The Structure of Semisimple Algebras: The Artin–Wedderburn Theorem

2018 ◽  
pp. 103-116
Author(s):  
Karin Erdmann ◽  
Thorsten Holm
Keyword(s):  
Semisimple Algebras

scholarly journals Centralizers in semisimple algebras, and descent spectrum in Banach algebras

2011 ◽  
Vol 347 (1) ◽  
pp. 214-223 ◽  
Author(s):  
A. Haïly ◽  
A. Kaidi ◽  
A. Rodríguez Palacios
Keyword(s):  
Banach Algebras ◽  
Semisimple Algebras

scholarly journals Gradings on semisimple algebras

2018 ◽  
Vol 559 ◽  
pp. 145-171 ◽  
Author(s):  
Alejandra S. Córdova-Martínez ◽  
Alberto Elduque
Keyword(s):  
Semisimple Algebras

scholarly journals Maximal subalgebras of finite-dimensional algebras

2019 ◽  
Vol 31 (5) ◽  
pp. 1283-1304 ◽  
Author(s):  
Miodrag Cristian Iovanov ◽  
Alexander Harris Sistko
Keyword(s):  
Associative Algebra ◽  
Complete Classification ◽  
Maximal Subalgebras ◽  
Incidence Algebras ◽  
Finite Dimensional ◽  
Semisimple Algebras ◽  
Closed Fields ◽  
Representations Of Algebras ◽  
Algebraically Closed Fields ◽  
Full Classification

AbstractWe study maximal associative subalgebras of an arbitrary finite-dimensional associative algebra B over a field {\mathbb{K}} and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.

scholarly journals Equality of the algebraic and geometric ranks of Cartan subalgebras and applications to linearization of a system of ordinary differential equations

2017 ◽  
Vol 28 (11) ◽  
pp. 1750080
Author(s):  
Hassan Azad ◽  
Indranil Biswas ◽  
Fazal M. Mahomed
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vector Fields ◽  
Symmetry Algebra ◽  
Second Order ◽  
Semisimple Lie Algebra ◽  
Semisimple Algebras ◽  
Generic Orbit ◽  
Cartan Subalgebras ◽  
Local Vector

If [Formula: see text] is a semisimple Lie algebra of vector fields on [Formula: see text] with a split Cartan subalgebra [Formula: see text], then it is proved here that the dimension of the generic orbit of [Formula: see text] coincides with the dimension of [Formula: see text]. As a consequence one obtains a local canonical form of [Formula: see text] in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates — for a suitable choice of coordinate system. This result is used to classify semisimple algebras of local vector fields on [Formula: see text] and to determine all representations of [Formula: see text] as local vector fields on [Formula: see text]. These representations are in turn used to find linearizing coordinates for any second-order ordinary differential equation that admits [Formula: see text] as its symmetry algebra and for a system of two second-order ordinary differential equations that admits [Formula: see text] as its symmetry algebra.

ON COMMUTANTS, DUAL PAIRS AND NON-SEMISIMPLE ALGEBRAS FROM STATISTICAL MECHANICS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 675-705 ◽  
Author(s):  
PAUL P. MARTIN ◽  
DAVID S. MCANALLY
Keyword(s):  
Dual Pair ◽  
Quantum Spin ◽  
Spin Chains ◽  
Quantum Spin Chains ◽  
Complex Vector ◽  
Dual Pairs ◽  
Morita Equivalent ◽  
Finite Dimensional ◽  
Semisimple Algebras ◽  
Special Case

For M a finite dimensional complex vector space and A a certain type of (unital) subalgebra of End(M) (including some specific types of physical significance in the field of quantum spin chains) we give an algorithm for constructing the centraliser or commutant B of A on M. We give examples, and discuss the conditions for centralising to be an involution, i.e. A, B a dual pair, and for B and A to be Morita equivalent. A special case of one example shows that Hn(q), Uq(sl2) act as a dual pair on the tensored vector representation for all q.

Symmetric amenability and the nonexistence of Lie and Jordan derivations

1996 ◽  
Vol 120 (3) ◽  
pp. 455-473 ◽  
Author(s):  
B. E. Johnson
Keyword(s):  
Positive Result ◽  
Banach Algebra ◽  
Recent Work ◽  
Banach Algebras ◽  
Jordan Derivation ◽  
Symmetric Tensors ◽  
C Algebras ◽  
Semisimple Algebras ◽  
First Time ◽  
Semisimple Banach Algebra

A. M. Sinclair has proved that if is a semisimple Banach algebra then every continuous Jordan derivation from into is a derivation ([12, theorem 3·3]; ‘Jordan derivation’ is denned in Section 6 below). If is a Banach -bimodule one can consider Jordan derivations from into and ask whether Sinclair's theorem is still true. More recent work in this area appears in [1]. Simple examples show that it cannot hold for all modules and all semisimple algebras. However, for more restricted classes of algebras, including C*-algebras one does get a positive result and we develop two approaches. The first depends on symmetric amenability, a development of the theory of amenable Banach algebras which we present here for the first time in Sections 2, 3 and 4. A Banach algebra is symmetrically amenable if it has an approximate diagonal consisting of symmetric tensors. Most, but not all, amenable Banach algebras are symmetrically amenable and one can prove results for symmetric amenability similar to those in [8] for amenability. However, unlike amenability, symmetric amenability does not seem to have a concise homological characterisation. One of our results [Theorem 6·2] is that if is symmetrically amenable then every continuous Jordan derivation into an -bimodule is a derivation. Special techniques enable this result to be extended to other algebras, for example all C*-algebras. This approach to Jordan derivations appears in Section 6.

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