scholarly journals Invariant Hermitian forms on vertex algebras

2021 ◽  
pp. 2150059
Author(s):  
Victor G. Kac ◽  
Pierluigi Möseneder Frajria ◽  
Paolo Papi
Keyword(s):  
Simple Formula ◽  
Hermitian Form ◽  
Vertex Algebra ◽  
Vertex Algebras ◽  
Hermitian Forms

We study invariant Hermitian forms on a conformal vertex algebra and on their (twisted) modules. We establish existence of a non-zero invariant Hermitian form on an arbitrary [Formula: see text]-algebra. We show that for a minimal simple [Formula: see text]-algebra [Formula: see text] this form can be unitary only when its [Formula: see text]-grading is compatible with parity, unless [Formula: see text] “collapses” to its affine subalgebra.

On the axiomatic foundations of the theory of Hermitian forms

1970 ◽  
Vol 67 (2) ◽  
pp. 243-250 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Recent Work ◽  
Quadratic Forms ◽  
Hermitian Form ◽  
Major Objective ◽  
Hermitian Forms ◽  
Division Rings ◽  
General Terms ◽  
Definition Of ◽  
Simple Definition

In recent work on some topological problems (7), I was forced to adopt a complicated definition of ‘Hermitian form’ which differed from any in the literature. A recent paper by Tits(5) on quadratic forms over division rings contains a new and simple definition of these. A major objective of this paper is to formulate both these definitions in somewhat more general terms, and to show that they are equivalent.

TWISTED TENSOR PRODUCT MODULES OVER MÖBIUS TWISTED TENSOR PRODUCT NONLOCAL VERTEX ALGEBRAS

2013 ◽  
Vol 24 (05) ◽  
pp. 1350033 ◽  
Author(s):  
JIANCAI SUN ◽  
HENGYUN YANG
Keyword(s):  
Tensor Product ◽  
Vertex Algebra ◽  
Vertex Algebras ◽  
The Third ◽  
Twisted Tensor Product

This is the third part in a series of papers developing a twisted tensor product theory for nonlocal vertex algebras and its modules. In this paper we introduce and study twisted tensor product modules over Möbius twisted tensor product nonlocal vertex algebras. Among the main results, we find the isomorphic relation between the opposite Möbius twisted tensor product nonlocal vertex algebra and twisted tensor product of opposite Möbius nonlocal vertex algebras. And we also establish the isomorphism between two twisted tensor product contragredient modules. Furthermore, we study iterated twisted tensor product modules over iterated twisted tensor product nonlocal vertex algebras and find conditions for constructing an iterated twisted tensor product module of three factors.

scholarly journals Spinors and canonical hermitian forms

1988 ◽  
Vol 30 (3) ◽  
pp. 263-270 ◽  
Author(s):  
P. L. Robinson
Keyword(s):  
Clifford Algebra ◽  
Product Space ◽  
Hermitian Form ◽  
Inner Product Space ◽  
Inner Product ◽  
Spin Representation ◽  
Hermitian Forms ◽  
Real Inner Product Space ◽  
Adjoint Operation

The spaceSof spinors associated to a2m-dimensional real inner product space (V, B) carries a canonical Hermitian form 〈 〉 determined uniquely up to real multiples. This form arises as follows: the complex Clifford algebraC(V) of (V, B) is naturally provided with an antilinear involution; this induces an involution on EndSvia the spin representation; this is the adjoint operation corresponding to 〈 〉.

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 A SMASH PRODUCT CONSTRUCTION OF NONLOCAL VERTEX ALGEBRAS

2007 ◽  
Vol 09 (05) ◽  
pp. 605-637 ◽  
Author(s):  
HAISHENG LI
Keyword(s):  
Lie Algebras ◽  
Suitable Condition ◽  
Vertex Algebra ◽  
Vertex Algebras ◽  
Smash Product ◽  
Module Algebra ◽  
Primitive Elements ◽  
Product Construction

A notion of vertex bialgebra and a notion of nonlocal vertex module-algebra for a vertex bialgebra are studied and then a smash product construction of nonlocal vertex algebras is presented. For every nonlocal vertex algebra V satisfying a suitable condition, a canonical bialgebra B(V) is constructed such that primitive elements of B(V) are essentially pseudo-derivations and group-like elements are essentially pseudo-endomorphisms. As an application, vertex algebras associated with the Heisenberg Lie algebras as well as those associated with the nondegenerate even lattices are reconstructed through smash products, and furthermore, a different approach to the construction of modules for the lattice vertex algebras is given.

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 $-1$.

Mathieu–Zhao subspaces of vertex algebras

2019 ◽  
Vol 18 (12) ◽  
pp. 1950225
Author(s):  
Gaywalee Yamskulna
Keyword(s):  
Associative Algebra ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Vertex Operator Algebra ◽  
Quotient Space ◽  
Vertex Algebra ◽  
Jacobian Conjecture ◽  
Vertex Algebras ◽  
Text Type ◽  
Commutative Associative Algebra

We introduce a notion of Mathieu–Zhao subspaces of vertex algebras. Among other things, we show that for a vertex algebra [Formula: see text] and its subspace [Formula: see text] that contains [Formula: see text], [Formula: see text] is a Mathieu–Zhao subspace of [Formula: see text] if and only if the quotient space [Formula: see text] is a Mathieu–Zhao subspace of a commutative associative algebra [Formula: see text]. As a result, one can study the famous Jacobian conjecture in terms of Mathieu–Zhao subspaces of vertex algebras. In addition, for a [Formula: see text]-type vertex operator algebra [Formula: see text] that satisfies the [Formula: see text]-cofiniteness condition, we classify all Mathieu–Zhao subspaces [Formula: see text] that contain [Formula: see text].

General twisting of ϕ1-coordinated modules of nonlocal vertex algebras

2018 ◽  
Vol 29 (14) ◽  
pp. 1850098
Author(s):  
Jiancai Sun
Keyword(s):  
Tensor Products ◽  
Vertex Algebra ◽  
Vertex Algebras

We introduce a notion of [Formula: see text]-coordinated module twistor for a [Formula: see text]-coordinated module of a nonlocal vertex algebra. Furthermore, we study twisted tensor products and L-R-twisted tensor products of [Formula: see text]-coordinated modules of nonlocal vertex algebras and we unify these two constructions by [Formula: see text]-coordinated module twistors.

Twisted tensor products of φ-coordinated modules for nonlocal vertex algebras

2020 ◽  
Vol 32 (2) ◽  
pp. 433-446
Author(s):  
Jiancai Sun
Keyword(s):  
Tensor Products ◽  
Vertex Algebra ◽  
Vertex Algebras

AbstractWe study twisted tensor products of ϕ-coordinated modules of nonlocal vertex algebras. Furthermore, we introduce the notion of ϕ-coordinated module twistor for a ϕ-coordinated module of a nonlocal vertex algebra.

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