137 research outputs found
A categorification of the boson-fermion correspondence via representation theory of sl(∞)
On Einstein equations on manifolds and supermanifolds
The Einstein equations (EE) are certain conditions on the Riemann tensor on
the real Minkowski space M. In the twistor picture, after complexification and
compactification M becomes the Grassmannian of 2-dimensional
subspaces in the 4-dimensional complex one. Here we answer for which of the
classical domains considered as manifolds with G-structure it is possible to
impose conditions similar in some sense to EE. The above investigation has its
counterpart on superdomains: an analog of the Riemann tensor is defined for any
supermanifold with G-structure with any Lie supergroup G. We also derive
similar analogues of EE on supermanifolds. Our analogs of EE are not what
physicists consider as SUGRA (supergravity), for SUGRA see \cite{GL4,LP2}.Comment: arxiv version is already officia
Tensor representations of Mackey Lie algebras and their dense subalgebras
In this article we review the main results of the earlier papers [PStyr, PS] and [DPS], and establish related new results in considerably greater generality. We introduce a class of infinite-dimensional Lie algebras gM, which we call Mackey Lie algebras, and define monoidal categories TgM of tensor gM-modules. We also consider dense subalgebras a⊂gM and corresponding categories Ta. The locally finite Lie algebras sl(V,W),o(V),sp(V) are dense subalgebras of respective Mackey Lie algebras. Our main result is that if gM is a Mackey Lie algebra and a⊂gM is a dense subalgebra, then the monoidal category Ta is equivalent to Tsl(∞) or To(∞); the latter monoidal categories have been studied in detail in [DPS]. A possible choice of a is the well-known Lie algebra of generalized Jacobi matrices
Gelfand-Tsetlin modules for
We address the problem of classifying of irreducible Gelfand-Tsetlin modules for gl(m|n) and show that it reduces to the classification of Gelfand-Tsetlin modules for the even part. We also give an explicit tableaux construction and the irreducibility criterion for the class of quasi typical and quasi covariant Gelfand-Tsetlin modules which includes all essentially typical and covariant tensor finite dimensional modules. In the quasi typical case new irreducible representations are infinite dimensional gl(m|n)-modules which are isomorphic to the parabolically induced (Kac) modules. <br
Irreducible Characters of General Linear Superalgebra and Super Duality
We develop a new method to solve the irreducible character problem for a wide
class of modules over the general linear superalgebra, including all the
finite-dimensional modules, by directly relating the problem to the classical
Kazhdan-Lusztig theory. We further verify a parabolic version of a conjecture
of Brundan on the irreducible characters in the BGG category \mc{O} of the
general linear superalgebra. We also prove the super duality conjecture
The classification of almost affine (hyperbolic) Lie superalgebras
We say that an indecomposable Cartan matrix A with entries in the ground
field of characteristic 0 is almost affine if the Lie sub(super)algebra
determined by it is not finite dimensional or affine but the Lie (super)algebra
determined by any submatrix of A, obtained by striking out any row and any
column intersecting on the main diagonal, is the sum of finite dimensional or
affine Lie (super)algebras. A Lie (super)algebra with Cartan matrix is said to
be almost affine if it is not finite dimensional or affine, and all of its
Cartan matrices are almost affine.
We list all almost affine Lie superalgebras over complex numbers correcting
two earlier claims of classification and make available the list of almost
affine Lie algebras obtained by Li Wang Lai.Comment: 92 page
Super duality and irreducible characters of ortho-symplectic Lie superalgebras
We formulate and establish a super duality which connects parabolic
categories between the ortho-symplectic Lie superalgebras and classical Lie
algebras of types. This provides a complete and conceptual solution of
the irreducible character problem for the ortho-symplectic Lie superalgebras in
a parabolic category , which includes all finite-dimensional irreducible
modules, in terms of classical Kazhdan-Lusztig polynomials.Comment: 30 pages, Section 5 rewritten and shortene
- …