Related papers: On Griess Algebras

Let $A$ be a unital associative algebra over a field $k$. All unital associative algebras containing $A$ as a subalgebra of a given codimension $\mathfrak{c}$ are described and classified. For a fixed vector space $V$ of dimension…

The problem of the classification of the extensions of the Virasoro algebra is discussed. It is shown that all $H$-reduced $\hat{\cal G}_{r}$-current algebras belong to one of the following basic algebraic structures: local quadratic…

One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative surfaces, and this paper resolves a significant case of this problem. Specifically, let S denote the 3-dimensional Sklyanin algebra…

For any triple $(\mu,\lambda,\alpha)$ of complex numbers and an $\mathfrak a$-module ${V}$, a class of non-weight modules $\mathcal{M}\big(V,\mu,\Omega(\lambda,\alpha)\big)$ over the Virasoro algebra $\mathcal L$ is constructed in this…

Let V be a vertex operator algebra and g an automorphism of order T. We construct a sequence of associative algebras A_{g,n}(V) with n\in\frac{1}{T}\Z nonnegative such that A_{g,n}(V) is a quotient of A_{g,n+1/T}(V) and a pair of functors…

Let $V=\oC^n$ and let $T:=T(V)\otimes T(V^*)$ be the mixed tensor algebra over $V$. We characterize those subsets $A$ of $T$ for which there is a subgroup $G$ of the unitary group $\UU(n)$ such that $A=T^G$. They are precisely the…

Attached to a vertex algebra $\mathcal{V}$ are two geometric objects. The associated scheme of $\mathcal{V}$ is the spectrum of Zhu's Poisson algebra $R_{\mathcal{V}}$. The singular support of $\mathcal{V}$ is the spectrum of the associated…

For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge 1/2, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice…

In this paper, we study contragredient duals and invariant bilinear forms for modular vertex algebras (in characteristic $p$). We first introduce a bialgebra $\mathcal{H}$ and we then introduce a notion of $\mathcal{H}$-module vertex…

Let $F$ be an arbitrary field and let $f:V\times V\to F$ be a non-degenerate symmetric or alternating bilinear form defined on an $F$-vector space of finite dimension $m\geq 2$. Let $L(f)$ be the subalgebra of $gl(V)$ formed by all…

A standard combinatorial construction, due to Kontsevich, associates to any A-infinity algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We…

We find that a compatible graded left-symmetric algebra structure on the Witt algebra induces an indecomposable module of the Witt algebra with 1-dimensional weight spaces by its left multiplication operators. From the classification of…

In this work, we propose to study noncommutative geometry using the language of categories of sheaves of algebras with polynomial identities and their properties, introducing new (graded) noncommutative geometries. These include, for…

Let $F$ be an algebraically closed field of characteristic zero, and $G$ be a finite abelian group. If $A=\oplus_{g\in G} A_g$ is a $G$-graded algebra, we study degree-inverting involutions on $A$, i.e., involutions $*$ on $A$ satisfying…

In this paper, we shall give a way to construct a graded Lie algebra $L(\mathfrak{g},\rho,V,{\cal V},B_0)$ from a standard pentad $(\mathfrak{g},\rho,V,{\cal V},B_0)$ which consists of a Lie algebra $\mathfrak{g}$ which has a non-degenerate…

If $A$ is a finite-dimensional algebra graded by a group $G$, and $\sigma \in G$, we define a variant of paratrophic matrix associated with $A$ and $\sigma$, and we use it to characterize the $\sigma$-graded Frobenius property for $A$. We…

We show that a class of algebras is closed under the taking of homomorphic images and direct products if and only if the class consists of all algebras that satisfy a set of (generally simultaneous) equations. For classes of regular…

Let $A$ be a $K$-subalgebra of the polynomial ring $S=K[x_1,\ldots,x_d]$ of dimension $d$, generated by finitely many monomials of degree $r$. Then the Gauss algebra $\GG(A)$ of $A$ is generated by monomials of degree $(r-1)d$ in $S$. We…

We show that the grading of fields by conformal weight, when built into the initial group symmetry, provides a discrete, non-central conformal extension of any group containing dilatations. We find a faithful vector representation of the…

Let $\F$ be an algebraically closed field. Let $\V$ be a vector space equipped with a non-degenerate symmetric or symplectic bilinear form $B$ over $\F$. Suppose the characteristic of $\F$ is \emph{large}, i.e. either zero or greater than…