scholarly journals Amenability and weak amenability of tensor algebras and algebras of nuclear operators

1991 ◽  
Vol 51 (3) ◽  
pp. 483-488 ◽  
Author(s):  
Niels Grønbaek
Keyword(s):  
Tensor Algebra ◽  
Tensor Algebras ◽  
Canonical Map ◽  
Weak Amenability ◽  
Nuclear Operators ◽  
Finite Dimensional

AbstractLet E and F constitute a Banach pairing. We prove that the algebra of F-nuclear operators on E, Nf (E), is amenable if and only if E is finite dimensional and is weakly amenable if and only if dim KF ≦ 1, and the trace on E⊗F is injective on KF. Here KF is the kernel of the canonical map E⊗^F →NF(E). On the route we find the corresponding statements for the associated tensor algebra, E⊗^F.

scholarly journals CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

2021 ◽  
Author(s):  
MÁTYÁS DOMOKOS ◽  
VESSELIN DRENSKY
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Invariant Theory ◽  
Reductive Group ◽  
Symmetric Tensor ◽  
Free Associative Algebra ◽  
Tensor Algebra ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Degree Bounds

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

scholarly journals GRADED TWISTED CALABI–YAU ALGEBRAS ARE GENERALIZED ARTIN–SCHELTER REGULAR

2021 ◽  
pp. 1-54
Author(s):  
MANUEL L. REYES ◽  
DANIEL ROGALSKI
Keyword(s):  
Regularity Property ◽  
Graded Algebra ◽  
General Study ◽  
Locally Finite ◽  
Tensor Algebras ◽  
Finite Dimensional ◽  
Dimension 2 ◽  
Separable Algebras ◽  
Gk Dimension

Abstract This is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$ -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi–Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin–Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi–Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi–Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi–Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

Piecewise Hereditary Triangular Matrix Algebras

2021 ◽  
Vol 28 (01) ◽  
pp. 143-154
Author(s):  
Yiyu Li ◽  
Ming Lu
Keyword(s):  
Positive Integer ◽  
Triangular Matrix ◽  
Derived Categories ◽  
Matrix Algebras ◽  
Tensor Algebras ◽  
Finite Dimensional ◽  
Abelian Categories ◽  
Upper Triangular Matrix ◽  
Orbit Categories ◽  
Finite Dimensional Algebras

For any positive integer [Formula: see text], we clearly describe all finite-dimensional algebras [Formula: see text] such that the upper triangular matrix algebras [Formula: see text] are piecewise hereditary. Consequently, we describe all finite-dimensional algebras [Formula: see text] such that their derived categories of [Formula: see text]-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras [Formula: see text] for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

scholarly journals Multiplicity-free quotient tensor algebras

2001 ◽  
Vol 71 (2) ◽  
pp. 279-298
Author(s):  
G. E. Wall
Keyword(s):  
General Linear Group ◽  
Partial Solution ◽  
Tensor Algebra ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Tensor Algebras ◽  
Completely Reducible ◽  
Multiplicity Free ◽  
Free Quotient

AbstractLet V be an infinite-dimensional vector space ovre a field of characteristic 0. It is well known that the tensor algebra T on V is a completely reducible module for the general linear group G on V. This paper is concerned with those quotient algebras A of T that are at the same time modules for G. A partial solution is given to the problem of determinig those A in which no irreducible constitutent has multiplicity greater thatn 1.

scholarly journals On the algebraic K-theory of formal power series

2012 ◽  
Vol 10 (1) ◽  
pp. 165-189
Author(s):  
Ayelet Lindenstrauss ◽  
Randy McCarthy
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Natural Transformation ◽  
Tensor Algebra ◽  
Tensor Algebras ◽  
Taylor Tower ◽  
K Theory ◽  
Formal Power

AbstractIn this paper we extend the computation of the the typical curves of algebraic K-theory done by Lars Hesselholt and Ib Madsen to general tensor algebras. The models used allow us to determine the stages of the Taylor tower of algebraic K-theory as a functor of augmented algebras, as defined by Tom Goodwillie, when evaluated on derived tensor algebras.For R a discrete ring, and M a simplicial R-bimodule, we let R(M) denote the (derived) tensor algebra of M over R, and πR denote the ring of formal (derived) power series in M over R. We define a natural transformation of functors of simplicial R-bimodules Φ: which is closely related to Waldhausen's equivalence We show that Φ induces an equivalence on any finite stage of Goodwillie's Taylor towers of the functors at any simplicial bimodule. This is used to show that there is an equivalence of functors , and for connected bimodules, also an equivalence

scholarly journals Tensor Algebras in Finite Tensor Categories

2019 ◽  
Author(s):  
Pavel Etingof ◽  
Ryan Kinser ◽  
Chelsea Walton
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Classification Result ◽  
Tensor Categories ◽  
Tensor Algebras ◽  
Fusion Categories ◽  
Path Algebras ◽  
Finite Dimensional

