313 research outputs found
Hyperbolic Kac-Moody superalgebras
We present a classification of the hyperbolic Kac-Moody (HKM) superalgebras.
The HKM superalgebras of rank larger or equal than 3 are finite in number (213)
and limited in rank (6). The Dynkin-Kac diagrams and the corresponding simple
root systems are determined. We also discuss a class of singular
sub(super)algebras obtained by a folding procedure
Sylvester-t' Hooft generators of sl(n) and sl(n|n), and relations between them
Among the simple finite dimensional Lie algebras, only sl(n) possesses two
automorphisms of finite order which have no common nonzero eigenvector with
eigenvalue one. It turns out that these automorphisms are inner and form a pair
of generators that allow one to generate all of sl(n) under bracketing. It
seems that Sylvester was the first to mention these generators, but he used
them as generators of the associative algebra of all n times n matrices Mat(n).
These generators appear in the description of elliptic solutions of the
classical Yang-Baxter equation, orthogonal decompositions of Lie algebras, 't
Hooft's work on confinement operators in QCD, and various other instances. Here
I give an algorithm which both generates sl(n) and explicitly describes a set
of defining relations. For simple (up to center) Lie superalgebras, analogs of
Sylvester generators exist only for sl(n|n). The relations for this case are
also computed.Comment: 14 pages, 6 figure
Minkowski superspaces and superstrings as almost real-complex supermanifolds
In 1996/7, J. Bernstein observed that smooth or analytic supermanifolds that
mathematicians study are real or (almost) complex ones, while Minkowski
superspaces are completely different objects. They are what we call almost
real-complex supermanifolds, i.e., real supermanifolds with a non-integrable
distribution, the collection of subspaces of the tangent space, and in every
subspace a complex structure is given.
An almost complex structure on a real supermanifold can be given by an even
or odd operator; it is complex (without "always") if the suitable superization
of the Nijenhuis tensor vanishes. On almost real-complex supermanifolds, we
define the circumcised analog of the Nijenhuis tensor. We compute it for the
Minkowski superspaces and superstrings. The space of values of the circumcised
Nijenhuis tensor splits into (indecomposable, generally) components whose
irreducible constituents are similar to those of Riemann or Penrose tensors.
The Nijenhuis tensor vanishes identically only on superstrings of
superdimension 1|1 and, besides, the superstring is endowed with a contact
structure. We also prove that all real forms of complex Grassmann algebras are
isomorphic although singled out by manifestly different anti-involutions.Comment: Exposition of the same results as in v.1 is more lucid. Reference to
related recent work by Witten is adde
Cohomology of Lie superalgebras and of their generalizations
The cohomology groups of Lie superalgebras and, more generally, of color Lie
algebras, are introduced and investigated. The main emphasis is on the case
where the module of coefficients is non-trivial. Two general propositions are
proved, which help to calculate the cohomology groups. Several examples are
included to show the peculiarities of the super case. For L = sl(1|2), the
cohomology groups H^1(L,V) and H^2(L,V), with V a finite-dimensional simple
graded L-module, are determined, and the result is used to show that
H^2(L,U(L)) (with U(L) the enveloping algebra of L) is trivial. This implies
that the superalgebra U(L) does not admit of any non-trivial formal
deformations (in the sense of Gerstenhaber). Garland's theory of universal
central extensions of Lie algebras is generalized to the case of color Lie
algebras.Comment: 50 pages, Latex, no figures. In the revised version the proof of
Lemma 5.1 is greatly simplified, some references are added, and a pertinent
result on sl(m|1) is announced. To appear in the Journal of Mathematical
Physic
Orthogonal polynomials of discrete variable and Lie algebras of complex size matrices
We give a uniform interpretation of the classical continuous Chebyshev's and
Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie
algebra gl(N), where N is any complex number. One can similarly interpret
Chebyshev's and Hahn's q-polynomials and introduce orthogonal polynomials
corresponding to Lie superlagebras.
We also describe the real forms of gl(N), quasi-finite modules over gl(N),
and conditions for unitarity of the quasi-finite modules. Analogs of tensors
over gl(N) are also introduced.Comment: 25 pages, LaTe
Superconformal Primary Fields on a Graded Riemann Sphere
Primary superfields for a two dimensional Euclidean superconformal field
theory are constructed as sections of a sheaf over a graded Riemann sphere. The
construction is then applied to the N=3 Neveu-Schwarz case. Various quantities
in the N=3 theory are calculated and discussed, such as formal elements of the
super-Mobius group, and the two-point function.Comment: LaTeX2e, 23 pages; fixed typos, sorted references, modified
definition of primary superfield on page
Minimal deformations of the commutative algebra and the linear group GL(n)
We consider the relations of generalized commutativity in the algebra of
formal series , which conserve a tensor -grading and
depend on parameters . We choose the -preserving version of
differential calculus on . A new construction of the symmetrized tensor
product for -type algebras and the corresponding definition of minimally
deformed linear group and Lie algebra are proposed. We
study the connection of and with the special matrix
algebra \mbox{Mat} (n,Q) containing matrices with noncommutative elements.
A definition of the deformed determinant in the algebra \mbox{Mat} (n,Q) is
given. The exponential parametrization in the algebra \mbox{Mat} (n,Q) is
considered on the basis of Campbell-Hausdorf formula.Comment: 14 page
On Alternative Supermatrix Reduction
We consider a nonstandard odd reduction of supermatrices (as compared with
the standard even one) which arises in connection with possible extension of
manifold structure group reductions. The study was initiated by consideration
of the generalized noninvertible superconformal-like transformations. The
features of even- and odd-reduced supermatrices are investigated on a par. They
can be unified into some kind of "sandwich" semigroups. Also we define a
special module over even- and odd-reduced supermatrix sets, and the generalized
Cayley-Hamilton theorem is proved for them. It is shown that the odd-reduced
supermatrices represent semigroup bands and Rees matrix semigroups over a unit
group.Comment: 22 pages, Standard LaTeX with AmS font
Cohomologies of the Poisson superalgebra
Cohomology spaces of the Poisson superalgebra realized on smooth
Grassmann-valued functions with compact support on ($C^{2n}) are
investigated under suitable continuity restrictions on cochains. The first and
second cohomology spaces in the trivial representation and the zeroth and first
cohomology spaces in the adjoint representation of the Poisson superalgebra are
found for the case of a constant nondegenerate Poisson superbracket for
arbitrary n>0. The third cohomology space in the trivial representation and the
second cohomology space in the adjoint representation of this superalgebra are
found for arbitrary n>1.Comment: Comments: 40 pages, the text to appear in Theor. Math. Phys.
supplemented by computation of the 3-rd trivial cohomolog
On anomalies in classical dynamical systems
The definition of "classical anomaly" is introduced. It describes the
situation in which a purely classical dynamical system which presents both a
lagrangian and a hamiltonian formulation admits symmetries of the action for
which the Noether conserved charges, endorsed with the Poisson bracket
structure, close an algebra which is just the centrally extended version of the
original symmetry algebra. The consistency conditions for this to occur are
derived. Explicit examples are given based on simple two-dimensional models.
Applications of the above scheme and lines of further investigations are
suggested.Comment: arXiv version is already officia
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