Advanced Studies: Euro-Tbilisi Mathematical Journal

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^{R_-^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 +...+ a_r = b₀ + b₁ +...+ 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+s-d₀)d₀.


Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 15(4) (2022), pp. 155-160