Arbitrary Lie Algebra Research Articles

Unitarity of the KZ/Hitchin connection on conformal blocks in genus 0 for arbitrary Lie algebras

We prove that the vector bundles of conformal blocks, on suitable moduli spaces of genus zero curves with marked points, for arbitrary simple Lie algebras and arbitrary integral levels, carry unitary metrics of geometric origin which are preserved by the Knizhnik–Zamolodchikov/Hitchin connection (as conjectured by Gawedzki et al., 1991, in [7]). Our proof builds upon the work of Ramadas (2009) [22] who proved this unitarity statement in the case of the Lie algebra sl2 (and genus 0).

On an algorithm for finding derivations of Lie algebras

Let g be an arbitrary finite dimensional Lie algebra over the field R. We give as an additional alternative a detailed overview of an algorithm for finding derivations of g since such procedures are often of interest.

SOME PROPERTIES ON THE SCHUR MULTIPLIER OF A PAIR OF LIE ALGEBRAS

The aim of this paper is to introduce the concept of the Schur multiplier [Formula: see text] of a pair of Lie algebras and to obtain some inequalities for the dimension of [Formula: see text]. Also, we consider some of the features of central extension of an arbitrary Lie algebra. Moreover, we present a necessary and sufficient condition in which the Schur multiplier of a pair of Lie algebras can be embedded into the Schur multiplier of their factor Lie algebras.

Unitarity of the KZ/Hitchin connection on conformal blocks in genus 0 for arbitrary Lie algebras

Unitarity of the KZ/Hitchin connection on conformal blocks in genus 0 for arbitrary Lie algebras

On Covers of Perfect Lie Algebras

A Lie algebra is said to be perfect when it coincides with its derived subalgebra. The paper is devoted to give a complete structure of covers of perfect Lie algebras. Also, similar to a result of Alperin and Gorenstein (1966) in group theory, it is shown that every automorphism of a finite dimensional perfect Lie algebra may be lifted to an automorphism of its cover. Moreover, we present the concepts of irreducible and primitive extensions of an arbitrary Lie algebra and give some equivalent conditions for a central extension to be irreducible or primitive. Finally, we study the connection between the primitive extensions and the stem covers of finite dimensional perfect Lie algebras.

Quantum Integrable Model of an Arrangement of Hyperplanes

The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this paper a quantum integrable model is assigned to a weighted arrangement of affine hyperplanes. We show (under certain assumptions) that the algebra of Hamiltonians of the model is isomorphic to the algebra of functions on the critical set of the corresponding master function. For a discriminantal arrangement we show (under certain assumptions) that the symmetric part of the algebra of Hamiltonians is isomorphic to the Bethe algebra of the corresponding Gaudin model. It is expected that this correspondence holds in general (without the assumptions). As a byproduct of constructions we show that in a Gaudin model (associated to an arbitrary simple Lie algebra), the Bethe vector, corresponding to an isolated critical point of the master function, is nonzero.

Owner-reported breed-typical behavior and breed-group differences in the german pet dog population

We study the restricted one-dimensional central extensions of an arbitrary finite dimensional restricted simple Lie algebra for p≥5. For H2(g)=0, we explicitly describe the cocycles spanning H⁎2(g), and in the case H2(g)≠0, we give a procedure to describe a basis for H⁎2(g).

Maximal Abelian Dimensions in Some Families of Nilpotent Lie Algebras

This paper deals with the maximal abelian dimension of a Lie algebra, that is, the maximal value for the dimensions of its abelian Lie subalgebras. Indeed, we compute the maximal abelian dimension for every nilpotent Lie algebra of dimension less than 7 and for the Heisenberg algebra \(\mathfrak{H}_k\), with \(k\in\mathbb{N}\). In this way, an algorithmic procedure is introduced and applied to compute the maximal abelian dimension for any arbitrary nilpotent Lie algebra with an arbitrary dimension. The maximal abelian dimension is also given for some general families of nilpotent Lie algebras.

