Full field algebras, operads and tensor categories

Let 𝔤𝔦𝔪(A [2]) be the generalized intersection matrix algebra associated to the 2-fold affinization A [2] of a simply-laced Cartan matrix . The root system, nondegenerate invariant bilinear form, and Weyl group of 𝔤𝔦𝔪(A [2]) are discussed.

Decompositions in categories of highest weight modules

Given a non-trivial torsion-free abelian group (A,+,Q), a field F of characteristic 0, and a non-degenerate bi-additive skew-symmetric map $$\phi$$ : A $$\times$$ A $$\rightarrow$$ F, we define a Lie algebra $${\cal L}$$ = $${\cal L}$$ (A, $$\phi$$ ) over F with basis {ex | x $$\in$$ A/{0}} and Lie product [ex,ey] = $$\phi$$ (x,y)ex+y. We show that $${\cal L}$$ is endowed uniquely with a non-degenerate symmetric invariant bilinear form and the derivation algebra Der $${\cal L}$$ of $${\cal L}$$ is a complete Lie algebra. We describe the double extension D( $${\cal L}$$ , T) of $${\cal L}$$ by T, where T is spanned by the locally finite derivations of $${\cal L}$$ , and determine the second cohomology group H2(D( $${\cal L}$$ , T),F) using anti-derivations related to the form on D( $${\cal L}$$ , T). Finally, we compute the second Leibniz cohomology groups HL2( $${\cal L}$$ , F) and HL2(D( $${\cal L}$$ , T), F).

A Lie algebra is said to be metric if it admits a symmetric, invariant, and non-degenerate bilinear form. The harmonic oscillator algebra, which arises in the quantum mechanical description of a harmonic oscillator, is the smallest solvable non-abelian metric example. This algebra is the first step of a countable series of solvable Lie algebras that support invariant Lorentzian forms. Generalizing this situation, in this paper, we arrive at the oscillator Lie K -algebras as double extensions of metric spaces. The aim of this paper is to present some structural features, invariant metrics, and derivations of this class of algebras and to explore their possibilities of being extended to mixed metric Lie algebras.

A system of relativistic Snyder particles with mutual two-body interaction that lives in a Non-Commutative Snyder geometry is studied. The underlying novel symplectic structure is a coupled and extended version of (single particle) Snyder algebra. In a recent work by Casalbuoni and Gomis, Phys.Rev. D90, 026001 (2014), a system of interacting conventional particles (in commutative spacetime) was studied with special emphasis on it's Conformal Invariance. Proceeding along the same lines we have shown that our interacting Snyder particle model is also conformally invariant. Moreover, the conformal Killing vectors have been constructed. Our main emphasis is on the Hamiltonian analysis of the conformal symmetry generators. We demonstrate that the Lorentz algebra remains undeformed but validity of the full conformal algebra requires further restrictions.

In this paper, we give the classification of the irreducible nonsolvable Lie algebras of dimensions \documentclass[12pt]{minimal}\begin{document}$\le\hspace*{-2.5pt}13$\end{document}≤13 with nondegenerate, symmetric, and invariant bilinear forms.

In this paper we study Hom Leibniz superalgebras endowed with a supersymmetric, nondegenerate and invariant bilinear form. Such Hom Leibniz superalgebras are called quadratic Hom Leibniz superalgebras. We show that every quadratic Hom Leibniz superalgebra is symmetric. After introducing representation of Hom Leibniz superalgebras, we prove that Hom Leibniz superalgebras are quadratic if and only if the adjoint representation and its dual representation are equivalent. We also extend the notion of T*-extension to Hom Leibniz superalgebras. Finally, by using double extension, we describe inductively quadratic regular Hom Leibniz superalgebras.

We study the structure of a metric n-Lie algebra \(\mathcal{G}\) over the complex field ℂ. Let \(\mathcal{G} = \mathcal{S} \oplus \mathcal{R}\) be the Levi decomposition, where \(\mathcal{R}\) is the radical of \(\mathcal{G}\) and \(\mathcal{S}\) is a strong semisimple subalgebra of \(\mathcal{G}\). Denote by \(m\left( \mathcal{G} \right)\) the number of all minimal ideals of an indecomposable metric n-Lie algebra and \(\mathcal{R}^ \bot\) the orthogonal complement of R. We obtain the following results. As \(\mathcal{S}\)-modules, \(\mathcal{R}^ \bot\) is isomorphic to the dual module of \({\mathcal{G} \mathord{\left/ {\vphantom {\mathcal{G} \mathcal{R}}} \right. \kern-\nulldelimiterspace} \mathcal{R}}\). The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on \(\mathcal{G}\) is equal to that of the vector space of certain linear transformations on \(\mathcal{G}\); this dimension is greater than or equal to \(m\left( \mathcal{G} \right) + 1\). The centralizer of \(\mathcal{R}\) in \(\mathcal{G}\) is equal to the sum of all minimal ideals; it is the direct sum of \(\mathcal{R}^ \bot\) and the center of \(\mathcal{G}\). Finally, \(\mathcal{G}\) has no strong semisimple ideals if and only if \(\mathcal{R}^ \bot \subseteq \mathcal{R}\).

