25 research outputs found
The quotient Unimodular Vector group is nilpotent
Jose-Rao introduced and studied the Special Unimodular Vector group
and , its Elementary Unimodular Vector subgroup. They
proved that for , is a normal subgroup of . The
Jose-Rao theorem says that the quotient Unimodular Vector group,
, for , is a subgroup of the orthogonal quotient
group . The latter group is known to be
nilpotent by the work of Hazrat-Vavilov, following methods of A. Bak; and so is
the former.
In this article we give a direct proof, following ideas of A. Bak, to show
that the quotient Unimodular Vector group is nilpotent of class . We also use the Quillen-Suslin theory, inspired by A. Bak's method,
to prove that if , with a local ring, then the quotient
Unimodular Vector group is abelian
Algebraic K-theory of endomorphism rings
We establish formulas for computation of the higher algebraic -groups of
the endomorphism rings of objects linked by a morphism in an additive category.
Let be an additive category, and let Y\ra X be a covariant
morphism of objects in . Then for all , where is the
quotient ring of the endomorphism ring of modulo the
ideal generated by all those endomorphisms of which factorize through .
Moreover, let be a ring with identity, and let be an idempotent element
in . If is homological and has a finite projective resolution
by finitely generated projective -modules, then for all . This reduces calculations of the higher
algebraic -groups of to those of the quotient ring and the corner
ring , and can be applied to a large variety of rings: Standardly
stratified rings, hereditary orders, affine cellular algebras and extended
affine Hecke algebras of type .Comment: 21 pages. Representation-theoretic methods are used to study the
algebraic K-theory of ring
Open Problems on Central Simple Algebras
We provide a survey of past research and a list of open problems regarding
central simple algebras and the Brauer group over a field, intended both for
experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered,
compared to v
K-Decompositions and 3d Gauge Theories
This paper combines several new constructions in mathematics and physics.
Mathematically, we study framed flat PGL(K,C)-connections on a large class of
3-manifolds M with boundary. We define a space L_K(M) of framed flat
connections on the boundary of M that extend to M. Our goal is to understand an
open part of L_K(M) as a Lagrangian in the symplectic space of framed flat
connections on the boundary, and as a K_2-Lagrangian, meaning that the
K_2-avatar of the symplectic form restricts to zero. We construct an open part
of L_K(M) from data assigned to a hypersimplicial K-decomposition of an ideal
triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic
geometry, and combining them with the cluster coordinates for framed flat
PGL(K)-connections on surfaces. Using a canonical map from the complex of
configurations of decorated flags to the Bloch complex, we prove that any
generic component of L_K(M) is K_2-isotropic if the boundary satisfies some
topological constraints (Theorem 4.2). In some cases this implies that L_K(M)
is K_2-Lagrangian. For general M, we extend a classic result of Neumann-Zagier
on symplectic properties of PGL(2) gluing equations to reduce the
K_2-Lagrangian property to a combinatorial claim.
Physically, we use the symplectic properties of K-decompositions to construct
3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to
the compactification of K M5-branes on M. This extends known constructions for
K=2. Just as for K=2, the theories T_K[M] are described as IR fixed points of
abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead
to abelian mirror symmetries that are all generated by the elementary duality
between N_f=1 SQED and the XYZ model. In the large K limit, we find evidence
that the degrees of freedom of T_K[M] grow cubically in K.Comment: 121 pages + 2 appendices, 80 figures; Version 2: reorganized
mathematical perspective, swapped Sections 3 and