207,508 research outputs found
Dual Formulation of the Lie Algebra S-expansion Procedure
The expansion of a Lie algebra entails finding a new, bigger algebra G,
through a series of well-defined steps, from an original Lie algebra g. One
incarnation of the method, the so-called S-expansion, involves the use of a
finite abelian semigroup S to accomplish this task. In this paper we put
forward a dual formulation of the S-expansion method which is based on the dual
picture of a Lie algebra given by the Maurer-Cartan forms. The dual version of
the method is useful in finding a generalization to the case of a gauge free
differential algebra, which in turn is relevant for physical applications in,
e.g., Supergravity. It also sheds new light on the puzzling relation between
two Chern-Simons Lagrangians for gravity in 2+1 dimensions, namely the
Einstein-Hilbert Lagrangian and the one for the so-called "exotic gravity".Comment: 12 pages, no figure
Short Distance Expansion from the Dual Representation of Infinite Dimensional Lie Algebras
We compute the short distance expansion of fields or operators that live in
the coadjoint representation of an infinite dimensional Lie algebra by using
only properties of the adjoint representation and its dual. We explicitly
compute the short distance expansion for the duals of the Virasoro algebra,
affine Lie Algebras and the geometrically realized N-extended supersymmetric GR
Virasoro algebra.Comment: 19 pages, LaTeX twice, no figure, replacement has corrected Lie
algebr
A Three Parameter Hopf Deformation of the Algebra of Feynman-like Diagrams
We construct a three-parameter deformation of the Hopf algebra \LDIAG. This
is the algebra that appears in an expansion in terms of Feynman-like diagrams
of the {\em product formula} in a simplified version of Quantum Field Theory.
This new algebra is a true Hopf deformation which reduces to \LDIAG for some
parameter values and to the algebra of Matrix Quasi-Symmetric Functions
(\MQS) for others, and thus relates \LDIAG to other Hopf algebras of
contemporary physics. Moreover, there is an onto linear mapping preserving
products from our algebra to the algebra of Euler-Zagier sums
-Conformal Algebras
-conformal algebra is an axiomatic description of the operator
product expansion of chiral fields with simple poles at finitely many points.
We classify these algebras and their representations in terms of Lie algebras
and their representations with an action of the group . To every
-conformal algebra and a character of we associate a Lie
algebra generated by fields with the OPE with simple poles. Examples include
twisted affine Kac-Moody algebras, the sin algebra (which is a
``-conformal'' analodue of the general linear algebra) and its
analogues, the algebra of pseudodifferential operators on the circle, etc.Comment: 23 pages, AMSLatex Repotr-no: ITEP-TH-28/9
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