A classification of direct sums of closed groups

1966 ◽  
Vol 17 (3-4) ◽  
pp. 263-266 ◽  
Author(s):  
P. Hill
Keyword(s):  
Direct Sums ◽  
Closed Groups

scholarly journals Inductive limit of direct sums of simple TAI algebras

2019 ◽  
pp. 1-26
Author(s):  
Bo Cui ◽  
Chunlan Jiang ◽  
Liangqing Li
Keyword(s):  
Inductive Limit ◽  
Direct Sums ◽  
Rank Zero ◽  
C Algebra ◽  
C Algebras ◽  
Rank One ◽  
Tracial Rank ◽  
Ideal Property ◽  
Funct Anal

An ATAI (or ATAF, respectively) algebra, introduced in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (or in [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], respectively) is an inductive limit [Formula: see text], where each [Formula: see text] is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404], the second author classified all ATAI algebras by an invariant consisting orderd total [Formula: see text]-theory and tracial state spaces of cut down algebras under an extra restriction that all element in [Formula: see text] are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic [Formula: see text]-group as an addition to the invariant used in [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404]. The theorem is proved by reducing the class to the classification theorem of [Formula: see text] algebras with ideal property which is done in [G. Gong, C. Jiang and L. Li, A classification of inductive limit C*-algebras with ideal property, preprint (2016), arXiv:1607.07681]. Our theorem generalizes the main theorem of [X. C. Fang, The classification of certain non-simple C*-algebras of tracial rank zero, J. Funct. Anal. 256 (2009) 3861–3891], [C. Jiang, A classification of non simple C*-algebras of tracial rank one: Inductive limit of finite direct sums of simple TAI C*-algebras, J. Topol. Anal. 3 (2011) 385–404] (see Corollary 4.3).

scholarly journals Classification of irreducible integrable highest weight modules for current Kac–Moody algebras

2016 ◽  
Vol 16 (07) ◽  
pp. 1750123 ◽  
Author(s):  
S. Eswara Rao ◽  
Punita Batra
Keyword(s):  
Lie Algebras ◽  
Direct Sums ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Highest Weight Modules ◽  
Weight Modules

This paper classifies irreducible, integrable highest weight modules for “current Kac–Moody Algebras” with finite-dimensional weight spaces. We prove that these modules turn out to be modules of appropriate direct sums of finitely many copies of Kac–Moody Lie algebras.

The classification of certain ASH C*-algebras of real rank zero

2020 ◽  
pp. 1-20
Author(s):  
Qingnan An ◽  
George A. Elliott ◽  
Zhiqiang Li ◽  
Zhichao Liu
Keyword(s):  
Real Rank ◽  
Inductive Limits ◽  
Real Rank Zero ◽  
Direct Sums ◽  
Rank Zero ◽  
The Real ◽  
C Algebras ◽  
Generalized Dimension ◽  
Theoretic Classification

In this paper, using ordered total K-theory, we give a K-theoretic classification for the real rank zero inductive limits of direct sums of generalized dimension drop interval algebras.

Lexicographic Direct Sums of Elementary C*-Algebras

1987 ◽  
Vol 39 (2) ◽  
pp. 257-296
Author(s):  
Horst Behncke ◽  
George A. Elliott
Keyword(s):  
Ordered Set ◽  
Large Body ◽  
Large Degree ◽  
Chain Condition ◽  
Direct Sums ◽  
C Algebras

Besides the simple ones, there are several other kinds of C*-algebras which it has proved interesting to try to classify. For instance, a large body of results relates to the extensions of one given C*-algebra, possibly simple, by another. Extrapolating in this direction, we have considered the class of C*-algebras which can be decomposed in the strongest possible nontrivial sense in terms of their simple subquotients, and such that these simple subquotients in turn are as uncomplicated as possible.We have found that the classification of these C*-algebras, namely, the lexicographic direct sums of elementary C*-algebras, is to a large degree tractable, and yet involves an interesting new invariant in the antiliminary case, which is the case of no minimal ideals. Even the postliminary case, which is the case that the ordered set of simple subquotients satisfies the decreasing chain condition, is not without interest as an extension of the case of finitely many simple subquotients, analysed in the earlier papers [1] and [2].

