In this paper, the structure of cocommutative vertex bialgebras is investigated. For a general vertex bialgebra $V$, it is proved that the set $G(V)$ of group-like elements is naturally an abelian semigroup, whereas the set $P(V)$ of primitive elemen
ts is a vertex Lie algebra. For $gin G(V)$, denote by $V_g$ the connected component containing $g$. Among the main results, it is proved that if $V$ is a cocommutative vertex bialgebra, then $V=oplus_{gin G(V)}V_g$, where $V_{bf 1}$ is a vertex subbialgebra which is isomorphic to the vertex bialgebra ${mathcal{V}}_{P(V)}$ associated to the vertex Lie algebra $P(V)$, and $V_g$ is a $V_{bf 1}$-module for $gin G(V)$. In particular, this shows that every cocommutative connected vertex bialgebra $V$ is isomorphic to ${mathcal{V}}_{P(V)}$ and hence establishes the equivalence between the category of cocommutative connected vertex bialgebras and the category of vertex Lie algebras. Furthermore, under the condition that $G(V)$ is a group and lies in the center of $V$, it is proved that $V={mathcal{V}}_{P(V)}otimes C[G(V)]$ as a coalgebra where the vertex algebra structure is explicitly determined.
In this paper, we study nullity-2 toroidal extended affine Lie algebras in the context of vertex algebras and their $phi$-coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vert
ex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and $phi$-coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of irreducible modules for the variant Lie algebras.
Person re-identification (Re-ID) is a challenging task as persons are often in different backgrounds. Most recent Re-ID methods treat the foreground and background information equally for person discriminative learning, but can easily lead to potenti
al false alarm problems when different persons are in similar backgrounds or the same person is in different backgrounds. In this paper, we propose a Foreground-Guided Texture-Focused Network (FTN) for Re-ID, which can weaken the representation of unrelated background and highlight the attributes person-related in an end-to-end manner. FTN consists of a semantic encoder (S-Enc) and a compact foreground attention module (CFA) for Re-ID task, and a texture-focused decoder (TF-Dec) for reconstruction task. Particularly, we build a foreground-guided semi-supervised learning strategy for TF-Dec because the reconstructed ground-truths are only the inputs of FTN weighted by the Gaussian mask and the attention mask generated by CFA. Moreover, a new gradient loss is introduced to encourage the network to mine the texture consistency between the inputs and the reconstructed outputs. Our FTN is computationally efficient and extensive experiments on three commonly used datasets Market1501, CUHK03 and MSMT17 demonstrate that the proposed method performs favorably against the state-of-the-art methods.
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 algebra and a noti
on of $(V,mathcal{H})$-module for an $mathcal{H}$-module vertex algebra $V$. Then we give a modular version of Frenkel-Huang-Lepowskys theory and study invariant bilinear forms on an $mathcal{H}$-module vertex algebra. As the main results, we obtain an explicit description of the space of invariant bilinear forms on a general $mathcal{H}$-module vertex algebra, and we apply our results to affine vertex algebras and Virasoro vertex algebras.
In this paper, we study Virasoro vertex algebras and affine vertex algebras over a general field of characteristic $p>2$. More specifically, we study certain quotients of the universal Virasoro and affine vertex algebras by ideals related to the $p$-
centers of the Virasoro algebra and affine Lie algebras. Among the main results, we classify their irreducible $mathbb{N}$-graded modules by explicitly determining their Zhu algebras and show that these vertex algebras have only finitely many irreducible $mathbb{N}$-graded modules and they are $C_2$-cofinite.
This paper is about $phi$-coordinated modules for weak quantum vertex algebras. Among the main results, several canonical connections among $phi$-coordinated modules for different $phi$ are established. For vertex operator algebras, a reinterpretatio
n of Frenkel-Huang-Lepowskys theorem on contragredient module is given in terms of $phi$-coordinated modules.
We study $N$-graded $phi$-coordinated modules for a general quantum vertex algebra $V$ of a certain type in terms of an associative algebra $widetilde{A}(V)$ introduced by Y.-Z. Huang. Among the main results, we establish a bijection between the set
of equivalence classes of irreducible $N$-graded $phi$-coordinated $V$-modules and the set of isomorphism classes of irreducible $widetilde{A}(V)$-modules. We also show that for a vertex operator algebra, rationality, regularity, and fusion rules are independent of the choice of the conformal vector.
In this paper, we study in the context of quantum vertex algebras a certain Clifford-like algebra introduced by Jing and Nie. We establish bases of PBW type and classify its $mathbb N$-graded irreducible modules by using a notion of Verma module. On
the other hand, we introduce a new algebra, a twin of the original algebra. Using this new algebra we construct a quantum vertex algebra and we associate $mathbb N$-graded modules for Jing-Nies Clifford-like algebra with $phi$-coordinated modules for the quantum vertex algebra. We also show that the adjoint module for the quantum vertex algebra is irreducible.
This is a paper in a series systematically to study toroidal vertex algebras. Previously, a theory of toroidal vertex algebras and modules was developed and toroidal vertex algebras were explicitly associated to toroidal Lie algebras. In this paper,
we study twisted modules for toroidal vertex algebras. More specifically, we introduce a notion of twisted module for a general toroidal vertex algebra with a finite order automorphism and we give a general construction of toroidal vertex algebras and twisted modules. We then use this construction to establish a natural association of toroidal vertex algebras and twisted modules to twisted toroidal Lie algebras. This together with some other known results implies that almost all extended affine Lie algebras can be associated to toroidal vertex algebras.
In this paper, we study Heisenberg vertex algebras over fields of prime characteristic. The new feature is that the Heisenberg vertex algebras are no longer simple unlike in the case of characteristic zero. We then study a family of simple quotient v
ertex algebras and we show that for each such simple quotient vertex algebra, irreducible modules are unique up to isomorphism and every module is completely reducible. To achieve our goal, we also establish a complete reducibility theorem for a certain category of modules over Heisenberg algebras.
