Non-degenerate Invariant Bilinear Form Research Articles

Invariant bilinear forms under the operator group of order p³ with odd prime p

For an odd prime p, we formulate the number of all degree n representations of a group of order p3. And calculating the dimension of space of invariant bilinear forms corresponding to degree n representation over a field F which contains a primitive p3 root of unity. Here we also explicitly discussed the existence of a non-degenerate invariant bilinear form of the same space.

Non-relativistic and ultra-relativistic expansions of three-dimensional spin-3 gravity theories

In this paper, we present novel and known non-relativistic and ultra-relativistic spin-3 algebras, by considering the Lie algebra expansion method. We start by applying the expansion procedure using different semigroups to the spin-3 extension of the AdS algebra, leading to spin-3 extensions of known non-relativistic and ultra-relativistic algebras. We then generalize the procedure considering an infinite-dimensional semigroup, which allows to obtain a spin-3 extension of two new infinite families of the Newton-Hooke type and AdS Carroll type. We also present the construction of the gravity theories based on the aforementioned algebras. In particular, the expansion method based on semigroups also allows to derive the (non-degenerate) invariant bilinear forms, ensuring the proper construction of the Chern-Simons gravity actions. Interestingly, in the vanishing cosmological constant limit we recover the spin-3 extensions of the infinite-dimensional Galilean and infinite-dimensional Carroll gravity theories.

A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of B_n

Generalizing a construction presented in Arsie and Lorenzoni (Lett Math Phys 107:1919–1961, 2017), we show that the orbit space of B_2 less the image of the coordinate lines under the quotient map is equipped with two Dubrovin-Frobenius manifold structures which are related respectively to the defocusing and the focusing nonlinear Schrödinger (NLS) equations. Motivated by this example, we study the case of B_n and we show that the defocusing case can be generalized to arbitrary n leading to a Dubrovin-Frobenius manifold structure on the orbit space of the group. The construction is based on the existence of a non-degenerate and non-constant invariant bilinear form that plays the role of the Euclidean metric in the Dubrovin–Saito standard setting. Up to n=4 the prepotentials we get coincide with those associated with the constrained KP equations discussed in Liu et al. (J Geom Phys 97:177–189, 2015).

On the Cohomology of Lie Algebras with an Invariant Inner Product

In this work we consider the moduli space of all noncommutative metric Lie algebras, having a nondegenerate symmetric invariant bilinear form, over \(\mathbb C\) and \(\mathbb R\) up to dimension 5 and all metric Lie algebras over \(\mathbb C\) in dimension 6. We introduce cyclic and reduced cyclic cohomology to identify their metric deformations.

SPACE OF INVARIANT BILINEAR FORMS UNDER REPRESENTATION OF A GROUP OF ORDER 8

Let G be a group of order 8 and F an algebraically closed field with char(F)≠2. In this paper we compute the number of n degree representations of G and subsequent dimensions of the corresponding spaces of invariant bilinear forms over the field F. We explicitly discuss about the existence of non-degenerate invariant bilinear forms.

Matter-free higher spin gravities in 3D: Partially-massless fields and general structure

We study the problem of interacting theories with (partially)-massless and conformal higher spin fields without matter in three dimensions. A new class of theories that have partially-massless fields is found, which significantly extends the well-known class of purely massless theories. More generally, it is proved that the complete theory has to have a form of the flatness condition for a connection of a Lie algebra, which, provided there is a non-degenerate invariant bilinear form, can be derived from the Chern-Simons action. We also point out the existence of higher spin theories without the dynamical graviton in the spectrum. As an application of a more general statement that the frame-like formulation can be systematically constructed starting from the metric one by employing a combination of the local BRST cohomology technique and the parent formulation approach, we also obtain an explicit uplift of any given metric-like vertex to its frame-like counterpart. This procedure is valid for general gauge theories while in the case of higher spin fields in d-dimensional Minkowski space one can even use as a starting point metric-like vertices in the transverse-traceless gauge. In particular, this gives the fully off-shell lift for transverse-traceless vertices.

Quantization of continuum Kac–Moody algebras

Continuum Kac-Moody algebras have been recently introduced by the authors and O. Schiffmann. These are Lie algebras governed by a continuum root system, which can be realized as uncountable colimits of Borcherds-Kac-Moody algebras. In this paper, we prove that any continuum Kac-Moody algebra is canonically endowed with a non-degenerate invariant bilinear form. The positive and negative Borel subalgebras form a Manin triple with respect to this pairing, inducing on the continuum Kac-Moody algebra a topological quasi-triangular Lie bialgebra structure. We then construct an explicit quantization, which we refer to as a continuum quantum group, and we show that the latter is similarly realized as an uncountable colimit of Drinfeld-Jimbo quantum groups.

