truncated induced modules over transitive lie

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TRUNCATED INDUCED MODULES OVER TRANSITIVE LIE ALGEBRAS OF CHARACTERISTIC

This content has been downloaded from IOPscience. Please scroll down to see the full text. 1990 Math. USSR Izv. 34 575 (http://iopscience.iop.org/0025-5726/34/3/A04) View the table of contents for this issue, or go to the journal homepage for more

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H3B. AKaa. HayK CCCP Cep. ΜβτβΜ. TOM 53(1989), N> 3

Math. USSR Izvestiya Vol. 34(1990), No. 3

TRUNCATED INDUCED MODULES OVER TRANSITIVE LIE ALGEBRAS OF CHARACTERISTIC ρ UDC 512.554 M. I. KUZNETSOV ABSTRACT. By constructing truncated coinduced modules a theorem is proved on the minimal imbedding of a transitive Lie algebra over a perfect field into the Lie algebra W(&~) . A formula for cohomology with coefficients in a truncated induced module is obtained. A description is given of filtered Lie algebras over a perfect field associated with graded Lie algebras of Cartan type and their derivations. Cartan prolongations of truncated induced modules are investigated. Bibliography: 29 titles.

Since the papers of Kostrikin and Shafarevich [l]-[3] transitive Lie algebras have occupied an important position in research on the problem of classifying finitedimensional simple Lie algebras over fields of characteristic ρ > 0. The well-known imbedding theorems of Guillemin and Sternberg [4], Rim [5], and Blattner [6] give a realization of the transitive Lie algebra (_?, Sf0) as a subalgebra of the Lie algebra of differential operators that act on infinite-dimensional commutative algebras. The latest result in this direction is the imbedding theorem of Block and Wilson [7]. They observed that over a field of characteristic ρ > 0 Blattner's construction leads to an imbedding of ( J ? , Jz^,) into the Lie algebra of special derivations of a complete divided power algebra. If Sf is a finite-dimensional Lie algebra, then there is special interest in the imbedding of (Jz?, -S^) into the finite-dimensional Lie algebra of special derivations W{&~) [2]. For filtered Lie algebras Sf such that %xS? is a graded Lie algebra of Cartan type with the standard grading in the case of an algebraically closed field of characteristic ρ > 3 such an imbedding was obtained by Wilson ([8], Proposition 5.1). In the general case, if an imbedding of 5f into W{3r) is possible, then the problem of choosing a minimal flag 9~ arises. Here one should take note of an assertion of Kats ([9], Theorem 0), which, because of omissions in the proof, can be regarded as a plausible conjecture. Truncated induced modules over graded Lie algebras were investigated by Krylyuk [10], [11] and Ermolaev [12]. They turned out to be convenient for describing irreducible representations of height 1 (in the sense of Rudakov [13]). Moreover, it turned out that in the case when 5? = W{^), S{&~), or β?{&~), in the truncated 1 induced module indF there exists an if (J?")-module structure relative to which 3 1980 Mathematics Subject Classification (1985 Revision). Primary 17B05, 17B20. 17B40, 17B5O, 17B70; Secondary 17B10, 17B35, 17B56, 17B65. © 1990 American Mathematical Society 0025-5726/90 $1.00+$.25 per page 575