A new description of convex bases of PBW type for untwisted quantum affine algebras

In [8] we classified all ``convex orders'' on the positive root system $\Delta_+$ of an arbitrary untwisted affine Lie algebra ${\mathfrak g}$ and gave a concrete method of constructing all convex orders on $\Delta_+$. The aim of this paper is to give a new description of ``convex bases'' of PBW type of the positive subalgebra $U^+$ of the quantum affine algebra $U=U_q({\mathfrak g})$ by using the concrete method of constructing all convex orders on $\Delta_+$. Applying convexity properties of the convex bases of $U^+$, for each convex order on $\Delta_+$, we construct a pair of dual bases of $U^+$ and the negative subalgebra $U^-$ with respect to a $q$-analogue of the Killing form, and then present the multiplicative formula for the universal $R$-matrix of $U$.

A differential equation for diagonalizing complex semisimple Lie algebra elements

In this paper, we consider a generalization of Ebenbauer’s differential equation for non-symmetric matrix diagonalization to a flow on arbitrary complex semisimple Lie algebras. The flow is designed in such a way that the desired diagonalizations are precisely the equilibrium points in a given Cartan subalgebra. We characterize the set of all equilibria and establish a Morse–Bott type property of the flow. Global convergence to single equilibrium points is shown, starting from any semisimple Lie algebra element. For strongly regular initial conditions, we prove that the flow converges to an element of the Cartan subalgebra and thus achieves asymptotic diagonalization.

On the structure of graded Lie algebras

We study the structure of arbitrary graded Lie algebras. We show that any of such algebras L with a symmetric G-support is of the form L=U+∑jIj with U a subspace of L1 and any Ij a well described graded ideal of L, satisfying [Ij,Ik]=0 if j≠k. Under certain conditions, the gr-simplicity of L is characterized and it is shown that L is the direct sum of the family of its minimal graded ideals, each one being a gr-simple graded Lie algebra.

Algorithm to compute the maximal abelian dimension of Lie algebras

In this paper, the maximal abelian dimension is computationally obtained for an arbitrary finite-dimensional Lie algebra, defined by its nonzero brackets. More concretely, we describe and implement an algorithm which computes such a dimension by running it in the symbolic computation package MAPLE. Finally, we also show a computational study related to this implementation, regarding both the computing time and the memory used.

On multidimensional analogs of Melvin’s solution for classical series of Lie algebras

A multidimensional generalization of Melvin’s solution for an arbitrary simple Lie algebra $$ \mathcal{G} $$ is presented. The gravitational model contains n 2-forms and l ≥ n scalar fields, where n is the rank of $$ \mathcal{G} $$ . The solution is governed by a set of n functions obeying n ordinary differential equations with certain boundary conditions. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). A program (in Maple) for calculating these polynomials for classical series of Lie algebras is suggested. The polynomials corresponding to the Lie algebra D 4 are obtained. It is conjectured that the polynomials for A n -, B n - and C n -series may be obtained from polynomials for D n+1-series by using certain reduction formulas.

Some properties of the space of n-dimensional Lie algebras

Some general properties of the space of n-dimensional Lie algebras are studied. This space is defined by the system of Jacobi's quadratic equations. It is proved that these equations are linearly independent and equivalent to each other (more precisely, the quadratic forms defining these equations are affinely equivalent). Moreover, the problem on the closures of some orbits of the natural action of the group on is considered. Two Lie algebras are indicated whose orbits are closed in the projectivization of the space . The intersection of all irreducible components of the space is also treated. It is proved that this intersection is nontrivial and consists of nilpotent Lie algebras. Two Lie algebras belonging to this intersection are indicated. Some other results concerning arbitrary Lie algebras and the space formed by these algebras are presented.Bibliography: 17 titles.

A sort-Jacobi algorithm for semisimple lie algebras

A structure preserving sort-Jacobi algorithm for computing eigenvalues or singular values is presented. It applies to an arbitrary semisimple Lie algebra on its (−1)-eigenspace of the Cartan involution. Local quadratic convergence for arbitrary cyclic schemes is shown for the regular case. The proposed method is independent of the representation of the underlying Lie algebra and generalizes well-known normal form problems such as e.g. the symmetric, Hermitian, skew-symmetric, symmetric and skew-symmetric R -Hamiltonian eigenvalue problem and the singular value decomposition.

