Rationality of the vertex algebra VL+ when L is a non-degenerate even lattice of arbitrary rank

In this paper we prove that the vertex algebra $V_L^{+}$ is rational if L is a negative definite even lattice of finite rank, or if L is a non-degenerate even lattice of a finite rank that is neither positive definite nor negative definite. In particular, for such even lattices L, we show that the Zhu algebras of the vertex algebras $V_L^{+}$ are semisimple. This extends the result of Abe from [T. Abe, Rationality of the vertex operator algebra $V_L^{+}$ for a positive definite even lattice L, Math. Z. 249 (2) (2005) 455–484] which establishes the rationality of $V_L^{+}$ when L is a positive definite even lattice of finite rank.     

Connected quantized Weyl algebras and quantum cluster algebras

For an algebraically closed field $\mathbb{K}$, we investigate a class of noncommutative $\mathbb{K}$-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators $\{x_1,\dots ,x_n\}$ such that each pair satisfies a relation of the form $x_i{x}_j=q_{ij}{x}_j{x}_i+r_{ij}$, where $q_{ij}\in {\mathbb{K}}^{?}$ and $r_{ij}\in \mathbb{K}$, with, in some sense, sufficiently many pairs for which $r_{ij}\ne 0$. For such an algebra it turns out that there is a single parameter q such that each $q_{ij}=q^{\pm 1}$. Assuming that $q\ne \pm 1$, we classify connected quantized Weyl algebras, showing that there are two types linear and cyclic. When q is not a root of unity we determine the prime spectra for each type. The linear case is the easier, although the result depends on the parity of n, and all prime ideals are completely prime. In the cyclic case, which can only occur if n is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank.We apply connected quantized Weyl algebras to obtain presentations of two classes of quantum cluster algebras. Let $n\ge 3$ be an odd integer. We present the quantum cluster algebra of a Dynkin quiver of type $A_{n?1}$ as a factor of a linear connected quantized Weyl algebra by an ideal generated by a central element. We also consider the quiver $P_{n+1}^{(1)}$ identified by Fordy and Marsh in their analysis of periodic quiver mutation. When n is odd, we show that the quantum cluster algebra of this quiver is generated by a cyclic connected quantized Weyl algebra in n variables and one further generator. We also present it as the factor of an iterated skew polynomial algebra in $n+2$ variables by an ideal generated by a central element. For both classes, the quantum cluster algebras are simple noetherian.We establish Poisson analogues of the results on prime ideals and quantum cluster algebras. We determine the Poisson prime spectra for the semiclassical limits of the linear and cyclic connected quantized Weyl algebras and show that, when n is odd, the cluster algebras of $A_{n?1}$ and $P_{n+1}^{(1)}$ are simple Poisson algebras that can each be presented as a Poisson factor of a polynomial algebra, with an appropriate Poisson bracket, by a principal ideal generated by a Poisson central element.     

What does a group algebra of a free group “know” about the group?

We describe solutions to the problem of elementary classification in the class of group algebras of free groups. We will show that unlike free groups, two group algebras of free groups over infinite fields are elementarily equivalent if and only if the groups are isomorphic and the fields are equivalent in the weak second order logic. We will show that the set of all free bases of a free group F is 0-definable in the group algebra $K(F)$ when K is an infinite field, the set of geodesics is definable, and many geometric properties of F are definable in $K(F)$. Therefore $K(F)$ “knows” some very important information about F. We will show that similar results hold for group algebras of limit groups.     

Equations des algèbres Lie triple qui sont des algèbres train

In this paper, we consider equations of Lie triple algebras that are train algebras. We obtain two different types of equations depending on assuming the existence of an idempotent or a pseudo-idempotent.In general Lie triple algebras are not power-associative. However we show that their train equation with an idempotent is similar to train equations of power-associative algebras that are train algebras and we prove that Lie triple algebras that are train algebras of rank $\le 4$ with an idempotent are Jordan algebras.Moreover, the set of non-trivial idempotents has the same expression in Peirce decomposition as that of $e$-stable power-associative algebras.We also prove that the algebra obtained by $2$-gametization process of a Lie triple algebra is a Lie triple one.     

Universal quantum semigroupoids

We introduce the concept of a universal quantum linear semigroupoid (UQSGd), which is a weak bialgebra that coacts on a (not necessarily connected) graded algebra A universally while preserving grading. We restrict our attention to algebraic structures with a commutative base so that the UQSGds under investigation are face algebras (due to Hayashi). The UQSGd construction generalizes the universal quantum linear semigroups introduced by Manin in 1988, which are bialgebras that coact on a connected graded algebra universally while preserving grading. Our main result is that when A is the path algebra $\mathbb{k}Q$ of a finite quiver Q, each of the various UQSGds introduced here is isomorphic to the face algebra attached to Q. The UQSGds of preprojective algebras and of other algebras attached to quivers are also investigated.     

