Invariant Bilinear Form Research Articles

Representations of quantum groups defined over commutative rings II

In this article we study the structure of highest weight modules for quantum groups defined over a commutative ring with particular emphasis on the structure theory for invariant bilinear forms on these modules.

Central extensions of groups of sections

If q : P -> M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact forms. In the present paper we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context we provide sufficient conditions for integrability in terms of data related only to the group K.

Quasifields of symplectic translation planes

We show that a translation plane is symplectic if and only at least one of its associated quasifields admits a non-degenerate invariant symmetric bilinear form. As an application we prove that a proper desarguesian, Moufang or nearfield plane can never be symplectic. Moreover, we give a purely algebraic characterization of the quasifields which coordinatize symplectic translation planes.

Shapovalov determinant for loop superalgebras

For the Kac-Moody superalgebra associated with the loop superalgebra with values in a finite-dimensional Lie superalgebra g, we show what its quadratic Casimir element is equal to if the Casimir element for g is known (if g has an even invariant supersymmetric bilinear form). The main tool is the Wick normal form of the even quadratic Casimir operator for the Kac-Moody superalgebra associated with g; this Wick normal form is independently interesting. If g has an odd invariant supersymmetric bilinear form, then we compute the cubic Casimir element. In addition to the simple Lie superalgebras g = g(A) with a Cartan matrix A for which the Shapovalov determinant was known, we consider the Poisson Lie superalgebra poi(0|n) and the related Kac-Moody superalgebra.

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.

Pseudo-Riemannian Novikov algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. Pseudo-Riemannian Novikov algebras denote Novikov algebras with non-degenerate invariant symmetric bilinear forms. In this paper, we find that there is a remarkable geometry on pseudo-Riemannian Novikov algebras, and give a special class of pseudo-Riemannian Novikov algebras.

Deformed Kac–Moody algebras and their representations

A class of Lie algebras G ( A ) associated to generalized Cartan matrices A is studied. The Lie algebras G ( A ) have much simpler structure than Kac–Moody algebras, but have the same root spaces with g ( A ) . In particular, G ( A ) has an abelian subalgebra of “half size.” We show that, G ( A ) has a non-degenerate invariant symmetric bilinear form if and only if A is symmetrizable; G ( X 1 ) ≅ G ( X 2 ) if and only if the GCMs X 1 and X 2 are the same up to a permutation of rows and columns. We study the lowest (respectively highest) weight Verma module V ¯ ( λ ) (respectively V ˜ ( λ ) ) over G ( A ) , and obtain the necessary and sufficient conditions for V ¯ ( λ ) to be irreducible, and also find its maximal proper submodule when V ¯ ( λ ) is reducible. Then using graded dual module of V ¯ ( λ ) we deduce the necessary and sufficient conditions for V ˜ ( λ ) to be irreducible.

Enveloping algebras of Slodowy slices and the Joseph ideal

Let G be a simple algebraic group over an algebraically closed field \mathbb k of characteristic 0 , and \mathfrak g=\text{Lie}\,G . Let (e,h,f) be an \mathfrak{sl}_2 -triple in \mathfrak g with e being a long root vector in \mathfrak g . Let (\,\cdot\,,\,\cdot\,) be the G -invariant bilinear form on \mathfrak g with (e,f)=1 and let \chi\in\mathfrak g^* be such that \chi(x)=(e,x) for all x\in\mathfrak g . Let {\mathcal S} be the Slodowy slice at e through the adjoint orbit of e and let H be the enveloping algebra of {\mathcal S} ; see [31]. In this note we give an explicit presentation of H by generators and relations. As a consequence we deduce that H contains an ideal of codimension 1 which is unique if \mathfrak g is not of type \mathrm A . Applying Skryabin's equivalence of categories we then construct an explicit Whittaker model for the Joseph ideal of U(\mathfrak g) . Inspired by Joseph's Preparation Theorem we prove that there exists a homeomorphism between the primitive spectrum of H and the spectrum of all primitive ideals of infinite codimension in U(\mathfrak g) which respects Goldie rank and Gelfand–Kirillov dimension. We study highest weight modules for the algebra H and apply earlier results of Miličić–Soergel and Backelin to express the composition multiplicities of the Verma modules for H in terms of some inverse parabolic Kazhdan–Lusztig polynomials. Our results confirm in the minimal nilpotent case the de Vos–van Driel conjecture on composition multiplicities of Verma modules for finite {\mathcal W} -algebras. We also obtain some general results on the enveloping algebras of Slodowy slices and determine the associated varieties of related primitive ideals of U(\mathfrak g) . A sequel to this paper will treat modular aspects of this theory.

Invariant Forms on Lie Algebra Extensions

For a perfect ideal 𝔥 of the Lie algebra 𝔤 with a 𝔤-invariant symmetric bilinear form ⟨·, ·⟩, we consider the continuous cohomology class in defined by the 2-cocycles of the form ⟨[X, ·], ·⟩ on 𝔥, X ∈ 𝔤. We determine the obstruction for extending this class to 𝔤. Invariant symmetric bilinear forms on corresponding Abelian extensions of 𝔤 by (𝔤/𝔥)* are constructed. The result is applied to central extensions of the Lie algebra of symplectic vector fields and of the Lie algebra of divergence free vector fields.

