21,051 research outputs found
Rationality of Hilbert series in noncommutative invariant theory
It is a fundamental result in commutative algebra and invariant theory that a
finitely generated graded module over a commutative finitely generated graded
algebra has rational Hilbert series, and consequently the Hilbert series of the
algebra of polynomial invariants of a group of linear transformations is
rational, whenever this algebra is finitely generated. This basic principle is
applied here to prove rationality of Hilbert series of algebras of invariants
that are neither commutative nor finitely generated. Our main focus is on
linear groups acting on certain factor algebras of the tensor algebra that
arise naturally in the theory of polynomial identities.Comment: Examples both from commutative and noncommutative invariant theory
are included, a problem is formulated and references are added. Comments for
v3: references added, minor revisio
Quasideterminants
The determinant is a main organizing tool in commutative linear algebra. In
this review we present a theory of the quasideterminants defined for matrices
over a division algebra. We believe that the notion of quasideterminants should
be one of main organizing tools in noncommutative algebra giving them the same
role determinants play in commutative algebra.Comment: amstex; final version; to appear in Advances in Mat
Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs
Building on a result by W. Rump, we show how to exploit the right-cyclic law
(x.y).(x.z) = (y.x).(y.z) in order to investigate the structure groups and
monoids attached with (involutive nondegenerate) set-theoretic solutions of the
Yang-Baxter equation. We develop a sort of right-cyclic calculus, and use it to
obtain short proofs for the existence both of the Garside structure and of the
I-structure of such groups. We describe finite quotients that exactly play for
the considered groups the role that Coxeter groups play for Artin-Tits groups
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