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 ⊕ L^{R+S} ⊕ L^{RT}, 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 d_{0} of the center of B is equal to the number of simple roots
which is not contained in S ⋃ T. If n = a_{0} + a_{1} +...+ a_{r} = b_{0} + b_{1} +...+ b_{s}
where a_{i}, b_{i} ∊ 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+sd_{0})d_{0}.
Advanced Studies: EuroTbilisi Mathematical Journal, Vol. 15(4) (2022), pp. 155160
