Algorithm for Constructing Integrable Lie Algebras

The results of this subsection rest on the following idea. Suppose a Lie algebra is give, on the orbits of which there exists a maximal linear commutative algebra of functions (that is, a complete involutive family). 

A commutative, associative, graduated algebra A = Aq 0 + with a unit element ei 6 Aq will be called a connected algebra with PoincarS duality if dimAo = 1 and if on the algebra A there exists a linear functional a, identically equal to zero on A for t n, such that the bilinear symmetric form a(a 6) (where a, 6 6 A) is nondegenerate. 

Let G be a Lie algebra and A a connected algebra with Poincar duality. Let a be a functional such that the quadratic form 0((a ), a 6 A is nondegenerate. Examine the restriction of this form to the space An/2 If n is even, then replacing 

Therefore, there exists a basis 7i.7ib of the space iln/2 that ci(lilT i)) = Sii for all t,y = 1.A , where r is a permutation such that is an identical permutation. Since the scalar product a ah) (where a, 6 A) is nondegenerate, it follows that dim Ai = dim An i for any t = 1. n. Choose a special basis B in the algebra A. First choose a basis in Aq. Since dimAo = 1 then li = forms a basis in Aq which consists of a single element. Next choose arbitrary bases ( f) in the spaces Ap for p(. If n = 2p is even then in the space Ap choose the basis (7 ) so that In the spaces Ap with graduation p f 

Henceforth we widely use multiindex notation / = (t l. is a multiindex, the product of linear functions // = / ... a constant coefficient at the monimial x/ in the sum (2). Besides, summation sign over repeated indices and multiindices is omitted everywhere. The short notation is / = fjxj. For any element a G A define the polynomial function 

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