Lattice Algebra Research Articles

Representations of quantum lattice vertex algebras

Let Q be a non-degenerate even lattice, let VQ be the lattice vertex algebra associated to Q, and let VQη be a quantum lattice vertex algebra ([10]). In this paper, we prove that every VQη-module is completely reducible, and the set of simple VQη-modules are in one-to-one correspondence with the set of cosets of Q in its dual lattice.

Completely fixed point free isometry and cyclic orbifold of lattice vertex operator algebras

We continue our study of cyclic orbifolds of lattice vertex operator algebras and their full automorphism groups. We consider some special isometry g∈O(L) such that gi is fixed point free on L for any 1≤i≤|g|−1. We show that when L(2)=∅ and gi is fixed point free on L for any 1≤i≤|g|−1, VLgˆ has extra automorphisms implies either (1) the order of g is a prime or (2) L is isometric to the Leech lattice or some coinvariant sublattices of the Leech lattice.

The classification of semi-conformal subalgebras of an even lattice vertex operator algebras with rank two

This work is to understand structures of even lattice vertex operator algebras by using the geometry of the varieties of their semi-conformal vectors. In this paper, we shall describe the varieties of semi-conformal vectors of a class of even lattice vertex operator algebras with rank two to characterize these vertex operator algebras. Moreover, we shall give the G-orbit decomposition of the fixed vertex operator subalgebra VL+ of a rank-two lattice vertex operator algebra VL, where G is the automorphism group preserving the conformal vector of VL+. Based on this decomposition, we classify all maximal semi-conformal subalgebras of VL+.

Modified bosonic integrable hierarchy

In this paper, we construct a new integrable hierarchy by using charged free boson and Friedan–Martinec–Shenker (FMS) bosonization, called modified bosonic integrable hierarchy, which is an anologue of the classical (l−l′)–th modified KP hierarchy. Firstly, the representation of glˆ∞ is given by using charged free bosons. Then the modified bosonic integrable hierarchy is proposed. Next after discussing FMS bosonizations in the language of lattice vertex algebra, bilinear equations of modified bosonic integrable hierarchy are constructed by FMS bosonization, which are rewritten into Hirota bilinear equations by using Hirota derivatives. At last, we consider solutions of this modified bosonic integrable hierarchy.

MBJ-Neutrosophic WI Ideals in Lattice Wajsberg Algebra

In this study, we introduce the concepts of MBJ-Neutrosophic WI-ideal and MBJ-Neutrosophic lattice ideal of lattice Wajsberg algebras. We demonstrate that every MBJ-Neutrosophic WI-ideal of lattice Wajsberg algebra is an MBJ-Neutrosophic lattice ideal of lattice Wajsberg algebra. Additionally, we talk about its opposite. Furthermore, we discover that in lattice H-Wajsberg algebra, every MBJ-Neutrosophic lattice ideal is an MBJ-Neutrosophic WI-ideal.

Orbifolds of lattice vertex algebras

Orbifolds of lattice vertex algebras

Extra automorphisms of cyclic orbifolds of lattice vertex operator algebras

In this article, we study automorphisms of the cyclic orbifold of a vertex operator algebra associated with a rootless even lattice for a lift of a fixed-point free isometry of odd prime order p. We prove that such a cyclic orbifold contains extra automorphisms, not induced from automorphisms of the lattice vertex operator algebra, if and only if the rootless even lattice can be constructed by Construction B from a code over Zp or is isometric to the coinvariant lattice of the Leech lattice associated with a certain isometry of order p.

Multiplicative generalized Lie n-derivations on completely distributive commutative subspace lattice algebras

Multiplicative generalized Lie n-derivations on completely distributive commutative subspace lattice algebras

Vertex algebraic construction of modules for twisted affine Lie algebras of type [formula omitted

Let g˜ be the affine Lie algebra of type A2l(2). The integrable highest weight g˜-module L(kΛ0) called the standard g˜-module is realized by a tensor product of the twisted module VLT for the lattice vertex operator algebra VL. By using such vertex algebraic construction, we construct bases of the standard module, its principal subspace and the parafermionic space. As a consequence, we obtain their character formulas and settle the conjecture for vacuum modules stated in [15].

