Cyclic division algebras: a tool for space-time coding

Multiple antennas at both the transmitter and receiver ends of a wireless digital transmission channel may increase both data rate and reliability. Reliable high rate transmission over such channels can only be achieved through Space–Time coding. Rank and determinant code design criteria have been proposed to enhance diversity and coding gain. The special case of full-diversity criterion requires that the difference of any two distinct codewords has full rank. Extensive work has been done on Space–Time coding, aiming at finding fully diverse codes with high rate. Division algebras have been proposed as a new tool for constructing Space–Time codes, since they are non-commutative algebras that naturally yield linear fully diverse codes. Their algebraic properties can thus be further exploited to improve the design of good codes. The aim of this work is to provide a tutorial introduction to the algebraic tools involved in the design of codes based on cyclic division algebras. The different design criteria involved will be illustrated, including the constellation shaping, the information lossless property, the non-vanishing determinant property, and the diversity multiplexing trade-off. The final target is to give the complete mathematical background underlying the construction of the Golden code and the other Perfect Space–Time block codes

Inner Ideals of Simple Locally Finite Lie Algebras

Inner ideals of simple locally finite dimensional Lie algebras over an algebraically closed field of characteristic 0 are described. In particular, it is shown that a simple locally finite dimensional Lie algebra has a non-zero proper inner ideal if and only if it is of diagonal type. Regular inner ideals of diagonal type Lie algebras are characterized in terms of left and right ideals of the enveloping algebra. Regular inner ideals of finitary simple Lie algebras are described

Kazhdan and Haagerup Properties in algebraic groups over local fields

Given a Lie algebra \s, we call Lie \s-algebra a Lie algebra endowed with a reductive action of \s. We characterize the minimal \s-Lie algebras with a nontrivial action of \s, in terms of irreducible representations of \s and invariant alternating forms. As a first application, we show that if \g is a Lie algebra over a field of characteristic zero whose amenable radical is not a direct factor, then \g contains a subalgebra which is isomorphic to the semidirect product of sl_2 by either a nontrivial irreducible representation or a Heisenberg group (this was essentially due to Cowling, Dorofaeff, Seeger, and Wright). As a corollary, if G is an algebraic group over a local field K of characteristic zero, and if its amenable radical is not, up to isogeny, a direct factor, then G(K) has Property (T) relative to a noncompact subgroup. In particular, G(K) does not have Haagerup's property. This extends a similar result of Cherix, Cowling and Valette for connected Lie groups, to which our method also applies. We give some other applications. We provide a characterization of connected Lie groups all of whose countable subgroups have Haagerup's property. We give an example of an arithmetic lattice in a connected Lie group which does not have Haagerup's property, but has no infinite subgroup with relative Property (T). We also give a continuous family of pairwise non-isomorphic connected Lie groups with Property (T), with pairwise non-isomorphic (resp. isomorphic) Lie algebras.Comment: 11 pages, no figur

Homology of perfect complexes

It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective dimension, obstructions to realizing R as a closed fiber of some flat local homomorphism. Other applications include, as special cases, uniform proofs of known results on free actions of elementary abelian groups and of tori on finite CW complexes. The arguments use numerical invariants of objects in general triangulated categories, introduced here and called levels. They allow one to track, through changes of triangulated categories, homological invariants like projective dimension, as well as structural invariants like Loewy length. An intermediate result sharpens, with a new proof, the New Intersection Theorem for commutative algebras over fields. Under additional hypotheses on the ring $R$ stronger estimates are proved for Loewy lengths of modules of finite projective dimension.Comment: This version corrects an error in the statement (and proof) of Theorem 7.4 in the published version of the paper [Adv. Math. 223 (2010) 1731--1781]. These changes do not affect any other results or proofs in the paper. A corrigendum has been submitted

Resolutions for unit groups of orders

We present a general algorithm for constructing a free resolution for unit groups of orders in semisimple rational algebras. The approach is based on computing a contractible $G$-complex employing the theory of minimal classes of quadratic forms and Opgenorth's theory of dual cones. The information from the complex is then used together with Wall's perturbation lemma to obtain the resolution