Combinatorics of free vertex algebras

This paper illustrates the combinatorial approach to vertex algebra - study of vertex algebras presented by generators and relations. A necessary ingredient of this method is the notion of free vertex algebra. Borcherds \cite{bor} was the first to note that free vertex algebras do not exist in general. The reason for this is that vertex algebras do not form a variety of algebras, because the locality axiom (see sec 2 below) is not an identity. However, a certain subcategory of vertex algebras, obtained by restricting the order of locality of generators, has a universal object, which we call the free vertex algebra corresponding to the given locality bound. In [J. of Algebra, 217(2):496-527] some free vertex algebras were constructed and in certain special cases their linear bases were found. In this paper we generalize this construction and find linear bases of an arbitrary free vertex algebra. It turns out that free vertex algebras are closely related to the vertex algebras corresponding to integer lattices. The latter algebras play a very important role in different areas of mathematics and physics. Here we explore the relation between free vertex algebras and lattice vertex algebras in much detail. These results comply with the use of the word "free" in physical literature refering to some elements of lattice vertex algebras, like in "free field", "free bozon" or "free fermion". Among other things, we find a nice presentation of lattice vertex algebras in terms of generators and relations, thus giving an alternative construction of these algebras without using vertex operators. We remark that our construction works in a very general setting; we do not assume the lattice to be positive definite, neither non-degenerate, nor of a finite rank

On free conformal and vertex algebras

Any variety of classical algebras has a so-called conformal counterpart. For example one can consider Lie conformal or associative conformal algebras. Lie conformal algebras are closely related to vertex algebras. We define free objects in the categories of conformal and vertex algebras. In some cases we can explicitly build their Groebner bases.Comment: AMS-LaTeX, 19 pages. To process, run "latex freecv.tex

On twisted representations of vertex algebras

In this paper we develop a formalism for working with twisted realizations of vertex and conformal algebras. As an example, we study realizations of conformal algebras by twisted formal power series. The main application of our technique is the construction of a very large family of representations for the vertex superalgebra \goth V_\Lambda corresponding to an integer lattice $\Lambda$. For an automorphism \^\sigma:\goth V_\Lambda\to\goth V_\Lambda coming from a finite order automorphism $\sigma:\Lambda \to \Lambda$ we define a category $\cal O_{\^\sigma}$ of twisted representations of \goth V_\Lambda and show that this category is semisimple with finitely many isomorphism classes of simple objects.Comment: Some corrections and addition

On Griess Algebras

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 $V = V_0 \oplus V_2 \oplus V_3\oplus ...$, such that $\dim V_0 = 1$ and $V_2$ 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.Comment: This is a contribution to the Special Issue on Kac-Moody Algebras and Applications, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Hybridity, Mestizaje, and Montubios in Ecuador

The 'Montubio' ethnic identity has recently gained notoriety in Ecuador. This paper analyses how this identity emerges from and falls within Ecuador's construction of 'mestizaje' or mixture as a tool for national integration. Given the exclusionary and limited nature of mestizaje in Ecuador, it is argued that as far as Montubios are uncritically constructed in relation to such mestizaje, they cannot serve as a progressive hybrid identity able to overcome essentialisms and existent ethnic structures. This paper starts by briefly reviewing how mestizaje has been constructed in Ecuador and then examines how the Montubio identity emerges from this mestizaje. It then explores different ways in which mestizaje may be conceptualized, and examines how these different models disguise or address power dynamics within heterogeneous populations. It concludes by briefly noting how 'translocational positionality' might provide a way to conceptualize the most progressive promises of mestizaje that Montubios might access.

Well-centered overrings of an integral domain

Let A be an integral domain with field of fractions K. We investigate the structure of the overrings B of A (contained in K) that are well-centered on A in the sense that each principal ideal of B is generated by an element of A. We consider the relation of well-centeredness to the properties of flatness, localization and sublocalization for B over A. If B = A[b] is a simple extension of A, we prove that B is a localization of A if and only if B is flat and well-centered over A. If the integral closure of A is a Krull domain, in particular, if A is Noetherian, we prove that every finitely generated flat well-centered overring of A is a localization of A. We present examples of (non-finitely generated) flat well-centered overrings of a Dedekind domain that are not localizations.Comment: Example 3.11 was replace