By John Stillwell

During this new textbook, acclaimed writer John Stillwell provides a lucid creation to Lie idea appropriate for junior and senior point undergraduates. for you to accomplish that, he specializes in the so-called "classical groups'' that catch the symmetries of actual, complicated, and quaternion areas. those symmetry teams could be represented by way of matrices, which permits them to be studied via trouble-free equipment from calculus and linear algebra. This naive method of Lie conception is initially as a result of von Neumann, and it's now attainable to streamline it through the use of typical result of undergraduate arithmetic. To catch up on the constraints of the naive technique, finish of bankruptcy discussions introduce vital effects past these proved within the ebook, as a part of a casual caricature of Lie conception and its heritage. John Stillwell is Professor of arithmetic on the collage of San Francisco. he's the writer of numerous very hot books released by means of Springer, together with The 4 Pillars of Geometry (2005), components of quantity idea (2003), arithmetic and Its heritage (Second variation, 2002), Numbers and Geometry (1998) and components of Algebra (1994).

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**Example text**

42 2 Groups Exercises If we let x1 , x2 , x3 , x4 be the coordinates along mutually orthogonal axes in R4 , then it is possible to “rotate” the x1 and x2 axes while keeping the x3 and x4 axes fixed. 1 Write a 4 × 4 matrix for the transformation that rotates the (x1 , x2 )-plane through angle θ while keeping the x3 - and x4 -axes fixed. 2 Write a 4 × 4 matrix for the transformation that rotates the (x3 , x4 )-plane through angle φ while keeping the x1 - and x2 -axes fixed. 2, and deduce that SO(4) contains a subgroup isomorphic to T2 .

If we drop the requirement that orientation be preserved, then we get a larger group of transformations of Rn called the orthogonal group O(n). 52 3 Generalized rotation groups An example of a transformation that is in O(n), but not in SO(n), is reflection in the hyperplane orthogonal to the x1 -axis, (x1 , x2 , x3 , . . , xn ) → (−x1 , x2 , x3 , . . , xn ), which has the matrix ⎛ ⎞ −1 0 . . 0 ⎜ 0 1 . . 0⎟ ⎜ ⎟ ⎜ .. ⎟, ⎝ . ⎠ 0 0 ... 1 obviously of determinant −1. We notice that the determinant of a matrix A ∈ O(n) is ±1 because (as mentioned in the previous section) AAT = 1 ⇒ 1 = det(AAT ) = det(A) det(AT ) = det(A)2 .

The field properties of multiplication, on the other hand, depend on the curious definition of product of pairs, which has no obvious generalization to a product of n-tuples for n > 2. This raises the question; is it possible to define a “product” operation on n R that, together with the vector sum operation, makes Rn a field? Hamilton hoped to find such a product for each n. 6 Discussion 21 where the absolute value of u = (x1 , x2 , . . , xn ) is |u| = x21 + x22 + · · · + x2n . As we have seen, for n = 2 this property is equivalent to the Diophantus identity for sums of two squares, so a multiplicative absolute value in general implies an identity for sums of n squares.