Algebraic Realisation of the Zamolodchikov Metric in Narain Theories

Introduction. Let L be a graded Lie algebra (see ?2) over a commutative ring R with unit. We shall deal with graded representations of L. In ?2 we define the universal enveloping algebra U of L and prove a PoincareBirkhoff-Witt theorem for U. In ?3 we prove that if L is a finite dimensional graded Lie algebra over a field of characteristic : 2, then L has a faithful finite dimensional representation. A key lemma for this proof is a result on (nongraded) Lie algebras which states that if L is a finite dimensional Lie algebra over a field of characteristic 0 then L has a faithful, finite dimensional representation f such that f (x) is nilpotent whenever ad x is nilpotent. This work is the first section of a doctoral thesis written under the direction of Professor Gerhard Hochschild. The author takes this opportunity to thank him for his generous advice and instruction.

A well known result in the theory of Lie algebras, due to H. Zassenhaus, states that if is a finite dimensional Lie algebra over the field K such that the killing form of is non-degenerate, then the derivations of are all inner, [3, p. 74]. In particular, this applies to the finite dimensional split simple Lie algebras over fields of characteristic zero. In this paper we extend this result to a class of Lie algebras which generalize the split simple Lie algebras, and which are defined by Cartan matrices (for a definition see § 1). Because of the fact that the algebras we consider are usually infinite dimensional, the method we employ in our investigation is quite different from the standard one used in the finite dimensional case, and makes no reference to any associative bilinear form on the algebras.

We study the notion of the Killing form for Frobenius Lie algebras of dimension . The Killing form is a symmetric bilinear form on a finite dimensional Lie algebra over a field defined by where is denoted the trace and is an adjoint representation of . A Lie algebras is said to be semisimple if it has the nondegenerate Killing form. The research aims to consider the criterion for semisimplicity of Frobenius Lie algebras of dimension by using the Killing form. The results show that each Frobenius Lie algebra of dimension and is not semisimple since the the Killing form is degenerate or in other words, a determinant of a representation matrix of the Killing form is equal to zero. For the future research, it is still an open problem to consider the general formulas of the Killing form for higher dimensional Frobenius Lie algebra whether degenerate or nondegenerate such that the semisimplicity of a Lie algebra can be considered. We conjecture that each finite dimensional real Frobenius Lie algebra is not semisimple.

Two finite dimensional simple Lie algebras over an algebraically closed field are isomorphic if and only if they satisfy the same polynomial identities (A. Koshkulei, Y. Razmyslov, 1983). As an alternative approach to the characterization of finite dimensional simple Lie algebras, some numerical invariants of the algebra identities can be used. We associate with a finite dimensional Lie algebra $L$ a sequence of integers $c_n(L)$, called the $n$-th codimensions of $L$. It appears that these quantities grow asymptotically like $k^n$, for some nonnegative integer $k\leq$ dim $L$ (A. Giambruno, M. Zaicev, 1999). Moreover, $k =$ dim $L$ iff algebra $L$ is simple. * This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Croatia, September, 22-24, 2016.

Let H be a finite or infinite dimensional Lie algebra. Barnes [2] and Towers [5] considered the case when H is a finite-dimensional Lie algebra over an arbitrary field, and all maximal subalgebras of H have codimension 1. Barnes, using the cohomology theory of Lie algebras, investigated solvable algebras, and Towers extended Barnes's results to include all Lie algebras. In [4] complex finite-dimensional Lie algebras were considered for the case when all the maximal subalgebras of H are not necessarily of codimension 1 but whenwhere S(H) is the set of all Lie subalgebras in H of codimension 1. Amayo [1]investigated the finite-dimensional Lie algebras with core-free subalgebras of codimension 1 and also obtained some interesting results about the structure of infinite dimensional Lie algebras with subalgebras of codimension 1.

It is proved that, any finite dimensional complex Lie algebra L = [L, L], hence, over a field of characteristic zero, any finite dimensional Lie algebra L = [L, L] possessing a basis with complex structure constants, should be a weak co-split Lie algebra. A class of non-semi-simple co-split Lie algebras is given.

We give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of characteristic p>0 and announce that for p>3 the classification of finite dimensional simple Lie algebras is complete. Any such Lie algebra is up to isomorphism either classical (i.e. comes from characteristic 0) or a filtered Lie algebra of Cartan type or a Melikian algebra of characteristic 5.

