Z-graded Lie Algebras Research Articles

A Characterization of the Block Lie Algebra and Itsq-Forms in Characteristic 0

The Block algebraLreferred to here is the Lie algebra over a fieldFof characteristic 0 with basis {er,s∣(r,s)∈Z×Z\\{(0,0)}} and subject to the commutation relations [eh,k,er,s]=(rk−sh)eh+r,k+s. Let 0,1≠q∈F. Theq-formL(q) ofLis the Lie algebra with basis {er,s(q)∣(r,s)∈Z×Z\\{(0,0)}} and subject to the commutation relations [eh,k(q),er,s(q)]=(qrk−qsh)eh+r,k+s(q). We study here aZ×Z-graded Lie algebra[formula] withA0,0=0 and dimAi,j≤1 for all (i,j)∈Z×Z\\{(0,0)}. In this paper, we show thatAis necessarily isomorphic toLorL′(q)=[L(q),L(q)] for someq∈FifAsatisfies a certain nondegeneracy condition.

Structure theory of finite conformal algebras

After the seminal paper [BPZ] of Belavin, Polyakov and Zamolodchikov, conformal field theory has become by now a large field with many remarkable ramifications to other fields of mathematics and physics. A rigorous mathematical definition of the “chiral part” of a conformal field theory, called a vertex (= chiral) algebra, was proposed by Borcherds [Bo] more than ten years ago and continued in [DL], [FHL], [FLM], [K], [L] and in numerous other works. However, until now a classification of vertex algebras, similar, for example, to the classification of finite-dimensional Lie algebras, seems to be far away. In the present paper we give a solution to the special case of this problem when the chiral algebra is generated by a finite number of quantum fields, closed under the operator product expansion (in the sense that only derivatives of the generating fields may occur). In this situation the adequate tool is the notion of a conformal algebra [K] which, to some extent, is related to a chiral algebra in the same way a Lie algebra is related to its universal enveloping algebra. At the same time, the theory of conformal algebras sheds a new light on the problem of classification of infinite-dimensional Lie algebras. About thirty years ago one of the authors posed (and partially solved) the problem of classification of simple Z-graded Lie algebras of finite Gelfand-Kirillov dimension [K1]. This problem was completely solved by Mathieu [M1]-[M3] in a remarkable tour de force. The point of view of the present paper is that the condition of locality (which is the most basic axiom of quantum field theory) along with a finiteness condition, are more natural conditions, which are also much easier to handle. In this paper we develop a structure theory of finite rank conformal algebras. Applications of this theory are two-fold. On the one hand, the conformal algebra structure is an axiomatic description [K] of the operator product expansion (OPE) of chiral fields in a conformal field theory [BPZ]. Hence the theory of finite conformal algebras provides a classification of finite systems of fields closed under the OPE. On the other hand, the category of finite conformal algebras is (more or less) equivalent to the category of infinite-dimensional Lie algebras spanned by Fourier coefficients of a finite number of pairwise local fields (or rather formal distributions)

Canonical structure of classical field theory in the polymomentum phase space

Canonical structure of classical field theory in n dimensions is studied within the covariant polymomentum Hamiltonian formulation of De Donder-Weyl (DW). The bi-vertical ( n + 1)-form, called polysymplectic, is put forward as a generalization of the symplectic form in mechanics. Although not given in intrinsic geometric terms differently than a certain coset, it gives rise to the invariantly defined map between horizontal forms playing the role of dynamical variables and the so-called vertical multivectors generalizing Hamiltonian vector fields. The analogue of the Poisson bracket on forms is defined which leads to the structure of Z-graded Lie algebra on the so-called Hamiltonian forms for which the map above exists. A generalized Poisson structure appears in the form of what we call a “higher order” and a right Gerstenhaber algebra. The equations of motion of forms are formulated in terms of the Poisson bracket with the DW Hamiltonian n-form H vol ( vol ) is the space-time volume form, H is the DW Hamiltonian function) which is found to be related to the operation of the total exterior differentiation of forms. A few applications and a relation to the standard Hamiltonian formalism in field theory are briefly discussed.

Integrable equations on ℤ-graded Lie algebrasLie algebras

We consider evolution systems admitting L-A-pairs in ℤ-graded Lie algebras. We relate several hierarchies of integrable systems to a single L operator. The different hierarchies corresponds to different decompositions of the zero component of a ℤ-graded algebra into the sum of two subalgebras. This allows us to construct new examples of multi-component integrable system following the Burgers, mKdV, NLS and Boussinesq equations.

Generalized Spencer cohomology and filtered deformations of $\mathbb{Z}$-graded Lie superalgebras

In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)algebras. We develop a method of finding filtered deformations of such Z-graded Lie (super)algebras based on this cohomology. As an application we determine all simple filtered deformations of certain Z-graded Lie superalgebras classified in [K3], thus completing the last step in the classification of simple infinite-dimensional linearly compact Lie superalgebras.