263 research outputs found
On the nature of the Virasoro algebra
The multiplication in the Virasoro algebra comes from the commutator in a quasiassociative algebra with the multiplication
\renewcommand{\theequation}{} \be \ba{l} \ds e_p * e_q = - {q (1 + \epsilon
q) \over 1 + \epsilon (p + q)} e_{p+q} + {1 \over 2} \theta \left[p^3 - p +
\left(\epsilon - \epsilon^{-1} \right) p^2 \right] \delta^0_{p+q},
\vspace{3mm}\\ \ds e_p * \theta = \theta* e_p = 0. \ea \ee The multiplication
in a quasiassociative algebra satisfies the property
\renewcommand{\theequation}{} \be a * (b * c) - (a * b) * c = b * (a * c) -
(b * a) * c, \qquad a, b, c \in {\cal R}. \ee This property is necessary and
sufficient for the Lie algebra {\it Lie} to have a phase space. The
above formulae are put into a cohomological framework, with the relevant
complex being different from the Hochschild one even when the relevant
quasiassociative algebra becomes associative. Formula above
also has a differential-variational counterpart
Cyclic elements in semisimple lie algebras
We develop a theory of cyclic elements in semisimple Lie algebras. This notion was introduced by Kostant, who associated a cyclic element with the principal nilpotent and proved that it is regular semisimple. In particular, we classfiy all nilpotents giving rise to semisimple and regular semisimple cyclic elements. As an application, we obtain an explicit construction of all regular elements in Weyl groups
Faces of weight polytopes and a generalization of a theorem of Vinberg
The paper is motivated by the study of graded representations of Takiff
algebras, cominuscule parabolics, and their generalizations. We study certain
special subsets of the set of weights (and of their convex hull) of the
generalized Verma modules (or GVM's) of a semisimple Lie algebra \lie g. In
particular, we extend a result of Vinberg and classify the faces of the convex
hull of the weights of a GVM. When the GVM is finite-dimensional, we ask a
natural question that arises out of Vinberg's result: when are two faces the
same? We also extend the notion of interiors and faces to an arbitrary subfield
\F of the real numbers, and introduce the idea of a weak \F-face of any
subset of Euclidean space. We classify the weak \F-faces of all lattice
polytopes, as well as of the set of lattice points in them. We show that a weak
\F-face of the weights of a finite-dimensional \lie g-module is precisely
the set of weights lying on a face of the convex hull.Comment: Statement changed in Section 4. Typos fixed and some proofs updated.
Submitted to "Algebra and Representation Theory." 18 page
Reflection groups in hyperbolic spaces and the denominator formula for Lorentzian Kac--Moody Lie algebras
This is a continuation of our "Lecture on Kac--Moody Lie algebras of the
arithmetic type" \cite{25}.
We consider hyperbolic (i.e. signature ) integral symmetric bilinear
form (i.e. hyperbolic lattice), reflection group
, fundamental polyhedron \Cal M of and an acceptable
(corresponding to twisting coefficients) set P({\Cal M})\subset M of vectors
orthogonal to faces of \Cal M (simple roots). One can construct the
corresponding Lorentzian Kac--Moody Lie algebra {\goth g}={\goth
g}^{\prime\prime}(A(S,W,P({\Cal M}))) which is graded by .
We show that \goth g has good behavior of imaginary roots, its denominator
formula is defined in a natural domain and has good automorphic properties if
and only if \goth g has so called {\it restricted arithmetic type}. We show
that every finitely generated (i.e. P({\Cal M}) is finite) algebra {\goth
g}^{\prime\prime}(A(S,W_1,P({\Cal M}_1))) may be embedded to {\goth
g}^{\prime\prime}(A(S,W,P({\Cal M}))) of the restricted arithmetic type. Thus,
Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type is a
natural class to study.
Lorentzian Kac--Moody Lie algebras of the restricted arithmetic type have the
best automorphic properties for the denominator function if they have {\it a
lattice Weyl vector }. Lorentzian Kac--Moody Lie algebras of the
restricted arithmetic type with generalized lattice Weyl vector are
called {\it elliptic}Comment: Some corrections in Sects. 2.1, 2.2 were done. They don't reflect on
results and ideas. 31 pages, no figures. AMSTe
Hypermatrix factors for string and membrane junctions
The adjoint representations of the Lie algebras of the classical groups
SU(n), SO(n), and Sp(n) are, respectively, tensor, antisymmetric, and symmetric
products of two vector spaces, and hence are matrix representations. We
consider the analogous products of three vector spaces and study when they
appear as summands in Lie algebra decompositions. The Z3-grading of the
exceptional Lie algebras provide such summands and provides representations of
classical groups on hypermatrices. The main natural application is a formal
study of three-junctions of strings and membranes. Generalizations are also
considered.Comment: 25 pages, 4 figures, presentation improved, minor correction
Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits
We give a criterion for a Dynkin diagram, equivalently a generalized Cartan
matrix, to be symmetrizable. This criterion is easily checked on the Dynkin
diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of
compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin
diagram of compact hyperbolic type is 4. Building on earlier classification
results of Kac, Kobayashi-Morita, Li and Sa\c{c}lio\~{g}lu, we present the 238
hyperbolic Dynkin diagrams in ranks 3-10, 142 of which are symmetrizable. For
each symmetrizable hyperbolic generalized Cartan matrix, we give a
symmetrization and hence the distinct lengths of real roots in the
corresponding root system. For each such hyperbolic root system we determine
the disjoint orbits of the action of the Weyl group on real roots. It follows
that the maximal number of disjoint Weyl group orbits on real roots in a
hyperbolic root system is 4.Comment: J. Phys. A: Math. Theor (to appear
A geometrical angle on Feynman integrals
A direct link between a one-loop N-point Feynman diagram and a geometrical
representation based on the N-dimensional simplex is established by relating
the Feynman parametric representations to the integrals over contents of
(N-1)-dimensional simplices in non-Euclidean geometry of constant curvature. In
particular, the four-point function in four dimensions is proportional to the
volume of a three-dimensional spherical (or hyperbolic) tetrahedron which can
be calculated by splitting into birectangular ones. It is also shown that the
known formula of reduction of the N-point function in (N-1) dimensions
corresponds to splitting the related N-dimensional simplex into N rectangular
ones.Comment: 47 pages, including 42 pages of the text (in plain Latex) and 5 pages
with the figures (in a separate Latex file, requires axodraw.sty) a note and
three references added, minor problem with notation fixe
Algebraic invariants of five qubits
The Hilbert series of the algebra of polynomial invariants of pure states of
five qubits is obtained, and the simplest invariants are computed.Comment: 4 pages, revtex. Short discussion of quant-ph/0506073 include
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