Abstract. We classify unital associative conformal algebras of linear growth and provide new examples of such.

Invariant bilinear forms on vertex algebras have been around for quite some time now. They were mentioned by Borcherds in [1] and were used in many early works on vertex algebras, especially in relation with the vertex algebras associated with lattices [2, 3, 8]. The first systematic study of invariant forms on vertex algebras is due to Frenkel, Huang and Lepowsky [7]. This theory was

Abstract. We describe simple finitely generated associative conformal algebras of Gel’fand–Kirillov dimension one. 1.

Conformal algebras. A conformal algebra is, roughly speaking, a linear space A with infinitely many bilinear products (n) : A × A → A, parameterized by an non-negative integer n, and a derivation D: A → A. An important property of these products is that for any fixed a, b ∈ A we have a(n)b = 0 when n is large enough. See section §1 below for formal definitions.

The notion of vertex algebra due to Borcherds [3] and Frenkel-Lepowsky-Meurman [8] is one of the fundamental concepts of modern mathematics. The definition is purely algebraic, but its central point of the is an intricate Jacobi identity ([8]), which makes the definition hard to motivate from first principles. The identity can

Abstract. In this paper we prove that for any commutative (but in general non-associative) algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V0 ⊕ V2 ⊕ V3 ⊕ · · · , such that dim V0 = 1 and V2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V can be chosen with a non-degenerate invariant bilinear form, in which case it is simple. Key words: vertex algebra; Griess algebra

Abstract. In this paper we prove that for any commutative (but in general non-associative) algebra A with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra V = V0 ⊕ V1 ⊕ V2 ⊕..., such that dim V0 = 1, V1 = 0 and V2 contains A. We can choose V so that if A has a unit e, then 2e is the Virasoro element of V, and if G is a finite group of automorphisms of A, then G acts on V as well. In addition, the algebra V has a non-degenerate invariant bilinear form, and therefore is simple.

Invariant bilinear forms on vertex algebras have been around for quite some time now. They were mentioned by Borcherds in [1] and were used in many early works on vertex algebras, especially in relation with the vertex algebras associated with lattices [2, 3, 7]. The first systematic study of invariant forms on vertex algebras is due to Frenkel, Huang and Lepowsky [6]. This theory was