{U}_{mathfrak{q}}{mathfrak{sl}}_2 -invariant non-compact boundary conditions for the XXZ spin chain

We introduce new {U}_{mathfrak{q}}{mathfrak{sl}}_2 -invariant boundary conditions for the open XXZ spin chain. For generic values of mathfrak{q} we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for mathfrak{q}={e}^{frac{ipi}{p}} a 2p-th root of unity — to its p-dimensional analogue. Both cases are parametrised by a continuous “spin” α ∈ ℂ.To motivate our construction, we first specialise to mathfrak{q} = i, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the (η, ξ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations.Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if mathfrak{q} α ∉± mathfrak{q} ℤ, the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to {U}_{mathfrak{q}}{mathfrak{sl}}_2 . Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of mathfrak{q} and α. Finally, we conjecture the general form of the Schur-Weyl duality in this case.This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.

On the automatic regularity of derivations from Riesz subalgebras of [formula omitted

We investigate the automatic regularity of bounded derivations from a Banach lattice algebra of regular operators A into a Banach A-module with a Banach lattice structure compatible with the module operations.

Vertex operator algebras with positive central charges whose dimensions of weight one spaces are 8 and 16

We classify rational simple self-dual vertex operator algebras V of CFT and of finite type with positive central charges, where dV=dim⁡V1 is either 8 or 16, under the condition that the space spanned by characters of V-modules is equal to the solution space of some monic modular linear differential equation of the third order. It is shown that central charges are c=8 for dV=8, and c=4,16 for dim⁡dV=16, respectively. In fact, under the condition that V1 equipped with the 0th product of V are abelian Lie algebras for dV=8,c=8 and dV=16,c=16, the vertex operator algebras V are isomorphic to V2E8 and VΛ16, respectively. For dV=16 and c=4, V is isomorphic to the lattice vertex operator algebra VA2⊕A2.

Tensor decomposition, parafermions, level-rank duality, and reciprocity law for vertex operator algebras

For a semisimple Lie algebra s l n \frak {sl}_n , the basic representation L s l n ^ ( 1 , 0 ) L_{\widehat {\frak {sl}_{n}}}(1,0) of the affine Lie algebra s l n ^ \widehat {\frak {sl}_{n}} is a lattice vertex operator algebra. The first main result of the paper is to prove that the commutant vertex operator algebra of L s l n ^ ( l , 0 ) L_{\widehat {\frak {sl}_{n}}}(l,0) in the l l -fold tensor product L s l n ^ ( 1 , 0 ) ⊗ l L_{\widehat {\frak {sl}_{n}}}(1,0)^{\otimes l} is isomorphic to the parafermion vertex operator algebra K ( s l l , n ) K(\frak {sl}_{l},n) , which is the commutant of the Heisenberg vertex operator algebra L h ^ ( n , 0 ) L_{\widehat {\frak {h}}}(n,0) in L s l l ^ ( n , 0 ) L_{\widehat {\frak {sl}_l}}(n,0) . The result provides a version of level-rank duality. The second main result of the paper is to prove more general version of the first result that the commutant of L s l n ^ ( l 1 + ⋯ + l s , 0 ) L_{\widehat {\frak {sl}_{n}}}(l_1+\cdots +l_s, 0) in L s l n ^ ( l 1 , 0 ) ⊗ ⋯ ⊗ L s l n ^ ( l s , 0 ) L_{\widehat {\frak {sl}_{n}}}(l_1,0)\otimes \cdots \otimes L_{\widehat {\frak {sl}_{n}}}(l_s, 0) is isomorphic to the commutant of the vertex operator algebra generated by a Levi Lie subalgebra of s l l 1 + ⋯ + l s \frak {sl}_{l_1+\cdots +l_s} corresponding to the composition ( l 1 , ⋯ , l s ) (l_1, \cdots , l_s) in the rational vertex operator algebra L s l ^ l 1 + ⋯ + l s ( n , 0 ) L_{\widehat {\frak {sl}}_{l_1+\cdots +l_s}}(n,0) . This general version also resembles a version of reciprocity law discussed by Howe in the context of reductive Lie groups. In the course of the proof of the main results, certain Howe duality pairs also appear in the context of vertex operator algebras.