Abstract This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers and more generally on tensor algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework of finite (multi-)tensor categories $\mathcal{C}$, classifying tensor algebras in $\mathcal{C}$ in terms of $\mathcal{C}$-module categories. We obtain two classification results for actions of semisimple Hopf algebras: the first for actions that preserve the ascending filtration on tensor algebras and the second for actions that preserve the descending filtration on completed tensor algebras. Extending to more general fusion categories, we illustrate our classification result for tensor algebras in the pointed fusion categories $\textsf{Vec}_{G}^{\omega }$ and in group-theoretical fusion categories, especially for the representation category of the Kac–Paljutkin Hopf algebra.

scholarly journals Noncommutative classical invariant theory

1988 ◽  
Vol 112 ◽  
pp. 153-169 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Vector Space ◽  
Invariant Theory ◽  
Characteristic Zero ◽  
Tensor Algebra ◽  
Dimensional Vector ◽  
Classical Invariant Theory ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Classical Invariant

Let K be a field of characteristic zero, V a finite dimensional vector space and G a subgroup of GL(V). The action of G on V is extended to the symmetric algebra on V over K,and the tensor algebra on V over K,

scholarly journals A note concerning the ideal of nuclear operators

1996 ◽  
Vol 38 (2) ◽  
pp. 233-236
Author(s):  
Bruce A. Barnes
Keyword(s):  
Banach Algebra ◽  
Linear Operators ◽  
Nuclear Operator ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Nuclear Operators ◽  
Finite Dimensional ◽  
Dimensional Range ◽  
The Ideal

AbstractLet be a Banach algebra of bounded linear operators such that contains every operator with finite dimensional range. Then contains every nuclear operator.

scholarly journals Young diagrammatic methods in non-commutative invariant theory

1991 ◽  
Vol 121 ◽  
pp. 15-34
Author(s):  
Yasuo Teranishi
Keyword(s):  
Vector Space ◽  
Invariant Theory ◽  
Characteristic Zero ◽  
Symmetric Algebra ◽  
Tensor Algebra ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Invariant Ring

In this paper we will study some aspects of non-commutative invariant theory. Let V be a finite-dimensional vector space over a field K of characteristic zero and letK[V] = K⊕V⊕S2(V)⊕…, andK′V› = K⊕V⊕⊕2(V)⊕⊕3V⊕&be respectively the symmetric algebra and the tensor algebra over V. Let G be a subgroup of GL(V). Then G acts on K[V] and K′V›. Much of this paper is devoted to the study of the (non-commutative) invariant ring K′V›G of G acting on K′V›.In the first part of this paper, we shall study the invariant ring in the following situation.

scholarly journals THE DECOMPOSITION OF LIE POWERS

2006 ◽  
Vol 93 (1) ◽  
pp. 175-196 ◽  
Author(s):  
R. M. BRYANT ◽  
M. SCHOCKER
Keyword(s):  
Positive Integer ◽  
Free Associative Algebra ◽  
Free Lie Algebra ◽  
Tensor Algebra ◽  
Prime Characteristic ◽  
Characteristic P ◽  
Descent Algebra ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Solomon Descent Algebra

Let $G$ be a group, $F$ a field of prime characteristic $p$ and $V$ a finite-dimensional $FG$-module. Let $L(V)$ denote the free Lie algebra on $V$ regarded as an $FG$-submodule of the free associative algebra (or tensor algebra) $T(V)$. For each positive integer $r$, let $L^r (V)$ and $T^r (V)$ be the $r$th homogeneous components of $L(V)$ and $T(V)$, respectively. Here $L^r (V)$ is called the $r$th Lie power of $V$. Our main result is that there are submodules $B_1$, $B_2$, ... of $L(V)$ such that, for all $r$, $B_r$ is a direct summand of $T^r(V)$ and, whenever $m \geqslant 0$ and $k$ is not divisible by $p$, the module $L^{p^mk} (V)$ is the direct sum of $L^{p^m} (B_k)$, $L^{p^{m - 1}} (B_{pk})$, ..., $L^1 (B_{p^mk})$. Thus every Lie power is a direct sum of Lie powers of $p$-power degree. The approach builds on an analysis of $T^r (V)$ as a bimodule for $G$ and the Solomon descent algebra.

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