Finite irreducible modules of Lie conformal algebras 𝒲(a,b) and some Schrödinger–Virasoro type Lie conformal algebras

2019 ◽  
Vol 30 (06) ◽  
pp. 1950026 ◽  
Author(s):  
Lipeng Luo ◽  
Yanyong Hong ◽  
Zhixiang Wu
Keyword(s):  
Complete Classification ◽  
Conformal Algebra ◽  
Direct Sums ◽  
Conformal Algebras ◽  
Irreducible Modules ◽  
Rank One ◽  
Virasoro Type

Lie conformal algebras [Formula: see text] are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we first give a complete classification of all finite nontrivial irreducible conformal modules of [Formula: see text]. It is shown that all such modules are of rank one. Moreover, with a similar method, all finite nontrivial irreducible conformal modules of Schrödinger–Virasoro type Lie conformal algebras [Formula: see text] and [Formula: see text] are characterized.

Limits of Sums of Digraph Algebras with Reduced Digraphs Having Two Vertices

1997 ◽  
Vol 08 (01) ◽  
pp. 61-82 ◽  
Author(s):  
David Heffernan ◽  
Stephen C. Power
Keyword(s):  
Triangular Matrix ◽  
Direct Sums ◽  
Matrix Algebras ◽  
Upper Triangular Matrix ◽  
Theoretic Classification

We obtain a K-theoretic classification of limits of direct sums of 2 × 2 block upper triangular matrix algebras with respect to regular embeddings. An intrinsic characterisation of these algebras is also obtained.

Representations of the Bondi-Metzner—Sachs group III. Poincaré spin multiplicities and irreducibility

1973 ◽  
Vol 335 (1602) ◽  
pp. 301-311 ◽  
Keyword(s):  
Elementary Particles ◽  
Factor Group ◽  
Positive Mass ◽  
Negative Mass ◽  
Direct Sums ◽  
Group Iii ◽  
Group I

It has been conjectured that representations of the B.M.S. group may be of relevance to the classification of elementary particles. In an effort to examine this conjecture, the Poincaré spin multiplicities occurring in each induced B.M.S. representation are calculated. For positive mass squared, direct sums of discrete Poincaré spins occur. For non-positive mass-squared, direct integrals of continuous Poincaré spins (together with, possibly, direct sums as well for negative mass squared) occur, though the Bondi spins are always discrete. It is proved that all induced B.M.S. representations (and hence also those of Komar’s factor group I) are irreducible.

scholarly journals On finite GK-dimensional Nichols algebras over abelian groups

2021 ◽  
Vol 271 (1329) ◽  
Author(s):  
Nicolás Andruskiewitsch ◽  
Iván Angiono ◽  
István Heckenberger
Keyword(s):  
Hopf Algebras ◽  
Abelian Groups ◽  
Jordan Block ◽  
Vector Spaces ◽  
Direct Sums ◽  
Content Type ◽  
Nichols Algebra ◽  
Nichols Algebras ◽  
Kirillov Dimension

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GKdim \operatorname {GKdim} for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over Z \mathbb {Z} with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite GKdim \operatorname {GKdim} if and only if the size of the block is 2 and the eigenvalue is ± 1 \pm 1 ; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite GKdim \operatorname {GKdim} if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite GKdim \operatorname {GKdim} . Consequently we present several new examples of Nichols algebras with finite GKdim \operatorname {GKdim} , including two not in the class alluded to above. We determine which among these Nichols algebras are domains.

On Classification of Certain C*-Algebras

2000 ◽  
Vol 43 (3) ◽  
pp. 320-329
Author(s):  
George Elliott ◽  
Igor Fulman
Keyword(s):  
Inductive Limits ◽  
Direct Sums ◽  
C Algebras

AbstractWe consider C*-algebras which are inductive limits of finite direct sums of copies of . For such algebras, the lattice of closed two-sided ideals is proved to be a complete invariant.

A CLASSIFICATION OF NON-SIMPLE C*-ALGEBRAS OF TRACIAL RANK ONE: INDUCTIVE LIMITS OF FINITE DIRECT SUMS OF SIMPLE TAI C*-ALGEBRAS

2011 ◽  
Vol 03 (03) ◽  
pp. 385-404 ◽  
Author(s):  
CHUNLAN JIANG
Keyword(s):  
Compatibility Conditions ◽  
Inductive Limits ◽  
Direct Sums ◽  
State Spaces ◽  
Tracial State ◽  
C Algebras ◽  
Rank One ◽  
Tracial Rank ◽  
Unital Simple

In this paper, we will classify the class of C*-algebras which are inductive limits of finite direct sums of unital simple separable nuclear C*-algebras with tracial rank no more than one (or equivalently TAI algebras) with torsion K1-group which satisfy the UCT. The invariant consists of ordered total K-theory and the tracial state spaces of cutdown algebras (with certain compatibility conditions).

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