: Regarding $E(n)_*E(n)$, see "On the Structure of the Hopf Algebroid $E(n)_*E(n)$" by Keith Johnson. Johnson shows that $$E(n)_*E(n) \otimes \mathbb{Q} \simeq \mathbb{Q}[v_1,\cdots,v_{n-1},v_n^{\pm 1},\overline{v}_1,\cdots,\overline{v}_{n−1},\overline{v}_n^{\pm 1}],$$
where $v_i = \eta_L(v_i)$ and $\overline{v}_j = \eta_R(v_j)$.
Moreover, let $A$ be the ring of integers in an unramified degree $n$ extension of p-adic numbers, and let $\mathbb{S}_n$ be the $n$-th Morava stabilizer group. Johnson also shows that there is an embedding of $E(n)_*E(n)$ in $C(\mathbb{S}_n \times \mathbb{S}_n ,A)$.