Universal central extensions of current Lie superalgebras

We identify the universal central extension of g=A⊗k, where k is a finite dimensional perfect Lie superalgebra equipped with a nondegenerate homogeneous invariant supersymmetric bilinear form κ which is invariant under all derivations and A is a unital supercommutative associative (super)algebra.

2-Local Automorphisms on Basic Classical Lie Superalgebras

Let G be a basic classical Lie superalgebra except A(n, n) and D(2, 1, α) over the complex number field ℂ. Using existence of a non-degenerate invariant bilinear form and root space decomposition, we prove that every 2-local automorphism on G is an automorphism. Furthermore, we give an example of a 2-local automorphism which is not an automorphism on a subalgebra of Lie superalgebra spl(3, 3).

Double extensions of Lie superalgebras in characteristic 2 with nondegenerate invariant supersymmetric bilinear form

A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form will be called a NIS-Lie (super)algebra. The double extension of a NIS-Lie (super)algebra is the result of simultaneously adding to it a central element and an outer derivation so that the larger algebra has also a NIS. Affine loop algebras, Lie (super)algebras with symmetrizable Cartan matrix over any field, Manin triples, symplectic reflection (super)algebras are among the Lie (super)algebras suitable to be doubly extended.We consider double extensions of Lie superalgebras in characteristic 2, and concentrate on peculiarities of these notions related with the possibility for the bilinear form, the center, and the derivation to be odd. Two Lie superalgebras we discovered by this method are indigenous to the characteristic 2.

Minimal graded Lie algebras and representations of quadratic algebras

Let (g0,B0) be a quadratic Lie algebra (i.e. a Lie algebra g0 with a non-degenerate symmetric invariant bilinear form B0) and let (ρ,V) be a finite dimensional representation of g0. We define on Γ(g0,B0,V)=V⁎⊕g0⊕V a structure of local Lie algebra in the sense of Kac ([4]), where the bracket between g0 and V (resp. V⁎) is given by the representation ρ (resp. ρ⁎), and where the bracket between V and V⁎ depends on B0 and ρ. This implies the existence of two Z-graded Lie algebras gmax(Γ(g0,B0,V)) and gmin(Γ(g0,B0,V)) whose local part is Γ(g0,B0,V). We investigate these graded Lie algebras, more specifically in the case where g0 is reductive. Roughly speaking, the map (g0,B0,V)⟼gmin(Γ(g0,B0,V)) is a bijection between triplets and a class of graded Lie algebras. We show that the existence of “associated sl2-triples” is equivalent to the existence of non-trivial relative invariants on some orbit, and we define the “graded Lie algebras of polynomial type” which give rise to some dual airs.

Invariant bilinear forms of algebras given by faithfully flat descent

The existence of nondegenerate invariant bilinear forms is one of the most important tools in the study of Kac–Moody Lie algebras and extended affine Lie algebras. In practice, these forms are created, or shown to exist, either by assumption or in an ad hoc basis. The purpose of this work is to describe the nature of the space of invariant bilinear forms of certain algebras given by faithfully flat descent (which includes the affine Kac–Moody Lie algebras, as well as Azumaya algebras and multiloop algebras) within a functorial framework. This will allow us to conclude the existence, uniqueness and nature of invariant bilinear forms for many important classes of algebras.

On Generalized Intersection Matrix Algebras Arising from 2-Fold Affinization of Cartan Matrices

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.

Octo-bialgebras

The notion of octo-algebra was introduced by Leroux as a Loday algebra with 8 operations. In this paper, we introduce a notion of octo-bialgebra as a bialgebra theory of octo-algebras, which is equivalent to a double construction of a quadri-algebra with a nondegenerate 2-cocycle or a double construction of an octo-algebra with a nondegenerate invariant bilinear form. Some properties of octo-bialgebras are given, including the study of the coboundary cases which leads to a construction from an analogue of the classical Yang-Baxter equation in an octo-algebra.