Idempotent et cohomologie de Hochschild

Any idempotent element e of an (associative) algebra T defines an algebra $A=eTe$ with unit e. We show that the morphism which compares their Hochschild cohomology algebras is a Gerstenhaber algebras morphism. Moreover, this morphism factorizes through the cohomological algebras of many triangular algebras. To cite this article: B. Bendiffalah, D. Guin, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

The sl3 Jones polynomial of the trefoil: A case study of q-holonomic sequences

The ${\mathfrak{sl}}_3$ colored Jones polynomial of the trefoil knot is a q-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this q-holonomic sequence as a case study. On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, this work illustrates the applicability and computational power of the employed computer algebra methods.     

Deformation of central charges,vertex operator algebras whose Griess algebras are Jordan algebras

If a vertex operator algebra $V={\oplus }_{n=0}^{\infty }{V}_n$ satisfies $\dim V_0=1$, $V_1=0$, then $V_2$ has a commutative (nonassociative) algebra structure called Griess algebra. One of the typical examples of commutative (nonassociative) algebras is a Jordan algebra. For example, the set ${\mathrm{Sym}}_d(\mathbb{C})$ of symmetric matrices of degree d becomes a Jordan algebra. On the other hand, in the theory of vertex operator algebras, central charges influence the properties of vertex operator algebras. In this paper, we construct vertex operator algebras with central charge c and its Griess algebra is isomorphic to ${\mathrm{Sym}}_d(\mathbb{C})$ for any complex number c and a positive integer d.     

Sur l'algèbre enveloppante d'une algèbre pré-Lie

We construct an associative product on the symmetric module $S(L)$ of any pre-Lie algebra L. It turns $S(L)$ into a Hopf algebra which is isomorphic to the envelopping algebra of $L_{\mathrm{Lie}}$. Then we prove that in the case of rooted trees our construction is dual to that of Connes and Kreimer. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees. To cite this article: J.-M. Oudom, D. Guin, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Isolation number versus Boolean rank

Let $\mathbb{B}$ be the binary Boolean algebra. The Boolean rank, or factorization rank, of a matrix A in $M_{m\text{,}n}(\mathbb{B})$ is the smallest k such that A can be factored as an $m\times k$ times a $k\times n$ matrix. The isolation number of a matrix, A, is the largest number of entries equal to 1 in the matrix such that no two ones are in the same row, no two ones are in the same column, and no two ones are in a submatrix of A of the form $\left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$. It is known that the isolation number of A is always at most the Boolean rank. This paper investigates for each k, if the isolation number of A is k what are some of the possible values of the Boolean rank of A.     

Graded Steinberg algebras and partial actions

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space which is totally disconnected, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{?}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.     

Amenability and vanishing of L2-Betti numbers: An operator algebraic approach

We introduce a Følner condition for dense subalgebras in finite von Neumann algebras and prove that it implies dimension flatness of the inclusion in question. It is furthermore proved that the Følner condition naturally generalizes the existing notions of amenability and that the ambient von Neumann algebra of a Følner algebra is automatically injective. As an application, we show how our techniques unify previously known results concerning vanishing of $L^2$-Betti numbers for amenable groups, quantum groups and groupoids and moreover provide a large class of new examples of algebras with vanishing $L^2$-Betti numbers.     

Integrality and gauge dependence of Hennings TQFTs

We provide a general construction of integral TQFTs over a general commutative ring, $\mathbb{k}$, starting from a finite Hopf algebra over $\mathbb{k}$ which is Frobenius and double balanced. These TQFTs specialize to the Hennings invariants of the respective doubles on closed 3-manifolds.We show the construction applies to index 2 extensions of the Borel parts of Lusztig's small quantum groups for all simple Lie types, yielding integral TQFTs over the cyclotomic integers for surfaces with one boundary component.We further establish and compute isomorphisms of TQFT functors constructed from Hopf algebras that are related by a strict gauge transformation in the sense of Drinfeld. Formulas for the natural isomorphisms are given in terms of the gauge twist element.These results are combined and applied to show that the Hennings invariant associated to quantum-${\mathfrak{sl}}_2$ takes values in the cyclotomic integers. Using prior results of Chen et al. we infer integrality also of the Witten–Reshetikhin–Turaev $SO(3)$ invariant for rational homology spheres.As opposed to most other approaches the methods described in this article do not invoke calculations of skeins, knots polynomials, or representation theory, but follow a combinatorial construction that uses only the elements and operations of the underlying Hopf algebras.