scholarly journals On the representation theory of the vertex algebra L−5/2(sl(4))

2021 ◽  
Author(s):  
Dražen Adamović ◽  
Ozren Perše ◽  
Ivana Vukorepa
Keyword(s):  
Representation Theory ◽  
Tensor Category ◽  
Vertex Algebra ◽  
Hamiltonian Reduction ◽  
Conformal Weight ◽  
Conformal Embedding ◽  
Fusion Algebra ◽  
Minimal Reduction ◽  
Simple Quotient ◽  
Conformal Embeddings

We study the representation theory of non-admissible simple affine vertex algebra [Formula: see text]. We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra [Formula: see text], and show that it generates the maximal ideal in [Formula: see text]. We classify irreducible [Formula: see text]-modules in the category [Formula: see text], and determine the fusion rules between irreducible modules in the category of ordinary modules [Formula: see text]. It turns out that this fusion algebra is isomorphic to the fusion algebra of [Formula: see text]. We also prove that [Formula: see text] is a semi-simple, rigid braided tensor category. In our proofs, we use the notion of collapsing level for the affine [Formula: see text]-algebra, and the properties of conformal embedding [Formula: see text] at level [Formula: see text] from D. Adamovic et al. [Finite vs infinite decompositions in conformal embeddings, Comm. Math. Phys. 348 (2016) 445–473.]. We show that [Formula: see text] is a collapsing level with respect to the subregular nilpotent element [Formula: see text], meaning that the simple quotient of the affine [Formula: see text]-algebra [Formula: see text] is isomorphic to the Heisenberg vertex algebra [Formula: see text]. We prove certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction functor [Formula: see text]. It turns out that the properties of [Formula: see text] are more subtle than in the case of minimal reduction.

Strongly graded vertex algebras generated by vertex Lie algebras

2019 ◽  
Vol 21 (08) ◽  
pp. 1850069
Author(s):  
Yufeng Pei ◽  
Jinwei Yang
Keyword(s):  
Lie Algebras ◽  
Category Theory ◽  
Tensor Category ◽  
Vertex Algebra ◽  
Vertex Algebras ◽  
Intertwining Operators ◽  
Infinite Dimensional ◽  
Conformal Field ◽  
Convergence And Extension ◽  
Logarithmic Intertwining Operators

We construct three families of vertex algebras along with their modules from appropriate vertex Lie algebras, using the constructions in [Vertex Lie algebra, vertex Poisson algebras and vertex algebras, in Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory[Formula: see text] Proceedings of an International Conference at University of Virginia[Formula: see text] May 2000, in Contemporary Mathematics, Vol. 297 (American Mathematical Society, 2002), pp. 69–96] by Dong, Li and Mason. These vertex algebras are strongly graded vertex algebras introduced in [Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, in Conformal Field Theories and Tensor Categories[Formula: see text] Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, eds. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2 (Springer, New York, 2014), pp. 169–248] by Huang, Lepowsky and Zhang in their logarithmic tensor category theory and can also be realized as vertex algebras associated to certain well-known infinite dimensional Lie algebras. We classify irreducible [Formula: see text]-gradable weak modules for these vertex algebras by determining their Zhu’s algebras. We find examples of strongly graded generalized modules for these vertex algebras that satisfy the [Formula: see text]-cofiniteness condition introduced in [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009] by the second author. In particular, by a result of the second author [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009, 26 pp.], the convergence and extension property for products and iterates of logarithmic intertwining operators in [Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2011); arXiv:1110.1929 ] among such strongly graded generalized modules is verified.

scholarly journals An Application of Collapsing Levels to the Representation Theory of Affine Vertex Algebras

2018 ◽  
Vol 2020 (13) ◽  
pp. 4103-4143 ◽  
Author(s):  
Dražen Adamović ◽  
Victor G Kac ◽  
Pierluigi Möseneder Frajria ◽  
Paolo Papi ◽  
Ozren Perše
Keyword(s):  
Maximal Ideal ◽  
Vertex Algebra ◽  
Lie Superalgebras ◽  
Vertex Algebras ◽  
Locally Finite ◽  
Complete Reducibility ◽  
Finite Module ◽  
Nontrivial Ideal ◽  
Conformal Embeddings ◽  
Reducibility Result

