11,290 research outputs found
Fibred product of commutative algebras: generators and relations
The method of direct computation of universal (fibred) product in the
category of commutative associative algebras of finite type with unity over a
field is given and proven. The field of coefficients is not supposed to be
algebraically closed and can be of any characteristic. Formation of fibred
product of commutative associative algebras is an algebraic counterpart of
gluing algebraic schemes by means of some equivalence relation in algebraic
geometry. If initial algebras are finite-dimensional vector spaces the
dimension of their product obeys Grassmann-like formula. Finite-dimensional
case means geometrically the strict version of adding two collections of points
containing some common part.
The method involves description of algebras by generators and relations on
input and returns similar description of the product algebra. It is
"ready-to-eat" even for computer realization. The product algebra is
well-defined: taking another descriptions of the same algebras leads to
isomorphic product algebra. Also it is proven that the product algebra enjoys
universal property, i.e. it is indeed fibred product. The input data is a
triple of algebras and a pair of homomorphisms . Algebras and homomorphisms can be described in
any fashion. We prove that for computing the fibred product it is enough to
restrict to the case when are surjective and describe how to
reduce to surjective case. Also the way to choose generators and relations for
input algebras is considered.Comment: 15 pages, generalization of result of previous versio
Twisting algebras using non-commutative torsors
Non-commutative torsors (equivalently, two-cocycles) for a Hopf algebra can
be used to twist comodule algebras. After surveying and extending the
literature on the subject, we prove a theorem that affords a presentation by
generators and relations for the algebras obtained by such twisting. We give a
number of examples, including new constructions of the quantum affine spaces
and the quantum tori.Comment: 27 pages. Masuoka is a new coauthor. Introduction was revised.
Sections 1 and 2 were thoroughly restructured. The presentation theorem in
Section 3 is now put in a more general framework and has a more general
formulation. Section 4 was shortened. All examples (quantum affine spaces and
tori, twisting of SL(2), twisting of the enveloping algebra of sl(2)) are
left unchange
Bethe subalgebras in affine Birman--Murakami--Wenzl algebras and flat connections for q-KZ equations
Commutative sets of Jucys-Murphyelements for affine braid groups of
types were defined. Construction of
-matrix representations of the affine braid group of type and its
distinguish commutative subgroup generated by the -type Jucys--Murphy
elements are given. We describe a general method to produce flat connections
for the two-boundary quantum Knizhnik-Zamolodchikov equations as necessary
conditions for Sklyanin's type transfer matrix associated with the two-boundary
multicomponent Zamolodchikov algebra to be invariant under the action of the
-type Jucys--Murphy elements. We specify our general construction to
the case of the Birman--Murakami--Wenzl algebras. As an application we suggest
a baxterization of the Dunkl--Cherednik elements in the double affine
Hecke algebra of type
Noncommutative curves and noncommutative surfaces
In this survey article we describe some geometric results in the theory of
noncommutative rings and, more generally, in the theory of abelian categories.
Roughly speaking and by analogy with the commutative situation, the category
of graded modules modulo torsion over a noncommutative graded ring of
quadratic, respectively cubic growth should be thought of as the noncommutative
analogue of a projective curve, respectively surface. This intuition has lead
to a remarkable number of nontrivial insights and results in noncommutative
algebra. Indeed, the problem of classifying noncommutative curves (and
noncommutative graded rings of quadratic growth) can be regarded as settled.
Despite the fact that no classification of noncommutative surfaces is in sight,
a rich body of nontrivial examples and techniques, including blowing up and
down, has been developed.Comment: Suggestions by many people (in particular Haynes Miller and Dennis
Keeler) have been incorporated. The formulation of some results has been
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