Chapter 5. Irreducible representations of Lie superalgebras

Abstract

If x ti , . . . , x ir do not generate a free associative algebra in r variables then it follows from the condition of linear independence of a lt ..., a s+1 that the number under consideration is greater then (8). Now recall that d r (m) = r m . It is not difficult to verify that with m tending to infinity (7) and (8), up to scalars, are equivalent to r m+l and r m (m(r -1) -1), respectively. For m sufficiently large (8) exceeds (7), a contradiction. Hence (5) and (2) are equal to zero and the result follows. Π Corollary. Let L = @ g€ G^g be a colour Lie superalgebra over an arbitrary field. Suppose that [L_,L_]=0. Then J(U(L)) = U(L)L_. Moreover, J(U(L)) is a locally nilpotent algebra. Proof. By setting M = L_ and applying the theorem one has J(U(L)} r> i/(L)L_. The converse inclusion follows from the fact that U(L)/ C7(L)L_ s U(L + ) is a prime algebra (2.8 of Chapter 3). Π 1.2. Example. Let L = Γ η (1, 1) be the Heisenberg Lie superalgebra with where z is central and the ground field K is arbitrary. Then J(U(L)) = 0.Proof. Consider a localization of U(L) by Z -{z'\i e N} and set K = K(z\ which is a transcendental extension. Then it is easy to verify that l/(L) z is isomorphic to the Clifford algebra B of the vector space V = <^a l ,...,a", b 1 ,..., £>">£ with a bilinear formSince Φ is non-degenerate Β is a simple algebra. This implies J(U(L)) = 0 because Ζ is a central set. Π The Generic Flatness Lemma.Let η be an integer. For any tuple ν = (v ls . . . , v") ε I^J" we set | v| = Vj + · · · + v n . There exists a unique total order < on N" such that: 1. ν < μ provided that |v| < |μ|; 2. among all elements with |v| fixed < coincides with the lexicographical order. It is easy to see that for any v, v', v" e F^" one has ν < v' if, and only if, ν + ν" < ν' + ν". We will also use another order on N".