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AuthorCurtis, Morton L. author
TitleMatrix Groups [electronic resource] / by Morton L. Curtis
ImprintNew York, NY : Springer US, 1979
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Descript XII, 191 p. online resource


These notes were developed from a course taught at Rice Univ- sity in the spring of 1976 and again at the University of Hawaii in the spring of 1977. It is assumed that the students know some linear algebra and a little about differentiation of vector-valued functions. The idea is to introduce students to some of the concepts of Lie group theory--all done at the concrete level of matrix groups. As much as we could, we motivated developments as a means of deciding when two matrix groups (with different definitions) are isomorphie. In Chapter I "group" is defined and examples are given; ho- morphism and isomorphism are defined. For a field k denotes the algebra of n x n matrices over k We recall that A E Mn(k) has an inverse if and only if det A # 0 , and define the general linear group GL(n,k) We construct the skew-field E of quaternions and note that for A E Mn(E) to operate linearlyon Rn we must operate on the right (since we multiply a vector by a scalar n n on the left). So we use row vectors for Rn, c E and write xA , for the row vector obtained by matrix multiplication. We get a complex-valued determinant function on Mn (E) such that det A # 0 guarantees that A has an inverse


1 General Linear Groups -- A. Groups -- B. Fields, Quaternions -- C. Vectors and Matrices -- D. General Linear Groups -- E. Exercises -- 2 Orthogonal Groups -- A. Inner Products -- B. Orthogonal Groups -- C. The Isomorphism Question -- D. Reflections in Rn -- E. Exercises -- 3 Homomorphisms -- A. Curves in a Vector Space -- B. Smooth Homomorphisms -- C. Exercises -- 4 Exponential and Logarithm -- A. Exponential of a Matrix -- B. Logarithm -- C. One-parameter Subgroups -- D. Lie Algebras -- E. Exercises -- 5 SO(3) and Sp(1) -- A. The Homomorphism ? : S3 ? SO(3) -- B. Centers -- C. Quotient Groups -- D. Exercises -- 6 Topology -- A. Introduction -- B. Continuity of Functions, Open Sets, Closed Sets -- C. Connected Sets, Compact Sets -- D. Subspace Topology, Countable Bases -- E. Manifolds -- F. Exercises -- 7 Maximal Tori -- A. Cartesian Products of Groups -- B. Maximal Tori in Groups -- C. Centers Again -- D. Exercises -- 8 Covering by Maximal Tori -- A. General Remarks -- B. (โ{128} ) for U(n) and SU(n) -- C. (โ{128} ) for SO(n) -- D. (โ{128} ) for Sp(n) -- E. Reflections in Rn (again) -- F. Exercises -- 9 Conjugacy of Maximal Tori -- A. Monogenic Groups -- B. Conjugacy of Maximal Tori -- C. The Isomorphism Question Again -- D. Simple Groups, Simply-Connected Groups -- E. Exercises -- 10 Spin(k) -- A. Clifford Algebras -- B. Pin(k) and Spin(k) -- C. The Isomorphisms -- D. Exercises -- 11 Normalizers, Weyl Groups -- A. Normalizers -- B. Weyl Groups -- C. Spin(2n+1) and Sp(n) -- D. SO(n) Splits -- E. Exercises -- 12 Lie Groups -- A. Differentiable Manifolds -- B. Tangent Vectors, Vector Fields -- C. Lie Groups -- D. Connected Groups -- E. Abelian Groups

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