508 research outputs found
Polynomial identity rings as rings of functions
We generalize the usual relationship between irreducible Zariski closed
subsets of the affine space, their defining ideals, coordinate rings, and
function fields, to a non-commutative setting, where "varieties" carry a
PGL_n-action, regular and rational "functions" on them are matrix-valued,
"coordinate rings" are prime polynomial identity algebras, and "function
fields" are central simple algebras of degree n. In particular, a prime
polynomial identity algebra of degree n is finitely generated if and only if it
arises as the "coordinate ring" of a "variety" in this setting. For n = 1 our
definitions and results reduce to those of classical affine algebraic geometry.Comment: 24 pages. This is the final version of the article, to appear in J.
Algebra. Several proofs have been streamlined, and a new section on
Brauer-Severi varieties has been adde
The prime spectrum of algebras of quadratic growth
We study prime algebras of quadratic growth. Our first result is that if
is a prime monomial algebra of quadratic growth then has finitely many
prime ideals such that has GK dimension one. This shows that prime
monomial algebras of quadratic growth have bounded matrix images. We next show
that a prime graded algebra of quadratic growth has the property that the
intersection of the nonzero prime ideals such that has GK dimension 2
is non-empty, provided there is at least one such ideal. From this we conclude
that a prime monomial algebra of quadratic growth is either primitive or has
nonzero locally nilpotent Jacobson radical. Finally, we show that there exists
a prime monomial algebra of GK dimension two with unbounded matrix images
and thus the quadratic growth hypothesis is necessary to conclude that there
are only finitely many prime ideals such that has GK dimension 1.Comment: 23 page
PrrC-anticodon nuclease: functional organization of a prototypical bacterial restriction RNase
The tRNA(Lys) anticodon nuclease PrrC is associated in latent form with the type Ic DNA restriction endonuclease EcoprrI and activated by a phage T4-encoded inhibitor of EcoprrI. The activation also requires the hydrolysis of GTP and presence of dTTP and is inhibited by ATP. The N-proximal NTPase domain of PrrC has been implicated in relaying the activating signal to a C-proximal anticodon nuclease site by interacting with the requisite nucleotide cofactors [Amitsur et al. (2003) Mol. Microbiol., 50, 129–143]. Means described here to bypass PrrC's self-limiting translation and thermal instability allowed purifying an active mutant form of the protein, demonstrating its oligomeric structure and confirming its anticipated interactions with the nucleotide cofactors of the activation reaction. Mutagenesis and chemical rescue data shown implicate the C-proximal Arg(320), Glu(324) and, possibly, His(356) in anticodon nuclease catalysis. This triad exists in all the known PrrC homologs but only some of them feature residues needed for tRNA(Lys) recognition by the Escherichia coli prototype. The differential conservation and consistent genetic linkage of the PrrC proteins with EcoprrI homologs portray them as a family of restriction RNases of diverse substrate specificities that are mobilized when an associated DNA restriction nuclease is compromised
Parameter curves for the regular representations of tame bimodules
We present results and examples which show that the consideration of a
certain tubular mutation is advantageous in the study of noncommutative curves
which parametrize the simple regular representations of a tame bimodule. We
classify all tame bimodules where such a curve is actually commutative, or in
different words, where the unique generic module has a commutative endomorphism
ring. This extends results from [14] to arbitrary characteristic; in
characteristic two additionally inseparable cases occur. Further results are
concerned with autoequivalences fixing all objects but not isomorphic to the
identity functor.Comment: 13 pages, to appear in J. Algebra. Typos correcte
Polynomial identity rings as rings of functions, II
In characteristic zero, Zinovy Reichstein and the author generalized the
usual relationship between irreducible Zariski closed subsets of the affine
space, their defining ideals, coordinate rings, and function fields, to a
non-commutative setting, where "varieties" carry a PGL_n-action, regular and
rational "functions" on them are matrix-valued, "coordinate rings" are prime
polynomial identity algebras, and "function fields" are central simple algebras
of degree n. In the present paper, much of this is extended to prime
characteristic. In addition, a mistake in the earlier paper is corrected. One
of the results is that the finitely generated prime PI-algebras of degree n are
precisely the rings that arise as "coordinate rings" of "n-varieties" in this
setting. For n = 1 the definitions and results reduce to those of classical
affine algebraic geometry.Comment: 24 pages, LaTeX. Many changes. Theorem II.1.3 has been strengthened,
Sections II.6-II.8 have been rewritten, and Section II.9 is ne
Nonassociative differential extensions of characteristic p
Let F be a field of characteristic p. We define and investigate nonassociative differential extensions of F and of a finite-dimensional central division algebra over F and give a criterium for these algebras to be division. As special cases, we obtain classical results for associative algebras by Amitsur and Jacobson. We construct families of nonassociative division algebras which can be viewed as generalizations of associative cyclic extensions of a purely inseparable field extension of exponent one or a central division algebra. Division algebras which are nonassociative cyclic extensions of a purely inseparable field extension of exponent one are particularly easy to obtain
Algebraic curves for commuting elements in the q-deformed Heisenberg algebra
In this paper we extend the eliminant construction of Burchnall and Chaundy
for commuting differential operators in the Heisenberg algebra to the
q-deformed Heisenberg algebra and show that it again provides annihilating
curves for commuting elements, provided q satisfies a natural condition. As a
side result we obtain estimates on the dimensions of the eigenspaces of
elements of this algebra in its faithful module of Laurent series.Comment: 18 pages, 2 figures, LaTeX. Final version with some improvements in
presentation. To appear in Journal of Algebra
Revisiting the Equivalence Problem for Finite Multitape Automata
The decidability of determining equivalence of deterministic multitape
automata (or transducers) was a longstanding open problem until it was resolved
by Harju and Karhum\"{a}ki in the early 1990s. Their proof of decidability
yields a co_NP upper bound, but apparently not much more is known about the
complexity of the problem. In this paper we give an alternative proof of
decidability, which follows the basic strategy of Harju and Karhumaki but
replaces their use of group theory with results on matrix algebras. From our
proof we obtain a simple randomised algorithm for deciding language equivalence
of deterministic multitape automata and, more generally, multiplicity
equivalence of nondeterministic multitape automata. The algorithm involves only
matrix exponentiation and runs in polynomial time for each fixed number of
tapes. If the two input automata are inequivalent then the algorithm outputs a
word on which they differ
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