CHAPTER 2 CLASSICAL LIE SUPERALGEBRAS We present here the basic definitions and results concerning classical Lie su- peralgebras and their highest weight representations. For further details, see the articles [16] and [17]. Though a good deal of the following carries over to infinite dimensional Lie superalgebras, we will here only consider finite dimensional ones. A Lie superalgebra G is a complex vector space which is a direct sum of two subspaces GQ and G-f, called the even and the odd part, respectively, endowed with a super Lie bracket [.,.] : G x G G satisfying [x,y] = -(-l)ldOBx)l6°*v)bl,*h M M ] = [[*,»],*] + (-l)(desx)(degv)fo,M. Here, degx = 0 (1) if and only if x is even (odd). A bilinear form B on G is called invariant if (2.2.1) (i) B(x,y) = (~l)^x^^B(y,x). (2.2.2) (ii) B(x, y) = 0 for x G GQ and y G Gr. (2.2.3) (Hi) B([x,y],z) = B(x,\y,z]). The Lie superalgebra is then called (basic) classical if 2.3.a) G is simple. 2.3.b) GQ is reductive. 2.3.c) There exists a non-degenerate invariant bilinear form B on G. The list of (basic) classical Lie superalgebras is the following as determined by Kac : 2.4.1) Simple Lie algebras. 2.4.2) A(m,n),B(m,n),C(n),D(m,n),D(2,l,a),F(±), and G(3). We will describe the elements of the list in 2.4.2) in greater detail as we proceed. From now on we assume that G is a (basic) classical Lie superalgebra. We let H denote a fixed Cartan subalgebra of GQ, we let, as usual, A denote the set of roots of H in G, and for 7 G A we let ft7 G H be determined by 7(h) = B(h7,h), h G 4
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