This book offers a systematic treatment--the first in book form--of the development and use of cohomological induction to construct unitary representations. George Mackey introduced induction in 1950 ...as a real analysis construction for passing from a unitary representation of a closed subgroup of a locally compact group to a unitary representation of the whole group. Later a parallel construction using complex analysis and its associated co-homology theories grew up as a result of work by Borel, Weil, Harish-Chandra, Bott, Langlands, Kostant, and Schmid. Cohomological induction, introduced by Zuckerman, is an algebraic analog that is technically more manageable than the complex-analysis construction and leads to a large repertory of irreducible unitary representations of reductive Lie groups. The book, which is accessible to students beyond the first year of graduate school, will interest mathematicians and physicists who want to learn about and take advantage of the algebraic side of the representation theory of Lie groups. Cohomological Induction and Unitary Representations develops the necessary background in representation theory and includes an introductory chapter of motivation, a thorough treatment of the "translation principle, " and four appendices on algebra and analysis.
The Lie admissible non-associative algebra
N
A
n
,
m
,
s
¯
is defined in the papers Seul Hee Choi, Ki-Bong Nam, Derivations of a restricted Weyl type algebra I, Rocky Mountain J. Math. 37 (6) (2007) ...1813–1830; Seul Hee Choi, Ki-Bong Nam, Weyl type non-associative algebra using additive groups I, Algebra Colloq. 14 (3) (2007) 479–488; Ki-Bong Nam, On Some Non-associative Algebras using Additive Groups, Southeast Asian Bull. Math., vol. 27, Springer-Verlag, 2003, 493–500. We define in this work the algebra
N
A
n
i
n
,
m
,
s
¯
N
which generalizes the previous one and is not Lie admissible. We prove that the antisymmetrized Lie algebra
(
N
A
n
i
n
,
m
,
s
¯
N
)
-
is simple and contains the simple Lie algebra
sl
m
+
s
(
F
)
. We also prove that the matrix ring is embedded in
N
0
,
n
,
0
¯
.
Handbook of Tilting Theory Hügel, Lidia Angeleri; Happel, Dieter; Krause, Henning
2007., Volume:
v.Series Number 332
eBook
Tilting theory originates in the representation theory of finite dimensional algebras. Today the subject is of much interest in various areas of mathematics, such as finite and algebraic group ...theory, commutative and non-commutative algebraic geometry, and algebraic topology. The aim of this book is to present the basic concepts of tilting theory as well as the variety of applications. It contains a collection of key articles, which together form a handbook of the subject, and provide both an introduction and reference for newcomers and experts alike.
We prove that every finite-dimensional homogeneously simple associative algebra over an algebraically closed field is representable as the product of a full matrix algebra and a graded field.
The purpose of this book is to provide a self-contained account, accessible to the non-specialist, of algebra necessary for the solution of the integrability problem for transitive pseudogroup ...structures.
Originally published in 1981.
ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
In this paper we prove the existence and give precise descriptions of maximal algebras of Martindale like quotients for arbitrary strongly prime linear Jordan algebras. As a consequence, we show that ...Zelmanov's classification of strongly prime Jordan algebras can be viewed exactly as the description of their maximal algebras of Martindale-like quotients. As a side result, we show that the Martindale associative algebra of symmetric quotients can be expressed in terms of the symmetrized product, i.e., in purely Jordan terms.
We prove that the Lie algebra of skew-symmetric elements of the free associative algebra of rank 2 with respect to the standard involution is generated as a module by the elements
a, b
and
a, b
3
..., where
a
and
b
are Jordan polynomials. Using this result we prove that the Lie algebra of Jordan derivations of the free Jordan algebra of rank 2 is generated as a characteristic
F
-module by two derivations. We show that the Jordan commutator
s
-identities follow from the Glennie-Shestakov
s
-identity.
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