On commutative conformal algebras
by
Pavel S. Kolesnikov
Sobolev Institute of Mathematics, Novosibirsk, Russia

The theory of conformal algebras arose from the theory of vertex algebras, and the last one appeared from mathematical physics. Formal definition of conformal algebra was given by V. Kac [1].

We will investigate associative and associative-commutative (commutative for short) conformal algebras.

Definition. A linear space C endowed with a countable number of bilinear operations ·_(n)·: C×C --> C, n in Z₊, and with a linear operator D:C --> C is called associative conformal algebra, if it satisfies to the following conditions:

for allx, y in C existsN in Z+ : x(n) y=0 for n >= N,

(1)

D(x(n) y) = Dx(n) y + x(n) Dy,

(2)

Dx(n) y = -n x(n-1) y,

(3)

(x(n) y)(m) z = å s >= 0  (-1)s æ ç è n s ö ÷ ø x(n-s) (y(m+s) z).

(4)

Relations (1)-(3) are the common axioms of conformal algebras, (4) is called conformal associativity. Condition (1) defines locality function on C.

If in addition to (1)-(4) C satisfies to

x(n) y = y(n) x + å s >= 0  (-1)n+s  1 s! Ds (y(n+s) x),

(5)

then it is called commutative conformal algebra.

It seems to be interesting to develop the theory of commutative conformal algebras, similar to usual algebraic geometry (this idea was stated by V. Kac in verbal conversation in 1999). Here we give some results on this line, the main of them is

Theorem 1. Finitely generated commutative conformal algebra is nötherian.

The following theorem is analogous to the Cohen's theorem [2].

Theorem 2. Every commutative conformal algebra of bounded locality of generators satisfies to the arising chain condition for \Phi-ideals, i. e. ideals, invariant over substitution of generators.

But there is a difference between "classical" and conformal theory in the Specht-like questions.

Proposition 1. Let T be a system of identities T={T_n : x_(n) y = 0, n in Z₊} (conformal algebra, satisfying to T is trivial). Then for any n in Z₊ there exists 1-generated non trivial commutative conformal algebra, satisfying to every T_m, m <= n.

It means that the system of identities T is not equivalent to any its finite subsystem even for commutative conformal algebras. Here we have used Composition-diamond lemma for associative conformal algebras [3].

Some other questions of structure theory of conformal algebras occur to be similar to the "classical" ones.

Theorem 3. Let C be an associative conformal algebra. Then its lower nil-radical (Baer's radical, see, for example, [4]) coincides with the intersection of prime ideals of the algebra.

1. V. Kac. Vertex algebras for beginners. Second edition. University Lecture Series, 10. American Mathematical Society, Providence, RI, 1998.

2. D. E. Cohen. On the laws of a metabelian variety. J. Algebra. 1967. V. 5, N. 3. P. 267-273.

3. L. A. Bokut', Y. Fong, W.-F. Ke. Composition-Diamond lemma for associative conformal algebras, to appear in Contemporary Math., AMS.

4. N. Jacobson. Structure of rings. American Mathematical Society, Providence, R.I., 1956.

Date received: June 19, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafe-14.