Ghost-free superconformal action for multipleM2-branes

The Bagger--Lambert construction of N = 8 superconformal field theories (SCFT) in three dimensions is based on 3-algebras. Three groups of researchers recently realized that an arbitrary semisimple Lie algebra can be incorporated by using a suitable Lorentzian signature 3-algebra. The SU(N) case is a candidate for the SCFT describing coincident M2-branes. However, these theories contain ghost degrees of freedom, which is unsatisfactory. We modify them by gauging certain global symmetries. This eliminates the ghosts from these theories while preserving all of their desirable properties. The resulting theories turn out to be precisely equivalent to N = 8 super Yang--Mills theories.

Bagger-Lambert theory for general Lie algebras

We construct the totally antisymmetric structure constants f ABCD of a 3-algebra with a Lorentzian bi-invariant metric starting from an arbitrary semi-simple Lie algebra. The structure constants f ABCD can be used to write down a maximally superconformal 3d theory that incorporates the expected degrees of freedom of multiple M2 branes, including the ``center-of-mass mode described by free scalar and fermion fields. The gauge field sector reduces to a three dimensional BF term, which underlies the gauge symmetry of the theory. We comment on the issue of unitarity of the quantum theory, which is problematic, despite the fact that the specific form of the interactions prevent the ghost fields from running in the internal lines of any Feynman diagram. Giving an expectation value to one of the scalar fields leads to the maximally supersymmetric 3d Yang-Mills Lagrangian with the addition of two U(1) multiplets, one of them ghost-like, which is decoupled at large g YM.

A complex for the cohomology of restricted Lie algebras

Let $$\mathfrak{g} = (\mathfrak{g}, [p])$$ be a restricted Lie algebra of characteristic p and M a $$\mathfrak{g}$$ -module. If $$\mathfrak{g}$$ is abelian, we give an explicit description of the cochain spaces $$C^k(\mathfrak{g}; M)$$ and differentials for the computation of the restricted Lie algebra cohomology $$H^k(\mathfrak{g}; M)$$ for k < p. If $$\mathfrak{g}$$ is an arbitrary (non-abelian) restricted Lie algebra, we give explicit descriptions of $$C^k(\mathfrak{g}; M)$$ for k ≤ 3. We use our results to classify extensions of restricted modules and infinitesimal deformations of restricted Lie algebras.

Sort‐Jacobi and generalizations of the symmetric eigenvalue problem

AbstractIn this paper, it is sketched how the Sort‐Jacobi method for the symmetric eigenvalue problem extends to the (–1)‐eigenspace of the Cartan involution on an arbitrary semisimple Lie algebra. The proposed method is independent of the representation of the underlying Lie algebra and generalizes well‐known normal form problems such as the symmetric, Hermitian, skewsymmetric, symmetric and skew‐symmetric ℝ‐Hamiltonian eigenvalue problem and the singular value decomposition. It allows a unified treatment of the above eigenvalue methods, including local convergence analysis. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Lie algebra of formal vector fields which are extended by formal g-valued functions

In this paper, we consider the infinite-dimensional Lie algebra Wn⋉g⊗O n of formal vector fields on the n-dimensional plane which is extended by formal g-valued functions of n variables. Here g is an arbitrary Lie algebra. We show that the cochain complex of this Lie algebra is quasi-isomorphic to the quotient of the Weyl algebra of (gl n ⊕ g) by the (2n+1)st term of the standard filtration. We consider separately the case of a reductive Lie algebra g. We show how one can use the methods of formal geometry to construct characteristic classes of bundles. For every G-bundle on an n-dimensional complex manifold, we construct a natural homomorphism from the ring A of relative cohomologies of the Lie algebra Wn⋉g⊗O n to the ring of cohomologies of the manifold. We show that generators of the ring A are mapped under this homomorphism to characteristic classes of tangent and G-bundles. Bibliography: 10 titles.