2,732 research outputs found
PBW deformations of Koszul algebras over a nonsemisimple ring
Let be a generalized Koszul algebra over a finite dimensional algebra
. We construct a bimodule Koszul resolution of when the projective
dimension of equals 2. Using this we prove a Poincar\'e-Birkhoff-Witt
(PBW) type theorem for a deformation of a generalized Koszul algebra. When the
projective dimension of is greater than 2, we construct bimodule Koszul
resolutions for generalized smash product algebras obtained from braidings
between finite dimensional algebras and Koszul algebras, and then prove the PBW
type theorem. The results obtained can be applied to standard Koszul
Artin-Schelter Gorenstein algebras in the sense of Minamoto and Mori.Comment: Section 3 revised, to appear in Math.
Nakayama automorphisms of double Ore extensions of Koszul regular algebras
Let be a Koszul Artin-Schelter regular algebra and an algebra
homomorphism from to . We compute the Nakayama
automorphisms of a trimmed double Ore extension
(introduced in \cite{ZZ08}). Using a similar method, we also obtain the
Nakayama automorphism of a skew polynomial extension , where
is a graded algebra automorphism of . These lead to a
characterization of the Calabi-Yau property of , the
skew Laurent extension and with a diagonal type.Comment: The paper has been heavily revised including the title, and will
appear in Manuscripta Mathematic
Hopf algebra actions on differential graded algebras and applications
Let be a finite dimensional semisimple Hopf algebra, a differential
graded (dg for short) -module algebra. Then the smash product algebra
is a dg algebra. For any dg -module , there is a quasi-isomorphism of
dg algebras: \mathrm{RHom}_A(M,M)\#H\longrightarrow \mathrm{RHom}_{A\#H}(M\ot
H,M\ot H). This result is applied to -Koszul algebras, Calabi-Yau algebras
and AS-Gorenstein dg algebrasComment: to appear in Bull. Belg. Math So
Deformations of Koszul Artin-Schelter Gorenstein algebras
We compute the Nakayama automorphism of a PBW-deformation of a Koszul
Artin-Schelter Gorenstein algebra of finite global dimension, and give a
criterion for an augmented PBW-deformation of a Koszul Calabi-Yau algebra to be
Calabi-Yau. The relations between the Calabi-Yau property of augmented
PBW-deformations and that of non-augmented cases are discussed. The Nakayama
automorphisms of PBW-deformations of Koszul Artin-Schelter Gorenstein algebras
of global dimensions 2 and 3 are given explicitly. We show that if a
PBW-deformation of a graded Calabi-Yau algebra is still Calabi-Yau, then it is
defined by a potential under some mild conditions. Some classical results are
also recovered. Our main method used in this paper is elementary and based on
linear algebra. The results obtained in this paper will be applied in a
subsequent paper.Comment: to appear at Manuscripta Mat
Derived -module endomorphism rings
Let be a Hopf algebra, be an -Galois extension. Let and
be the derived categories of right -modules and of right -modules
respectively. An object may be regarded as an object in
via the restriction functor. We discuss the relations of the derived
endomorphism rings
E_A(M^\cdot)=\op_{i\in\mathbb{Z}}\Hom_{D(A)}(M^\cdot,M^\cdot[i]) and
E_B(M^\cdot)=\op_{i\in\mathbb{Z}}\Hom_{D(B)}(M^\cdot,M^\cdot[i]). If is a
finite dimensional semisimple Hopf algebra, then is a graded
subalgebra of . In particular, if is a usual -module, a
necessary and sufficient condition for to be an -Galois graded
extension of is obtained. As an application of the results, we show
that the Koszul property is preserved under Hopf Galois graded extensions.Comment: to appear at Glasgow Mathematical Journa
Probabilistically Safe Policy Transfer
Although learning-based methods have great potential for robotics, one
concern is that a robot that updates its parameters might cause large amounts
of damage before it learns the optimal policy. We formalize the idea of safe
learning in a probabilistic sense by defining an optimization problem: we
desire to maximize the expected return while keeping the expected damage below
a given safety limit. We study this optimization for the case of a robot
manipulator with safety-based torque limits. We would like to ensure that the
damage constraint is maintained at every step of the optimization and not just
at convergence. To achieve this aim, we introduce a novel method which predicts
how modifying the torque limit, as well as how updating the policy parameters,
might affect the robot's safety. We show through a number of experiments that
our approach allows the robot to improve its performance while ensuring that
the expected damage constraint is not violated during the learning process
The Green rings of pointed tensor categories of finite type
In this paper, we compute the Clebsch-Gordan formulae and the Green rings of
connected pointed tensor categories of finite type.Comment: 14 page
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