Bilinear forms on fermionic Novikov algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian super-operator in a super-variable. In this paper, we show that there is a remarkable geometry on fermionic Novikov algebras with non-degenerate invariant symmetric bilinear forms, which we call pseudo-Riemannian fermionic Novikov algebras. They are related to pseudo-Riemannian Lie algebras. Furthermore, we obtain a procedure to classify pseudo-Riemannian fermionic Novikov algebras. As an application, we give the classification in dimension ⩽4. Motivated by the one in dimension 4, we construct some examples in high dimensions.

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.

Lie algebras with quadratic dimension equal to 2

The quadratic dimension of a Lie algebra is defined as the dimension of the linear space spanned by all its invariant non-degenerate symmetric bilinear forms. We prove that a quadratic Lie algebra with quadratic dimension equal to 2 is a local Lie algebra, this is to say, it admits a unique maximal ideal. We describe local quadratic Lie algebras using the notion of double extension and characterize those with quadratic dimension equal to 2 by the study of the centroid of such Lie algebras. We also give some necessary or sufficient conditions for a Lie algebra to have quadratic dimension equal to 2. Examples of local Lie algebras with quadratic dimension larger than 2 are given.

Solvable quadratic Lie algebras

A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.

Metric Lie algebras and quadratic extensions

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant symmetric bilinear form. We show that any metric Lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module a in a canonical way. Identifying equivalence classes of quadratic extensions of l by a with a certain cohomology set H2 Q(l,a), we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.

"SQUARE ROOT" OF THE PROCA EQUATION: SPIN-3/2 FIELD EQUATION

New equations describing particles with spin-3/2 are derived. The nonlocal equation with the unique mass can be considered as "square root" of the Proca equation in the same sense as the Dirac equation is related to the Klein–Gordon–Fock equation. The local equation describes spin-3/2 particles with three mass states. The equations considered involve fields with spin-3/2 and spin-1/2, i.e. multispin 1/2, 3/2. The projection operators extracting states with definite energy, spin, and spin projections are obtained. All independent solutions of the local equation are expressed through projection matrices. The first order relativistic wave equation in the 20-dimensional matrix form, the relativistically invariant bilinear form and the corresponding Lagrangian are given. Two parameters characterizing nonminimal electromagnetic interactions of fermions are introduced, and the quantum-mechanical Hamiltonian is found. It is proved that there is only causal propagation of waves in the approach considered.

Berezin kernels and analysis on Makarevich spaces

Following ideas of van Dijk and Hille we study the link which exists between maximal degenerate representations and Berezin kernels. We consider the conformal group Conf( V) of a simple real Jordan algebra V. The maximal degenerate representations π s ( s ε ℂ) we shall study are induced by a character of a maximal parabolic subgroup P¯ of Conf( V). These representations π s can be realized on a space I s of smooth functions on V. There is an invariant bilinear form ℬ s on the space Is. The problem we consider is to diagonalize this bilinear form ℬ s , with respect to the action of a symmetric subgroup G of the conformal group Conf( V). This bilinear form can be written as an integral involving the Berezin kernel B v an invariant kernel on the Riemannian symmetric space G/K, which is a Makarevich symmetric space in the sense of Bertram. Then we can use results by van Dijk and Pevzner who computed the spherical Fourier transform of B v . From these, one deduces that the Berezin kernel satisfies a remarkable Bernstein identity: D( ν)B ν = b(ν)B ν+1 , where D( ν) is an invariant differential operator on G/K and b(ν) is a polynomial. By using this identity we compute a Hua type integral which gives the normalizing factor for an intertwining operator from I −s to I s . Furthermore, we obtain the diagonalization of the invariant bilinear form with respect to the action of the maximal compact group U of the conformal group Conf( V).

Supertrace and Superquadratic Lie Structure on the Weyl Algebra, and Applications to Formal Inverse Weyl Transform

Using the Moyal *-product and orthosymplectic supersymmetry, we construct a natural non trivial supertrace and an associated non degenerate invariant supersymmetric bilinear form for the Lie superalgebra structure of the Weyl algebra. We decompose adjoint and twisted adjoint actions. We define a renormalized supertrace and a formal inverse Weyl transform in a deformation quantization framework and develop some examples.

Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture

We study the representation theory of the superconformal algebra $W_k(g,f_{\theta})$ associated with a minimal gradation of $g$. Here, $g$ is a simple finite-dimensional Lie superalgebra with a non-degenerate, even supersymmetric invariant bilinear form. Thus, $W_k(g,f_{\theta})$ can be one of the well-known superconformal algebras including the Virasoro algebra, the Bershadsky-Polyakov algebra, the Neveu-Schwarz algebra, the Bershadsky-Knizhnik algebras, the N=2 superconformal algebra, the N=4 superconformal algebra, the N=3 superconformal algebra and the big N=4 superconformal algebra. We prove the conjecture of V. G. Kac, S.-S. Roan and M. Wakimoto for $W_k(g,f_{\theta})$. In fact, we show that any irreducible highest weight character of $W_k(g,f_{\theta})$ at any level $k\in C$ is determined by the corresponding irreducible highest weight character of the Kac-Moody affinization of $g$.

NON-DEGENERATE INVARIANT BILINEAR FORMS ON NONASSOCIATIVE TRIPLE SYSTEMS

A bilinear form f on a nonassociative triple system is said to be invariant if and only if f( ,d)=f(a, )=f(c, ) for all . is called a pseudo-metric triple system if f is non-degenerate and invariant. A decomposition theory for triple systems and pseudo-metric triple systems is established. Moreover, the finite-dimensional metric Lie triple systems are characterized in terms of the structure of the non-degenerate, invariant and symmetric bilinear forms on them.