Algebra isomorphic

The notions of Lie algebra homomorphism and Lie algebra isomorphism will be important to us. 

WhiD74 White, D. E. Linear and multilinear aspects of isomorph rejection. Linear and multilinear algebra (1974) 211-226. 

This is identical to Eq. (6.45) with A = D and X - 1 = N/2. One also notes that the spectrum of the Poschl-Teller potential in one dimension is identical to that of the Morse potential in one dimension. These two potentials are therefore called isospectral. This identity arises from the fact that, as mentioned in Chapter 3, the two algebras 0(2) and U(l) are isomorphic. The situation is different in three dimensions, where this is no longer the case. 

As mentioned already in Chapter 2, the algebras U(l) and 0(2) are isomorphic (and Abelian). A consequence of this statement is that in one-dimension there is a large number of potentials that correspond exactly to an algebraic structure with a dynamical symmetry. Of particular interest in molecular physics are ... 

Some algebras have identical commutation relations. They are therefore called isomorphic algebras. A list of isomorphic algebras (of low order) is shown in Table A.3. In this table, the sign denotes direct sums of the algebra, that is, addition of the corresponding operators. There is also the trivial case U(l) SO(2). 

The algebraic theory of molecules, discussed in Chapters 2,4, and 5, makes use of the algebra of SO(4). In order to do calculations, one therefore needs the Racah calculus of SO(4) (Lemus, 1988). Fortunately, in view of the isomorphism of Table A.3,... 

Since is commutative, the Fock representation is isomorphic to the symmetric algebra From now, we drop 1 in the above element and simply denote it by... 

A (locally) closed substack of an algebraic stack S is a full sub category T C S which is an algebraic stack, which contains together with any t 6 Ob([Pg.6]

The first two chapters are about moduli of abelian varieties, i.e. about classifying abelian varieties. An abelian variety is a proper variety endowed with a group structure. It turns out that any abelian variety A is projective there exists some ample line bundle on A. An ample line bundle determines an isogeny A A — A, which depends on C only up to algebraic equivalence. Such a morphism A is called a polarization, it is called a principal polarization if A is an isomorphism. 

Let (A, A) be a principally polarized abelian variety over an algebraically closed field k. If the characteristic of k is not equal to the prime p, then the kernel of multiplication by p on A(k) is a finite group isomorphic to (Z/pZ)29. The polarization A induces a nondegenerate alternating pairing A k)[p] x A( )[p] -+ pp(k). Hence, we can try to classify principally polarized abelian varieties with a symplectic basis for the group of points of order p. However, this no longer works in characteristic p. 

Proof. By the definition of isomorphic, there must be an invertible, surjective linear transformation T from V to W. If W is finite-dimensional, then we can apply the Fundamental Theorem of Linear Algebra (Proposition 2.5) to find... 

The reader should check that Definition 8.7 is satisfied and notice that those factors of 1 /2 are necessary. Thus Ti is a Lie algebra homomorphism. To see that it is an isomorphism, note that the matrices on the three right-hand sides of the defining equations for Ti form a basis of su(2). Similarly, defining 72 0Q 5o(3) by... 

When two or more Lie algebras are isomorphic, it is common practice to call them equal or the same. For example, we will refer to 0q, 5o(3) and 5m(2) as the same algebra and use the shorthand i for 72(1) or ri(i), etc., when the context precludes confusion. This Lie algebra shows up in yet another guise in many physics texts, where one encounters triples of operators, say,, Ja, , Ji, J , satisfying commutation relations... 

To confirm that this is an isomorphism of Lie algebras, note first that 5 is a well-defined linear transformation (by Proposition 2,3). Then check that it is a homomorphism of Lie algebras by checking all bracket relations between the matrices above. We leave this verification mostly to the reader, giving just one example ... 

Then we say that T is a homomorphism of (Lie algebra) representations. If in addition T is injective and surjective then we say that T is an isomorphism of (Lie algebra) representations and that p is isomorphic fo p. 

Exercise 8.3 Show that the Heisenberg Lie algebra H is isomorphic to... 

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