Cartesian sum

One way to combine vector spaces is to take a Cartesian sum. (Mathematicians sometimes call this a Cartesian product. Another common term is direct sum.)... 

Definition 2.11 Suppose Vi,. ..,Vn are vector spaces over the same scalar field. The Cartesian sum of these vector spaces, denoted Vi V or... 

Thus, for example, C" is equal (as a complex vector space) to the Cartesian sum of n copies of C ... 

We will use Cartesian sums and tensor products to build and decompose representations in Chapters 5 and 7. Tensor products are useful in combining different aspects of one particle. For instance, when we consider both the mobile and the spin properties of an electron (in Section 11.4) the state space is the tensor product of the mobile state space defined in Chapter 3)... 

Exercise 3.17 Show that any Cartesian sum V V of complex scalar product spaces has a complex scalar product defined by... 

Physically, expressing a vector space of wave functions as a Cartesian sum... 

The following consequence of Schur s lemma will be useful in the proof that every polynomial restricted to the two-sphere is equal to a harmonic polynomial restricted to the two-sphere (Proposition 7.3). The idea is that once we decompose a representation into a Cartesian sum of irreducibles, every irreducible subrepresentation appears as a term in the sum. 

We start with a convenient definition. Just as prime powers play a particular role in number theory, Cartesian sums of copies of one irreducible representation play a particular role in representation theory. 

This Cartesian sum representation is called the isotypic decomposition of V. The list of representations Wj and their multiplicities Cj is called the isotype of V. 

We know that is an irreducible invariant subspace of P by Proposition 7.2. By Proposition 6.5 and Proposition 7.1 we know that is not isomorphic to any subrepresentation of the Cartesian sum... 

By induction we know that is isomorphic to this Cartesian sum also, we know that is injective. Hence is not isomorphic to any subrepresentation of Hence by Proposition 6.7, the subspace must be... 

In this section we introduce Lie algebras, Lie algebra homomorphisms and Lie algebra Cartesian sums. In the examples we introduce all the Lie algebras we will need in our study of the hydrogen atom. 

Exercise 11.7 For each nonnegative integer , decompose the representation of S U (F) on 0 mro a Cartesian sum of its irreducible components. Conclude that this representation is reducible. Is there a meaningful physical consequence or interpretation of this reducibility ... 

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