New examples of infinite-dimensional n-Lie algebras

Author(s):  
Doston Jumaniyozov ◽  
Bakhrom Omirov
2011 ◽  
Vol 226 (2) ◽  
pp. 1911-1972 ◽  
Author(s):  
Maria Gorelik ◽  
Victor Kac

Author(s):  
C. J. Atkin

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].


2003 ◽  
Vol 12 (05) ◽  
pp. 589-604
Author(s):  
Hideaki Nishihara

Weight systems are constructed with solvable Lie algebras and their infinite dimensional representations. With a Heisenberg Lie algebra and its polynomial representations, the derived weight system vanishes on Jacobi diagrams with positive loop-degree on a circle, and it is proved that the derived knot invariant is the inverse of the Alexander-Conway polynomial.


1970 ◽  
Vol 22 (4) ◽  
pp. 839-846 ◽  
Author(s):  
Robert V. Moody

In [2], we considered a class of Lie algebras generalizing the classical simple Lie algebras. Using a field Φ of characteristic zero and a square matrix (Aij) of integers with the properties (1) Aii = 2, (2) Aij ≦ 0 if i ≠ j, (3) Aij = 0 if and only if Ajt = 0, and (4) is symmetric for some appropriate non-zero rational a Lie algebra E = E((Aij)) over Φ can be constructed, together with the usual accoutrements: a root system, invariant bilinear form, and Weyl group.For indecomposable (A ij), E is simple except when (Aij) is singular and removal of any row and corresponding column of (Aij) leaves a Cartan matrix. The non-simple Es, Euclidean Lie algebras, were our object of study in [3] as well as in the present paper. They are infinite-dimensional, have ascending chain condition on ideals, and proper ideals are of finite codimension.


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