An operator approach to the rational solutions of the classical Yang-Baxter equation

Motivated by the study of the operator forms of the constant classical Yang-Baxter equation given by Semonov-Tian-Shansky, Kupershmidt and the others, we try to construct the rational solutions of the classical Yang-Baxter equation with parameters by certain linear operators. The fact that the rational solutions of the CYBE for the simple complex Lie algebras can be interpreted in term of certain linear operators motivates us to give the notion of $\mathcal O$-operators such that these linear operators are the $\mathcal O$-operators associated to the adjoint representations. Such a study can be generalized to the Lie algebras with nondegenerate symmetric invariant bilinear forms. Furthermore we give a construction of a rational solution of the CYBE from an $\mathcal O$-operator associated to the coadjoint representation and an arbitrary representation with a trivial product in the representation space respectively.

On a class of Lie groups with a left invariant flat pseudo-metric

The study of the Lie groups with a left invariant flat pseudo-metric is equivalent to the study of the left-symmetric algebras with a nondegenerate left invariant bilinear form. In this paper, we consider such a structure satisfying an additional condition that there is a decomposition into a direct sum of the underlying vector spaces of two isotropic subalgebras. Moreover, there is a new underlying algebraic structure, namely, a special L-dendriform algebra and then there is a bialgebra structure which is equivalent to the above structure. The study of coboundary cases leads to a construction from an analogue of the classical Yang–Baxter equation.

Cardy Condition for Open-Closed Field Algebras

Let V be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that $${\mathcal{C}_V}$$ , the category of V-modules, is a modular tensor category. We study open-closed field algebras over V equipped with nondegenerate invariant bilinear forms for both open and closed sectors. We show that they give algebras over a certain $${\mathbb {C}}$$ -extension of the so-called Swiss-cheese partial dioperad, and we can obtain Ishibashi states easily in such algebras. The Cardy condition can be formulated as an additional condition on such open-closed field algebras in terms of the action of the modular transformation $${S: \tau \mapsto -\frac{1}{\tau}}$$ on the space of intertwining operators of V. We then derive a graphical representation of S in the modular tensor category $${\mathcal{C}_V}$$ . This result enables us to give a categorical formulation of the Cardy condition and the modular invariance condition for 1-point correlation functions on the torus. Then we incorporate these two conditions and the axioms of the open-closed field algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called the Cardy $${\mathcal{C}_V|\mathcal{C}_{V \otimes V}}$$ -algebra. In the end, we give a categorical construction of the Cardy $${\mathcal{C}_V|\mathcal{C}_{V \otimes V}}$$ -algebra in the Cardy case.

Classification of finite-dimensional solvable Lie algebras with nondegenerate invariant bilinear forms

In this paper, finite-dimensional solvable Lie algebras with nondegenerate invariant bilinear forms are classified. The classification is based on the investigation of the minimal generating sets, which are proved to be one of the three types: paired type, isolated type and mixed type.

Full field algebras, operads and tensor categories

We study the operadic and categorical formulations of (conformal) full field algebras. In particular, we show that a grading-restricted R × R -graded full field algebra is equivalent to an algebra over a partial operad constructed from spheres with punctures and local coordinates. This result is generalized to conformal full field algebras over V L ⊗ V R , where V L and V R are two vertex operator algebras satisfying certain finiteness and reductivity conditions. We also study the geometry interpretation of conformal full field algebras over V L ⊗ V R equipped with a nondegenerate invariant bilinear form. By assuming slightly stronger conditions on V L and V R , we show that a conformal full field algebra over V L ⊗ V R equipped with a nondegenerate invariant bilinear form exactly corresponds to a commutative Frobenius algebra with a trivial twist in the category of V L ⊗ V R -modules. The so-called diagonal constructions [Y.-Z. Huang, L. Kong, Full field algebras, arXiv: math.QA/0511328] of conformal full field algebras are given in tensor-categorical language.

Clover calculus for homology 3-spheres via basic algebraic topology

We present an alternative definition for the Goussarov--Habiro filtration of the Z-module freely generated by oriented integral homology 3-spheres, by means of Lagrangian-preserving homology handlebody replacements (LP-surgeries). Garoufalidis, Goussarov and Polyak proved that the graded space (G_n)_n associated to this filtration is generated by Jacobi diagrams. Here, we express elements associated to LP-surgeries as explicit combinations of these Jacobi diagrams in (G_n)_n. The obtained coefficient in front of a Jacobi diagram is computed like its weight system with respect to a Lie algebra equipped with a non-degenerate invariant bilinear form, where cup products in 3-manifolds play the role of the Lie bracket and the linking number replaces the invariant form. In particular, this article provides an algebraic version of the graphical clover calculus developed by Garoufalidis, Goussarov, Habiro and Polyak. This version induces splitting formulae for all finite type invariants of homology 3-spheres.