143 research outputs found
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
Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairs
We provide formulas for the denominator and superdenominator of a basic
classical type Lie superalgebra for any set of positive roots. We establish a
connection between certain sets of positive roots and the theory of reductive
dual pairs of real Lie groups. As an application of our formulas, we recover
the Theta correspondence for compact dual pairs. Along the way we give an
explicit description of the real forms of basic classical type Lie
superalgebras.Comment: Latex, 75 pages. Minor corrections. Final version, to appear in the
Japanese Journal of Mathematic
The return of the bursts: Thermonuclear flashes from Circinus X-1
We report the detection of 15 X-ray bursts with RXTE and Swift observations
of the peculiar X-ray binary Circinus X-1 during its May 2010 X-ray
re-brightening. These are the first X-ray bursts observed from the source after
the initial discovery by Tennant and collaborators, twenty-five years ago. By
studying their spectral evolution, we firmly identify nine of the bursts as
type I (thermonuclear) X-ray bursts. We obtain an arcsecond location of the
bursts that confirms once and for all the identification of Cir X-1 as a type I
X-ray burst source, and therefore as a low magnetic field accreting neutron
star. The first five bursts observed by RXTE are weak and show approximately
symmetric light curves, without detectable signs of cooling along the burst
decay. We discuss their possible nature. Finally, we explore a scenario to
explain why Cir X-1 shows thermonuclear bursts now but not in the past, when it
was extensively observed and accreting at a similar rate.Comment: Accepted for publication in The Astrophysical Journal Letters. Tables
1 & 2 merged. Minor changes after referee's comments. 5 pages, 4 Figure
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
Translation functors and decomposition numbers for the periplectic Lie superalgebra
We study the category of finite-dimensional integrable representations of the periplectic Lie superalgebra . We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter on the category by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for resembling those for . Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of . We also prove that indecomposable projective modules in this category are multiplicity-free
Translation functors and decomposition numbers for the periplectic Lie superalgebra
We study the category of finite-dimensional integrable representations of the periplectic Lie superalgebra . We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter on the category by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for resembling those for . Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of . We also prove that indecomposable projective modules in this category are multiplicity-free
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
Translation functors and decomposition numbers for the periplectic Lie superalgebra
We study the category of finite-dimensional integrable representations of the periplectic Lie superalgebra . We define an action of the Temperley-Lieb algebra with infinitely many generators and defining parameter on the category by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for resembling those for . Using the Temperley-Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of . We also prove that indecomposable projective modules in this category are multiplicity-free
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