Low dimensional cohomologies of biparabolic subalgebras
G. Rakviashvili
The dimensions of zero and first regular cohomologies of a biparabolic subalgebra
B of some simple Lie algebra are calculated. Namely, it is proved that if S and T
are subsets of simple roots such as
B = H ⊕ LR+S ⊕ LR-T, where H is
a splitting Cartan subalgebra and R+S and R+T are the positive (negative) roots generated by
S (by T respectively) then the dimension d0 of the center of B is equal to the number of simple roots
which is not contained in S ⋃ T. If n = a0 + a1 +...+ ar = b0 + b1 +...+ bs
where ai, bi ∊ N
are ordered partititions of n and B is the corresponding biparabolic subalgebra of sl(n),
then the dimension of outer derivations of B is equal to (r+s-d0)d0.
Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 15(4) (2022), pp. 155-160
|