**DIPS 02/2002**[tex source, PostScript, PDF file, dvi]-
**Title:**On n-ary generalizations of the Lie algebra sl_{2}()

**Author:**Arthemy V. KiselevPolynomials a

_{j}_{N}[x], 1__<__j__<__N, of degree deg a_{j}= N are shown to be closed w.r.t. action of the N-ary bracket [a_{1}, ..., a_{N}] = W(a_{1},...,a_{N}), where W denotes the Wronskian determinant. This bracket is proved to induce the homotopical N-Lie algebra structure on the polynomials_{N}[x] of degree N, so that an extended Jacobi identity (with 2^{N-1}summands) holds; the case N = 2 is the Lie bracket in sl_{2}() satisfying the Jacobi identity.The property of the Wronskian determinants to compose the homotopical N-Lie algebras is proved to hold for arbitrary analytic functions a

_{j}[[x]], thus extending the former result. The identity [[^{i1}...^{ik+1},^{j1}...^{jl+1}]]^{RN}= 0 is discussed, where is a derivation and [[^{.},^{.}]]^{RN}is the Richardson-Nijenhuis bracket.The notion of the Wronskian determinant is generalized to the case of n independent variables: (x

^{1},...,x^{n})^{n}, so that the resulting concept preserves the homotopical N-Lie Jacobi identity.

12 pages, LaTeX-2e (AmS-LaTeX).

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