The Cohomology of the Heisenberg Lie Algebras over Fields of Finite Characteristic

We give explicit formulas for the cohomology of the Heisenberg Lie algebras over fields of finite characteristic. We use this to show that in characteristic two, unlike all other cases, the Betti numbers are unimodal. The Heisenberg Lie algebra is the algebra hm with basis {x1, . . . , xm, y1, . . . , ym, z} and nonzero relations [xi, yi] = z, 1 ≤ i ≤ m. The cohomology (with trivial coefficients) H(hm) was one of the first explicit computations of the cohomology of a family of nilpotent Lie algebras. Louis Santharoubane [4] showed that over fields of characteristic zero, the Betti numbers are: dimH(hm) = ( 2m n )− ( 2m n−2 ), for all n ≤ m. Over fields of prime characteristic, the differential has larger kernel, and so one expects “more” cohomology. Recently, Emil Sköldberg [5] used algebraic Morse theory to compute the Poincaré polynomial Sm(t) = ∑ n dimH (hm)t of the Heisenberg Lie algebra hm over fields of characteristic two. He obtained Sm(t) = (1 + t)(1 + t) + (t+ t)(2t) 1 + t2 . In this paper we extend Sköldberg’s result to arbitrary characteristic by directly computing the Betti numbers. Theorem. Over fields of characteristic p, one has dimH(hm) = ( 2m n ) − ( 2m n− 2 ) + n+1 2p ∑ i=1 ( 2m+ 1 n− 2ip+ 1 ) − n−1 2p ∑ i=1 ( 2m+ 1 n− 2ip− 1 )