Unitary isomorphisms

Isomorphisms of unitary representations ought to preserve the unitary structure. When they do, they are called unitary isomorphisms of representations. 

The set of all such transformations constitutes the group U(2) which is isomorphic to the group of all unitary matrices of order 2. It is a 4 parameter, continuous, connected, compact, Lie group. The subgroup of U(2) which contains all the unitary matrices of order two with determinant +1, is the set of matrices whose general element is... 

There is a Lie group isomorphism between unit quaternions and the special unitary group 50 (2). Define a function T from the unit quaternions to 50 (2) by... 

In this section we define representations and give examples. We also define homomorphisms and isomorphisms of representations, as well as unitary representations and isomorphisms. 

Exercise 4.3 Show that the set ofly. diagonal special unitary matrices is a group and that it is isomorphic to the group T x T. (See Exercise 4.1 for the definition of the Cartesian product of groups.)... 

Exercise 5.7 Recall the representations R of SU (2) on homogeneous polynomials introduced in Section 4.6. Find a complex scalar product on the vector space of the representation 7 i 7 2 such that the representation is unitary. Consider the subspace Vi spanned by uxy — vx, uy — rxy and the subspace Vj spanned by [ux", 2uxy + vx, 2vxy + uy, vy". Use this complex scalar product to find Is your answer isomorphic to V- Is it equal to V3 ... 

Next, fix a natural number n and suppose that the result is known for all natural numbers k < n. Because every li nite-dimensional representation contains at least one irreducible representation, we can choose one and call it Wo-Set Co = dim Home (Wo, V). Then by Proposition 6.10 we know that Wq° is isomorphic to a subrepresentation U of V. Since the representation V is unitary, we can consider the complementary unitary representation [/- -, whose dimension is strictly less than n. 

Note that by Proposition A.I we know that R is unitary, while p is unitary by hypothesis. Hence we can apply Proposition 6.8 to find that the representation (SU(2), V, p) is isomorphic to the representation (SU(1), R ). 

Proposition 6.16 Every finite-dimensional, unitary, irreducible representation of SO(3) is isomorphic to Qn for some even n. In addition, Qn is isomorphic to Qn if and only ifn = n. ... 

Now W is a finite-dimensional, unitary, irreducible representation, so by Proposition 6.16 there must be a nonnegative even integer h and an isomorphism T P W of representations. Because T is an isomorphism, the fist of weights for P must be the same as the fist of weights for W. Hence... 

From Proposition 5.1 we know that y is an invariant subspace. Since the natural representation of S<9(3) on L (W ) is unitary. Proposition 5.4 implies that is a homomorphism of representations. Since V and y are irreducible, it follows from Schur s Lemma and the nontriviality of n [V] that fit gives an isomorphism of representations from V to y. ... 

Proof, (of Proposition 10.6) First we suppose that (S(/(2), V, p) is a linear irreducible unitary Lie group representation. By Proposition 6.14 we know that p is isomorphic to the representation R for some n. By Proposition 10.5 we know that R can be pushed forward to an irreducible projective representation of SO(3). Hence p can be pushed forward to an irreducible projective Lie group representation of SO(3). 

If each of the blocks in the matrices comprising the matrix system A cannot be reduced ftirther, the matrix system has been reduced completely and each of the matrix systems A1, A2, . .. in the direct sum is said to be irreducible. Matrix systems that are isomorphous to a group G are called matrix representations (Chapter 4). Irreducible representations (IRs) are of great importance in applications of group theory in physics and chemistry. A matrix representation in which the matrices are unitary matrices is called a unitary representation. Matrix representations are not necessarily unitary, but any representation of a finite group that consists of non-singular matrices is equivalent to a unitary representation, as will be demonstrated in Section A1.5. 

Four decades ago, Bell [3] introduced a particle-hole conjugation operator CB into nuclear shell theory. Its operator algebra is essentially isomorphic to that of Cq (for example, CB is unitary), the filled Dirac sea now corresponding to systems with half-filled shells. This was later extended to other areas of physics. For example,... 

These operators are used to define the isomorphism p(g) which characterizes the unitary irreducible and infinite-dimensional representation of H", through the exponential map... 

Clebsh-Gordan coefficient plays the role of matrix elements of a unitary transformation between the coupled and uncoupled schemes. In view of the isomorphism SUi(2)<8)SU2(2) —S0i(3)<8>S02(3), it is possible to recognize, starting from the product (3.1), two different coupling schemes. 

The operators R rotate functions without their deformation therefore, fliey preserve the scalar products in the Hilbert space and are unitary. They form a group isomorphic with the group of operators / . because they have the same multiplication table as the operators R if R = 1 2- Then H = R-i R.2, where Rif r) = /( j V) and R.2f(r) = fiR r). Indeed, Rf = (RiR2)f r) = f(R- R- r) = f(R h). 

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