Apoorva Khare

Representations of Complex Semi-simple Lie Groups and Lie Algebras

Lie groups and Lie algebras occupy a prominent and central place in mathematics, connecting differential geometry, representation theory, algebraic geometry, number theory, and theoretical physics. In some sense, the heart of (classical) representation theory is in the study of the semisimple Lie groups. Their study is simultaneously simple in its beauty, as well as complex in its richness. From Killing, Cartan, and Weyl, to Dynkin, Harish-Chandra, Bruhat, Kostant, and Serre, many mathematicians in the twentieth century have worked on building up the theory of semisimple Lie algebras and their universal enveloping algebras. Books by Borel, Bourbaki, Bump, Chevalley, Humphreys, Jacobson, Varadarajan, Vogan, and others form the texts for (introductory) graduate courses on the subject.

Category O over a deformation of the symplectic oscillator algebra

Abstract We discuss the representation theory of H f , which is a deformation of the symplectic oscillator algebra sp (2n)⋉ h n , where h n is the ((2 n +1)-dimensional) Heisenberg algebra. We first look at a more general algebra with a triangular decomposition. Assuming the PBW theorem, and one other hypothesis, we show that the BGG category O is abelian, finite length, and self-dual. We decompose O as a direct sum of blocks O (λ) , and show that each block is a highest weight category. In the second part, we focus on the case H f for n =1, where we prove all these assumptions, as well as the PBW theorem.

Complete characterization of Hadamard powers preserving Loewner positivity, monotonicity, and convexity ☆

Abstract Entrywise powers of symmetric matrices preserving positivity, monotonicity or convexity with respect to the Loewner ordering arise in various applications, and have received much attention recently in the literature. Following FitzGerald and Horn (1977) [8] , it is well-known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness. Similar phenomena have also recently been shown by Hiai (2009) to occur for monotonicity and convexity. In this paper, we complete the characterization of all the entrywise powers below and above the critical exponents that are positive, monotone, or convex on the cone of positive semidefinite matrices. We then extend the original problem by fully classifying the positive, monotone, or convex powers in a more general setting where additional rank constraints are imposed on the matrices. We also classify the entrywise powers that are super/sub-additive with respect to the Loewner ordering. Finally, we extend all the previous characterizations to matrices with negative entries. Our analysis consequently allows us to answer a question raised by Bhatia and Elsner (2007) regarding the smallest dimension for which even extensions of the power functions do not preserve Loewner positivity.

Quantized symplectic oscillator algebras of rank one

A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with Uq(sl2). We study its representation theory, and in particular, its category O.

Preserving positivity for rank-constrained matrices

Entrywise functions preserving the cone of positive semidefinite matrices have been studied by many authors, most notably by Schoenberg [Duke Math. J. 9, 1942] and Rudin [Duke Math. J. 26, 1959]. Following their work, it is well-known that entrywise functions preserving Loewner positivity in all dimensions are precisely the absolutely monotonic functions. However, there are strong theoretical and practical motivations to study functions preserving positivity in a fixed dimension

Faces and maximizer subsets of highest weight modules

n

Faces of Weight Polytopes and a Generalization of a Theorem of Vinberg

. Such characterizations for a fixed value of

Preserving positivity for matrices with sparsity constraints

n

Faces of polytopes and Koszul algebras

are difficult to obtain, and in fact are only known in the

Vector spaces as unions of proper subspaces

2 \times 2