Toral rank one Lie algebras

Let \( \mathfrak{L} \) be a finite dimensional Lie algebra over a field of characteristic p, let \( \mathfrak{A} \) be the universal associative algebra of \( \mathfrak{L} \) ([1] and [4]) and let \( \mathfrak{C} \) be the center of \( \mathfrak{A} \). In this note we prove that if a is a linear element of \( \mathfrak{A} \) then there exists a non-zero polynomial φ such that <j> (a) ε \( \mathfrak{C} \). We use this result to obtain the following: (1) a simple direct proof of Iwasawa’s theorem ([2], p. 420) that every finite dimensional Lie algebra of characteristic p has a faithful finite dimensional representation, (2) a proof of a conjecture of Chevalley that every finite dimensional Lie algebra of characteristic p has a representation which is not completely reducible, (3) a proof that \( \mathfrak{A} \) can be imbedded in a division algebra.

Lie algebras have many varied applications, both in mathematics and mathematical physics. This book provides a thorough but relaxed mathematical treatment of the subject, including both the Cartan-Killing-Weyl theory of finite dimensional simple algebras and the more modern theory of Kac-Moody algebras. Proofs are given in detail and the only prerequisite is a sound knowledge of linear algebra. The first half of the book deals with classification of the finite dimensional simple Lie algebras and of their finite dimensional irreducible representations. The second half introduces the theory of Kac-Moody algebras, concentrating particularly on those of affine type. A brief account of Borcherds algebras is also included. An Appendix gives a summary of the basic properties of each Lie algebra of finite and affine type.

Consider a finite dimensional restricted Lie algebra over a field of prime characteristic. Each linear form on this Lie algebra defines a finite dimensional quotient of its universal enveloping algebra, called a reduced enveloping algebra. This leads to a Cartan matrix recording the multiplicities as composition factors of the simple modules in the projective indecomposable modules for such a reduced enveloping algebra. In this paper we show how to compare such Cartan matrices belonging to distinct linear forms. As an application we rederive and generalise the reciprocity formula first discovered by Humphreys for Lie algebras of reductive groups. For simple Lie algebras of Cartan type we see, for example, that the Cartan matrices for linear forms of non-positive height are submatrices of the Cartan matrix for the zero linear form.

Cartan described some of the finite dimensional simple Lie algebras and three of the four series of simple infinite dimensional vectorial Lie algebras with polynomial coefficients as prolongs, which now bear his name. The rest of the simple Lie algebras of these two types (finite dimensional and vectorial) are, if the depth of their grading is greater than 1, results of generalized Cartan–Tanaka–Shchepochkina (CTS) prolongs. Here we are looking for new examples of simple finite dimensional modular Lie (super)algebras in characteristic 2 obtained as Cartan prolongs. We consider pairs (an (ortho-)orthogonal Lie (super)algebra or its derived algebra, its irreducible module) and compute the Cartan prolongs of such pairs. The derived algebras of these prolongs are simple Lie (super)algebras. We point out several amazing phenomena in characteristic 2: a supersymmetry of representations of certain Lie algebras, latent or hidden over complex numbers, becomes manifest; the adjoint representation of some simple Lie superalgebras is not irreducible.

The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite

Historically, the notion of the radical was a direct outgrowth of the notion of semisimplicity. It may be somewhat surprising, however, to remark that the radical was studied first in the context of nonassociative rings (namely, finite-dimensional Lie algebras) rather than associative rings. In the work of E. Cartan, the radical of a finite-dimensional Lie algebra A (say over ℂ) is defined to be the maximal solvable ideal of A: it is obtained as the sum of all the solvable ideals inA. The Lie algebra A is semisimple iff its radical is zero, i.e., iff it has no nonzero solvable ideals. Cartan characterized the semisimplicity of a Lie algebra in terms of the nondegeneracy of its Killing form, and showed that any semisimple Lie algebra is a finite direct sum of simple Lie algebras. Moreover, he classified the finite-dimensional simple Lie algebras (over ℂ). Therefore, the structure theory of finite-dimensional semisimple Lie algebras is completely determined.KeywordsDirect SummandCommutative RingLeft IdealGroup RingDivision RingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Historically, the notion of the radical was a direct outgrowth of the notion of semisimplicity. It may be somewhat surprising, however, to remark that the radical was studied first in the context of nonassociative rings (namely, finite-dimensional Lie algebras) rather than associative rings. In the work of E. Cartan, the radical of a finite-dimensional Lie algebra A (say over ℂ) is defined to be the maximal solvable ideal of A: it is obtained as the sum of all the solvable ideals in A. The Lie algebra A is semisimple iff its radical is zero, i.e., iff it has no nonzero solvable ideals. Cartan characterized the semisimplicity of a Lie algebra in terms of the nondegeneracy of its Killing form, and showed that any semisimple Lie algebra is a finite direct sum of simple Lie algebras. Moreover, he classified the finite-dimensional simple Lie algebras (over ℂ). Therefore, the structure theory of finite-dimensional semisimple Lie algebras is completely determined.KeywordsDirect SummandCommutative RingLeft IdealGroup RingNilpotent ElementThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.