On irreducible infinite conformal algebras

BOYALLIAN, CARINA; LIBERATI, JOSE

RESENHAS DO INSTITUTO DE MATEMATICA E ESTATISTICA DA UNIVERSIDADE DE SAO PAULO

Instituto de Matemática e Estatística da Universidade de São Paulo

Año: 2004 vol. 6 p. 129 - 140

0104-3854

The associative conformal algebra Cend$_N$ and the corresponding general Lieconformal algebra $gc_N$ are the most important examples of simple conformalalgebras which are not finite (see Sect. 2.10 in cite{K1}). One of themost important open problems of the theory of conformal algebras is theclassification of infinite   subalgebras of Cend$_N$ and of $gc_N$ whichact irreducibly on $cp^N$. (For a classification of such finite algebras, in the associative case seeTheorem~2.6 of the present paper, and in the (more difficult) Liecase see cite{CK} and cite{DK}.) The classical Burnside theorem states that any subalgebra of thematrix algebra $Mat_N CC$ that acts irreducibly on $CC^N$ isthe whole algebra $Mat_N CC$.  This is certainly not truefor  subalgebras of $Cend_N$ (which is the ``conformal´´analogue of $Mat_N CC$).  There is a family of infinitesubalgebras $Cend_{N,P}$ of $Cend_N$, where $P(x) in Mat_NCC [x]$, $det P (x) eq 0$, that still act irreducibly on $CC[partial]^N$.  One of the conjectures of cite{K2} states thatthere are no other infinite irreducible subalgebras of $Cend_N$. This conjecture was recently proved by Kolesnikov cite{Ko}. In the Lie conformal case, we have a conjecture  on the classification of infinite Lie conformal subalgebras ofgc$_N$  acting irreducibly on $Bbb C[partial]^N$, see Conjecture 4.4. This conjecture agrees with recent  results of E. Zelmanov [Z2] and  A. De Sole - V. Kac [DeK].