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Μ. 1. K.UZNETSOV

acts as special derivations, ^-modules with an additional @(9r)-structure are especially interesting to cohomology theory, since they are finite-dimensional models of modules of cross-sections of geometric bundles over manifolds on which Lie algebras of Cartan type act [14]. In this article for a perfect field Κ, char .Κ = ρ > 0, truncated induced and coinduced modules over the finite-dimensional transitive Lie algebra {2C, J2?o) are constructed. On the basis of this construction, we obtain a theorem on the minimal imbedding of ( J ? , Jz^) into the finite-dimensional Lie algebra W(9~), where y = &~{J?, J?Q) is some canonical flag in the space Ε = (Jz?/Jz?0)* (§1, Theorem 1.1). Here the techniques developed by Blattner in [6] are used in an essential way. Inthe main theorem of §2 (Theorem 2.1) it is established that the classes of truncated induced and coinduced modules over ( J ? , _S^) coincide with the class of if (y)-modules on which S? acts as special derivations (in the imbedding of J ? into WiSf), y = ^ ( J ? , 5?0)). Theorem 2.1 is used in the investigation of irreducible modules of height 1 over filtered Lie algebras (Proposition 2.3) and, in particular, over filtered Lie algebras of Cartan type (Corollary 2.4). In §3 the cohomology of the Lie algebra {SC, J?o) with coefficients in a truncated induced module are studied (Theorem 3.1). Here a formula is obtained that is analogous to the Gel'fand-Fuks formula for diagonal cohomologies of the Lie algebra of smooth vector fields [15]. This formula is then applied to filtered Lie algebras, and under additional restrictions we find that the flags ^{S1, -2^) and &~(g, Q0) coincide, where g = %x3> = ®\=_ Li and Q0 = 0 / > ο -ί- ( . The conditions of Theorem 3.2 hold, in particular, when Q is a graded Lie algebra of Cartan type X(&~) with X = W, S, CS, %?, C%f, or X (p > 2) with standard gradings. Exactly as in [9], it follows that £? is isomorphic to a Lie algebra of Cartan type Χ{9Γ, ω) for ρ > 3, and for X = W for any ρ. This result was obtained previously by Wilson for an algebraically closed field. Theorem 3.2 also gives a description of the Lie algebra Ocr£f . For graded Lie algebras of Cartan type such a description was obtained by Tselousov [16]. In §4 it is shown that (W[^), W{JF)^ is the only medium of transitive Lie algebras S? for which the first Cartan prolongation of the pair (indF, S?) is nontrivial; this is possible only in the case dim V = 1 (Theorem 4.1). When 5? = WiSF), S{3r), or ^f'{9r), this assertion was established in [11], and for an irreducible graded module indF of height 1 over a graded Lie algebra it was established in [17]. We also note that the remark to §0 provides an example of nonisomorphic Lie algebras J?j and ^ with isomorphic universal enveloping bialgebras ^ ( - 2 p , / = 1,2. The results of this article were presented at the seminar of A. L. Kostrikin, Yu. A. Bakhturin, and Yu. P. Razmyslov at Moscow State University. The author expresses his thanks to the seminar's leaders for their valuable observations. §0. Definitions and preliminary results

Henceforth, if no additional restrictions are indicated, the ground field Κ is assumed to be perfect, with charA^ = ρ > 0. DEFINITION 0.1. We shall call a Lie algebra J ? with distinguished subalgebra Ji?Q , i.e., the pair ( J ? , 2^), a transitive Lie algebra if 1) ^ contains no nontrivial ideals of J ? , and 2) N^(£f0) = Lo , where Ν^{£?0) is the normalizer of ^ in S? .

MODULES OVER TRANSITIVE LIE ALGEBRAS

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Condition 2) holds, for example, for a maximal subalgebra 20 . Let 2 be a transitive Lie algebra and 2' a subalgebra of 2 with distinguished subalgebra 2Q . The pair (2', 2$) is called a transitive subalgebra in [2, .2^) if 2' + 2Q = 2 and 2' Π ^ = 2^. We shall call a homomorphism (3: (.2*, 20) -• (yf, yKQ) of transitive Lie algebras transitive if the pair (φ(2), φ{20)) is a transitive subalgebra in {JV, J^). Let (2,20) be a finite-dimensional transitive Lie algebra, A = %f(2) the universal enveloping algebra, 2* the universal p-span of the algebra 2 , 2* = 2 + 2" + •••+ 2P" + • • • , and Jt = N^»(20) the normalizer of 20 in 2* . Β is the subalgebra with 1 in A generated by the set JK. We define a height function ν on 2:v(l) for I e 2 is equal to the smallest m such that

From the properties of p-powers in an associative algebra we obtain the following properties of the function ν : (0.1) (0.2)

v(al) = v(l), aeK*=K\{0}, v(/ 1 +/ 2 )