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## Über dieses Buch

The aim of this book is twofold: (i) to give an exposition of the basic theory of finite-dimensional algebras at a levelthat isappropriate for senior undergraduate and first-year graduate students, and (ii) to provide the mathematical foundation needed to prepare the reader for the advanced study of anyone of several fields of mathematics. The subject under study is by no means new-indeed it is classical­ yet a book that offers a straightforward and concrete treatment of this theory seems justified for several reasons. First, algebras and linear trans­ formations in one guise or another are standard features of various parts of modern mathematics. These include well-entrenched fields such as repre­ sentation theory, as well as newer ones such as quantum groups. Second, a study ofthe elementary theory offinite-dimensional algebras is particularly useful in motivating and casting light upon more sophisticated topics such as module theory and operator algebras. Indeed, the reader who acquires a good understanding of the basic theory of algebras is wellpositioned to ap­ preciate results in operator algebras, representation theory, and ring theory. In return for their efforts, readers are rewarded by the results themselves, several of which are fundamental theorems of striking elegance.

## Inhaltsverzeichnis

### 1. Linear Algebra

Abstract
Linear algebra is the branch of mathematics that is concerned with vector spaces and their linear transformations. This elegant and useful subject is a cornerstone of many disciplines, and the present chapter reviews those aspects of linear algebra that are particularly relevant to the theory of finite-dimensional algebras.
Douglas R. Farenick

### 2. Algebras

Abstract
The study of finite-dimensional algebras is in many ways a natural outgrowth of our thinking about numbers. Early in the 1800s, the representation of complex numbers as points in the real plane was widely known. To many at the time, this representation indicated how a geometric object (namely, the plane) could be made into a number field, and mathematicians began to question whether 3-dimensional space is in some way a number field. This proved to be problematic, however in a landmark paper published in 1844, W.R. Hamilton introduced a 4-dimensional “hypercomplex” number system that had all the arithmetical properties of ℝ or ℂ except for the commutative law of multiplication. Hamilton called these hypercomplex numbers quaternions. In seeking further constructions of this type, mathematicians were soon led conceptually from ordinary geometrical space into a realm where the number systems acquired a much greater level of abstraction. The progression from real and complex numbers to abstract, hypercomplex numbers culminates in the concept of an algebra.
Douglas R. Farenick

### 3. Invariant Subspaces

Abstract
If θ ∈ (0,2π) is fixed, then the linear transformation
$$R_\theta = \left( {\begin{array}{*{20}c} {\cos \theta } & { - \sin \theta } \\ {\sin \theta } & {\cos \theta } \\ \end{array} } \right)$$
acts as a rotation of the plane ℝ2 by θ radians in the counterclockwise direction. For example, R θ rotates the horizontal axis, namely, Span{e1}, to line
$$L_\theta = Span_\mathbb{R} \left\{ {\left( {\begin{array}{*{20}c} {cos \theta } \\ {sin \theta } \\ \end{array} } \right)} \right\}.$$
One thing is clear about this simple linear transformation: because R θ is rotating lines that pass through the origin, the only value of θ ∈ (0,2π) for which R θ maps a line back into itself is θ = π. In this case, the rotation transformation is particularly simple, for its action on each vector v ∈ ℝ2 is just multiplication by the scalar -1: that is, R π v = −v for all v ∈ ℝ2.
Douglas R. Farenick

### 4. Semisimple Algebras

Abstract
The year 1907 marks a turning point in the history of algebra, for it was in this year that J.M.H. Wedderburn published his acclaimed structure and isomorphism theorems for semisimple algebras. These important results form one foundation of the modern theory of algebras.
Douglas R. Farenick

### 5. Operator Algebras

Abstract
The theory of operator algebras was initiated by J. von Neumann in the early 1930s, in large part out of a desire to find a rigorous mathematical foundation for the newly emerging field of quantum physics. In very simple terms, the work of von Neumann was concerned with linear transformations, which we shall here call operators, on complex inner-product spaces (usually infinite-dimensional, and usually complete with respect to a natural metric). A related concept is that of a C*-algebra, introduced in 1943 by I. Gelfand and M.A. Neumark, which refers to algebras that are defined abstractly rather than as algebras of linear transformations. The term operator algebra generally applies to any algebra of operators that acts on an inner-product space and is closed under the canonical involution, or to any (possibly abstract) C*-algebra.
Douglas R. Farenick

### 6. Tensor Products

Abstract
If f and g are two polynomials in one variable with coefficients from $$\mathbb{F}$$, then there is a natural way in which f and g result in a new polynomial h in two variables, say x and y: one simply takes
$$h(x,y) = f(x)g(y).$$
Douglas R. Farenick

### Backmatter

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