Extensions of differential representations of SL(2) and tori

Linear differential algebraic groups (LDAGs) measure differential algebraic dependencies among solutions of linear differential and difference equations with parameters, for which LDAGs are Galois groups. The differential representation theory is a key to developing algorithms computing these groups. In the rational representation theory of algebraic groups, one starts with SL(2) and tori to develop the rest of the theory. In this paper, we give an explicit description of differential representations of tori and differential extensions of irreducible representation of SL(2). In these extensions, the two irreducible representations can be non-isomorphic. This is in contrast to differential representations of tori, which turn out to be direct sums of isotypic representations.Comment: 21 pages; few misprints corrected; Lemma 4.6 adde

Extensions and block decompositions for finite-dimensional representations of equivariant map algebras

Suppose a finite group acts on a scheme $X$ and a finite-dimensional Lie algebra $\mathfrak{g}$. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from $X$ to $\mathfrak{g}$. The irreducible finite-dimensional representations of these algebras were classified in previous work with P. Senesi, where it was shown that they are all tensor products of evaluation representations and one-dimensional representations. In the current paper, we describe the extensions between irreducible finite-dimensional representations of an equivariant map algebra in the case that $X$ is an affine scheme of finite type and $\mathfrak{g}$ is reductive. This allows us to also describe explicitly the blocks of the category of finite-dimensional representations in terms of spectral characters, whose definition we extend to this general setting. Applying our results to the case of generalized current algebras (the case where the group acting is trivial), we recover known results but with very different proofs. For (twisted) loop algebras, we recover known results on block decompositions (again with very different proofs) and new explicit formulas for extensions. Finally, specializing our results to the case of (twisted) multiloop algebras and generalized Onsager algebras yields previously unknown results on both extensions and block decompositions.Comment: 41 pages; v2: minor corrections, formatting changed to match published versio

On the representation theory of Virasoro Nets

We discuss various aspects of the representation theory of the local nets of von Neumann algebras on the circle associated with positive energy representations of the Virasoro algebra (Virasoro nets). In particular we classify the local extensions of the $c=1$ Virasoro net for which the restriction of the vacuum representation to the Virasoro subnet is a direct sum of irreducible subrepresentations with finite statistical dimension (local extensions of compact type). Moreover we prove that if the central charge $c$ is in a certain subset of $(1,\infty)$, including $[2,\infty)$, and $h \geq (c-1)/24$, the irreducible representation with lowest weight $h$ of the corresponding Virasoro net has infinite statistical dimension. As a consequence we show that if the central charge $c$ is in the above set and satisfies $c\leq 25$ then the corresponding Virasoro net has no proper local extensions of compact type.Comment: 34 page