Introduction. Let G1 and G be connected semisimple algebraic groups defined over a field K of characteristic zero and assume that there is an isomorphism f of G1 onto G which is defined over R, the algebraic closure of K. If p: G -? GL(V) is an absolutely irreducible (finite-dimensional) representation of G defined over K, then p of is an absolutely irreducible representation of G1 defined over K. Satake [7, p. 230] has shown that there is a field K1 which is a finite extension of K, a (unique) central simple division algebra K# defined over K1, a finite-dimensional right vector space V1 over K#, and a K1-homomorphism pi: G1 -GL(V1/K#) (the group of all nonsingular K#-linear endomorphisms of V1) such that (p of)(g) = 01(p1(g)) for all g E G1 where 01 is a unique absolutely irreducible representation of End (V1/K#) (the algebra of all K#-linear endomorphisms of V1) onto End (V). In this paper we are interested in the case where K= K1 and where there are invariant forms on V and V1. More precisely, we state the following two problems. PROBLEM 1. Assume that K#=K and that there are invariant bilinear forms B on V and B1 on V1 which are defined over K. What is the relationship between these two forms over K? Of course, if B is alternating, so is B1 and both are determined by dim V=dim V1. Hence, we shall always take B and B1 to be symmetric. PROBLEM 2. Assume that K# is a nontrivial division algebra over K (i.e., K# AK) and that there is an invariant bilinear form B on V and an invariant ?hermitian form F (c = + 1 or 1) on V1 both of which are defined over K. What is the relationship between these two forms over K? We are especially interested in the case K= Q,, a p-adic field. (In a future paper, we shall discuss the case K=R.) Here, some simplifications are immediately available. In Problem 2, it can be shown [7, p. 232] that K# has an involution of the first kind; but over Q,, it is known that the only such division algebra is the quaternion division algebra. Furthermore, it is known that a hermitian form on a finite-dimensional vector space over a quaternion division algebra defined over Q, is determined only by the dimension of the vector space. Therefore, in Problem 2 we shall always take F to be skew-hermitian; in the case where K# is a quaterion division algebra, this means that the form B is symmetric [7, p. 233]. If W is a vector space defined over K and if S is a symmetric form on W which is also defined over K, then three invariants can be associated with the pair (W, S),

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms, and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac–Moody algebras.

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V Superscript upper L"> <mml:semantics> <mml:msup> <mml:mi>V</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>L</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">V^{L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V Superscript upper R"> <mml:semantics> <mml:msup> <mml:mi>V</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>R</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">V^{R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be simple vertex operator algebras satisfying certain natural uniqueness-of-vacuum, complete reducibility and cofiniteness conditions and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a conformal full field algebra over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V Superscript upper L Baseline circled-times upper V Superscript upper R"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>V</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>L</mml:mi> </mml:mrow> </mml:msup> <mml:mo> ⊗ </mml:mo> <mml:msup> <mml:mi>V</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>R</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">V^{L}\otimes V^{R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q Subscript tau"> <mml:semantics> <mml:msub> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> τ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">q_{\tau }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> - <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q Subscript tau Baseline overbar"> <mml:semantics> <mml:mover> <mml:msub> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> τ </mml:mi> </mml:mrow> </mml:msub> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:annotation encoding="application/x-tex">\overline {q_{\tau }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -traces (natural traces involving <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q Subscript tau Baseline equals e Superscript 2 pi i tau"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> τ </mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi> π </mml:mi> <mml:mi>i</mml:mi> <mml:mi> τ </mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">q_{\tau }=e^{2\pi i\tau }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q Subscript tau Baseline overbar equals e Superscript 2 pi i tau Baseline overbar"> <mml:semantics> <mml:mrow> <mml:mover> <mml:msub> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> τ </mml:mi> </mml:mrow> </mml:msub> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mover> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi> π </mml:mi> <mml:mi>i</mml:mi> <mml:mi> τ </mml:mi> </mml:mrow> </mml:msup> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\overline {q_{\tau }}= \overline {e^{2\pi i\tau }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) of geometrically modified genus-zero correlation functions for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are convergent in suitable regions and can be extended to doubly periodic functions with periods <inline-formula content-type="math/mathml"> <mml

We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this construction also establishes the operator product expansion for intertwining operators, which correspond to chiral vertex operators, and more generally, it establishes the logarithmic operator product expansion for logarithmic intertwining operators. We review the main ideas in the construction of the tensor product bifunctors and the associativity isomorphisms. For rational and logarithmic conformal field theories, we review the precise results that yield braided tensor categories, and in the rational case, modular tensor categories as well. In the case of rational conformal field theory, we also briefly discuss the construction of the modular tensor categories for the Wess–Zumino–Novikov–Witten models and, especially, a recent discovery concerning the proof of the fundamental rigidity property of the modular tensor categories for this important special case. In the case of logarithmic conformal field theory, we mention suitable categories of modules for the triplet -algebras as an example of the applications of our general construction of the braided tensor category structure.

The two major approaches to chiral conformal field theory - one based on operator algebras and one based on vertex algebras - both lead to representation categories which are tensor categories and, in the case of rational chiral conformal field theories, more specifically modular tensor cat- egories. In this Arbeitsgemeinschaft, we have studied algebraic structures related to tensor categories arising in conformal field theory. The notion of a module category over this tensor category is central in the construction of a full local conformal field theory in various frameworks.

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

We describe the structure of the inclusions of factors A(E) contained in A(E')' associated with multi-intervals E of R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo-Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of A. As a consequence, the index of A(E) in A(E')' coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of A form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.