117 research outputs found

### Valuation Extensions of Filtered and Graded Algebras

In this note we relate the valuations of the algebras appearing in the
non-commutative geometry of quantized algebras to properties of sub-lattices in
some vector spaces. We consider the case of algebras with $PBW$-bases and prove
that under some mild assumptions the valuations of the ground field extend to a
non-commutative valuation. Later we introduce the notion of $F$-reductor and
graded reductor and reduce the problem of finding an extending non-commutative
valuation to finding a reductor in an associated graded ring having a domain
for its reduction.Comment: 12 page

### Differentiable functions of quaternion variables

We investigate differentiability of functions defined on regions of the real
quaternion field and obtain a noncommutative version of the Cauchy-Riemann
conditions. Then we study the noncommutative analog of the Cauchy integral as
well as criteria for functions of a quaternion variable to be analytic. In
particular, the quaternionic exponential and logarithmic functions are being
considered. Main results include quaternion versions of Hurwitz' theorem,
Mittag-Leffler's theorem and Weierstrass theorem.Comment: 48 pages, Late

### Cocommutative Calabi-Yau Hopf algebras and deformations

The Calabi-Yau property of cocommutative Hopf algebras is discussed by using
the homological integral, a recently introduced tool for studying infinite
dimensional AS-Gorenstein Hopf algebras. It is shown that the skew-group
algebra of a universal enveloping algebra of a finite dimensional Lie algebra
\g with a finite subgroup $G$ of automorphisms of \g is Calabi-Yau if and
only if the universal enveloping algebra itself is Calabi-Yau and $G$ is a
subgroup of the special linear group SL(\g). The Noetherian cocommutative
Calabi-Yau Hopf algebras of dimension not larger than 3 are described. The
Calabi-Yau property of Sridharan enveloping algebras of finite dimensional Lie
algebras is also discussed. We obtain some equivalent conditions for a
Sridharan enveloping algebra to be Calabi-Yau, and then partly answer a
question proposed by Berger. We list all the nonisomorphic 3-dimensional
Calabi-Yau Sridharan enveloping algebras

### Dualities of artinian coalgebras with applications to noetherian complete algebras

A duality theorem of the bounded derived category of quasi-finite comodules
over an artinian coalgebra is established. Let $A$ be a noetherian complete
basic semiperfect algebra over an algebraically closed field, and $C$ be its
dual coalgebra. If $A$ is Artin-Schelter regular, then the local cohomology of
$A$ is isomorphic to a shift of twisted bimodule ${}_1C_{\sigma^*}$ with
$\sigma$ a coalgebra automorphism. This yields that the balanced dualinzing
complex of $A$ is a shift of the twisted bimodule ${}_{\sigma^*}A_1$. If
$\sigma$ is an inner automorphism, then $A$ is Calabi-Yau

### Calabi-Yau coalgebras

We provide a construction of minimal injective resolutions of simple
comodules of path coalgebras of quivers with relations. Dual to Calabi-Yau
condition of algebras, we introduce the Calabi-Yau condition to coalgebras.
Then we give some descriptions of Calabi-Yau coalgebras with lower global
dimensions. An appendix is included for listing some properties of cohom
functors

### Quantum sections and Gauge algebras

Using quantum sections of filtered rings and the associated Rees rings one can lift the scheme structure on Proj of the associated graded ring to the Proj of the Rees ring . The algebras of interest here are positively filtered rings having a non-commutative regular quadratic algebra for the associated graded ring ; these are the socalled gauge algebras obtaining their name from special examples appearing in E. Witten's gauge theories . The paper surveys basic definitions and properties but concentrates on the development of several concrete examples

### q-Legendre Transformation: Partition Functions and Quantization of the Boltzmann Constant

In this paper we construct a q-analogue of the Legendre transformation, where
q is a matrix of formal variables defining the phase space braidings between
the coordinates and momenta (the extensive and intensive thermodynamic
observables). Our approach is based on an analogy between the semiclassical
wave functions in quantum mechanics and the quasithermodynamic partition
functions in statistical physics. The basic idea is to go from the
q-Hamilton-Jacobi equation in mechanics to the q-Legendre transformation in
thermodynamics. It is shown, that this requires a non-commutative analogue of
the Planck-Boltzmann constants (hbar and k_B) to be introduced back into the
classical formulae. Being applied to statistical physics, this naturally leads
to an idea to go further and to replace the Boltzmann constant with an infinite
collection of generators of the so-called epoch\'e (bracketing) algebra. The
latter is an infinite dimensional noncommutative algebra recently introduced in
our previous work, which can be perceived as an infinite sequence of
"deformations of deformations" of the Weyl algebra. The generators mentioned
are naturally indexed by planar binary leaf-labelled trees in such a way, that
the trees with a single leaf correspond to the observables of the limiting
thermodynamic system

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