Abstract We discover a large class of simple affine vertex algebras $V_{k} ({\mathfrak{g}})$, associated to basic Lie superalgebras ${\mathfrak{g}}$ at non-admissible collapsing levels $k$, having exactly one irreducible ${\mathfrak{g}}$-locally finite module in the category ${\mathcal O}$. In the case when ${\mathfrak{g}}$ is a Lie algebra, we prove a complete reducibility result for $V_k({\mathfrak{g}})$-modules at an arbitrary collapsing level. We also determine the generators of the maximal ideal in the universal affine vertex algebra $V^k ({\mathfrak{g}})$ at certain negative integer levels. Considering some conformal embeddings in the simple affine vertex algebras $V_{-1/2} (C_n)$ and $V_{-4}(E_7)$, we surprisingly obtain the realization of non-simple affine vertex algebras of types $B$ and $D$ having exactly one nontrivial ideal.

scholarly journals Vertex Algebras Associated with Hypertoric Varieties

2020 ◽  
Author(s):  
Toshiro Kuwabara
Keyword(s):  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Hamiltonian Reduction ◽  
Vertex Algebras ◽  
Torus Action ◽  
Level 1 ◽  
Hypertoric Varieties ◽  
Construction Works

Abstract We construct a family of vertex algebras associated with a family of symplectic singularity/resolution, called hypertoric varieties. While the hypertoric varieties are constructed by a certain Hamiltonian reduction associated with a torus action, our vertex algebras are constructed by (semi-infinite) BRST reduction. The construction works algebro-geometrically, and we construct sheaves of $\hbar $-adic vertex algebras over hypertoric varieties, which localize the vertex algebras. We determine when it is a vertex operator algebra by giving an explicit conformal vector. We also discuss the Zhu algebra of the vertex algebra and its relation with a quantization of the hypertoric variety. In certain cases, we obtain the affine ${\mathcal{W}}$-algebra associated with the subregular nilpotent orbit in $\mathfrak{s}\mathfrak{l}_N$ at level $N-1$ and simple affine vertex operator algebra for $\mathfrak{s}\mathfrak{l}_N$ at level $-1$.

scholarly journals Differential equations and logarithmic intertwining operators for strongly graded vertex algebras

2017 ◽  
Vol 19 (02) ◽  
pp. 1650009 ◽  
Author(s):  
Jinwei Yang
Keyword(s):  
Differential Equations ◽  
Category Theory ◽  
Tensor Category ◽  
Extension Property ◽  
Vertex Algebra ◽  
Intertwining Operators ◽  
Matrix Elements ◽  
Systems Of Differential Equations ◽  
Convergence And Extension ◽  
Logarithmic Intertwining Operators

We derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded vertex algebra under a certain finiteness condition and a condition related to the horizontal gradings. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory for such strongly graded generalized modules developed by Huang, Lepowsky and Zhang.

Extension of Vertex Operator Algebra $V_{\widehat{H}_{4}}(\ell,0)$

2014 ◽  
Vol 21 (03) ◽  
pp. 361-380 ◽  
Author(s):  
Cuipo Jiang ◽  
Song Wang
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Vertex Algebra ◽  
Irreducible Representations ◽  
Natural Conditions ◽  
Irreducible Modules ◽  
Heisenberg Algebras

We classify the irreducible restricted modules for the affine Nappi-Witten Lie algebra [Formula: see text] with some natural conditions. It turns out that the representation theory of [Formula: see text] is quite different from the theory of representations of Heisenberg algebras. We also study the extension of the vertex operator algebra [Formula: see text] by the even lattice L. We give the structure of the extension [Formula: see text] and its irreducible modules via irreducible representations of [Formula: see text] viewed as a vertex algebra.

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