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 multigraded generalizations of Kirillov-Reshetikhin modules

We study the category of Z^l-graded modules with finite-dimensional graded pieces for certain Z+^l-graded Lie algebras. We also consider certain Serre subcategories with finitely many isomorphism classes of simple objects. We construct projective resolutions for the simple modules in these categories and compute the Ext groups between simple modules. We show that the projective covers of the simple modules in these Serre subcategories can be regarded as multigraded generalizations of Kirillov-Reshetikhin modules and give a recursive formula for computing their graded characters

High-speed atomic force microscopy reveals a three-state elevator mechanism in the citrate transporter CitS

The secondary active transporter CitS shuttles citrate across the cytoplasmic membrane of gram-negative bacteria by coupling substrate translocation to the transport of two Na(+) ions. Static crystal structures suggest an elevator type of transport mechanism with two states: up and down. However, no dynamic measurements have been performed to substantiate this assumption. Here, we use high-speed atomic force microscopy for real-time visualization of the transport cycle at the level of single transporters. Unexpectedly, instead of a bimodal height distribution for the up and down states, the experiments reveal movements between three distinguishable states, with protrusions of ∼0.5 nm, ∼1.0 nm, and ∼1.6 nm above the membrane, respectively. Furthermore, the real-time measurements show that the individual protomers of the CitS dimer move up and down independently. A three-state elevator model of independently operating protomers resembles the mechanism proposed for the aspartate transporter Glt(Ph). Since CitS and Glt(Ph) are structurally unrelated, we conclude that the three-state elevators have evolved independently