Systematic orbifold constructions of Schellekens' vertex operator algebras from Niemeier lattices

We present a systematic, rigorous construction of all 70 strongly rational, holomorphic vertex operator algebras $V$ of central charge 24 with non-zero weight-one space $V_1$ as cyclic orbifold constructions associated with the 24 Niemeier lattice vertex operator algebras $V_N$ and certain 226 short automorphisms in $\operatorname{Aut}(V_N)$. We show that up to algebraic conjugacy these automorphisms are exactly the generalised deep holes, as introduced in arXiv:1910.04947, of the Niemeier lattice vertex operator algebras with the additional property that their orders are equal to those of the corresponding outer automorphisms. Together with the constructions in arXiv:1708.05990 and arXiv:1910.04947 this gives three different uniform constructions of these vertex operator algebras, which are related through 11 algebraic conjugacy classes in $\operatorname{Co}_0$. Finally, by considering the inverse orbifold constructions associated with the 226 short automorphisms, we give the first systematic proof of the result that each strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with non-zero weight-one space $V_1$ is uniquely determined by the Lie algebra structure of $V_1$.

Convergence structures and Hausdorff uo-Lebesgue topologies on vector lattice algebras of operators

A vector sublattice of the order bounded operators on a Dedekind complete vector lattice can be supplied with the convergence structures of order convergence, strong order convergence, unbounded order convergence, strong unbounded order convergence, and, when applicable, convergence with respect to a Hausdorff uo-Lebesgue topology and strong convergence with respect to such a topology. We determine the general validity of the implications between these six convergences on the order bounded operator and on the orthomorphisms. Furthermore, the continuity of left and right multiplications with respect to these convergence structures on the order bounded operators, on the order continuous operators, and on the orthomorphisms is investigated, as is their simultaneous continuity. A number of results are included on the equality of adherences of vector sublattices of the order bounded operators and of the orthomorphisms with respect to these convergence structures. These are consequences of more general results for vector sublattices of arbitrary Dedekind complete vector lattices. The special attention that is paid to vector sublattices of the orthomorphisms is motivated by explaining their relevance for representation theory on vector lattices.

First cohomologies of affine, Virasoro, and lattice vertex operator algebras

In this paper we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated to a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; (iii) the lattice VOA associated to a positive definite even lattice. We prove that in all these cases, the first cohomology $H^1(V, W)$ are given by the zero-mode derivations when $W$ is any $V$-module with an $\N$-grading (not necessarily by the operator $L(0)$). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. For negative energy representations of Virasoro VOA, the same conclusion holds when $W$ is $L(0)$-graded with lowest weight greater or equal to $-3$. Relationship between the first cohomology of the VOA and that of the associated Zhu's algebra is also discussed.

Eilenberg's variety theorem without Boolean operations

Eilenberg's variety theorem marked a milestone in the algebraic theory of regular languages by establishing a formal correspondence between properties of regular languages and properties of finite monoids recognizing them. Motivated by classes of languages accepted by quantum finite automata, we introduce basic varieties of regular languages, a weakening of Eilenberg's original concept that does not require closure under any boolean operations, and prove a variety theorem for them. To do so, we investigate the algebraic recognition of languages by lattice bimodules, generalizing Klíma and Polák's lattice algebras, and we utilize the duality between algebraic completely distributive lattices and posets.

Results on Topological Lattice Effect Algebras

In this paper, we define the notion of topological lattice effect algebras and investigate some of their properties. By using Sasaki arrows and F-balls, we construct two topology on lattice effect algebras. Then we study separation axioms on lattice effect algebras. Specifically, we find some conditions under which a topological lattice algebra is a $T_{0}$, $T_{1}$, and Hausdorff space. Finally, by using a strong filter and a quotient lattice effect algebra constructed by it, we investigate under what conditions this quotient lattice effect algebra will be a topological lattice effect algebra.

A Note of Jessen’s Inequality and Their Applications to Mean-Operators

A variant of Jessen’s type inequality for a semigroup of positive linear operators, defined on a Banach lattice algebra, is obtained. The corresponding mean value theorems lead to a new family of mean-operators.

Lattice structure of modular vertex algebras

In this paper we study the integral form of the lattice vertex algebra VL. We show that divided powers of general vertex operators preserve the integral lattice spanned by Schur functions indexed by partition-valued functions. We also show that the Garland operators, counterparts of divided powers of Heisenberg elements in affine Lie algebras, also preserve the integral form. These construe analogs of the Kostant Z-forms for the enveloping algebras of simple Lie algebras and the algebraic affine Lie groups in the situation of the lattice vertex algebras.