Group Theory in Quantum Mechanical Calculations

Quantum cloning inddimensions

“…Let Φ be the representation over End(H N sym ) associated with φ s . Since End(H N sym ) ∼ = H N sym ⊗ H N * sym one has Φ ∼ = φ s ⊗ φ * s , the tensor product or two totally symmetric SU (d)-irreps, therefore in the decomposition of Φ each SU (d)-irrep appears once [12]. As A U i = 0 simply means that the F i 's transform according to the adjoint representation, one must have…”

Lie superalgebras and the multiplet structure of the genetic code. I. Codon representations

“…The first step is to choose a Cartan subalgebra h of g, which is by definition just a Cartan subalgebra of its even part g0: its dimension is called the rank of g. (If g0 has a non-trivial center z 0 and a semisimple part g ss 0 , so that g0 = z0 ⊕ g ss 0 , then h = z0 ⊕ h ss , where h ss is a Cartan subalgebra of g ss 0 .) As in the case of ordinary semisimple Lie algebras, the specific choice of Cartan subalgebra is irrelevant, since they are all conjugate [14,15]. This gives rise to the root system ∆ = ∆ 0 ∪ ∆ 1 of g, where the set ∆ 0 of even roots is just the root system of g0, as an ordinary reductive Lie algebra, and the set ∆ 1 of odd roots is just the weight system of g1, as a g0-module.…”

“…Note, again, that this property is not guaranteed automatically, as it would be for ordinary semisimple Lie algebras, according to Cartan's criterion for semisimplicity. In fact, it turns out that an even, graded symmetric, invariant bilinear form on a simple Lie superalgebra is either non-degenerate or identically zero [14,15] and that, in particular, the Killing form of a simple Lie superalgebra defined by the supertrace operation in the adjoint representation may vanish identically. Moreover, there are simple Lie superalgebras whose Killing form vanishes identically but which are still basic because they admit some other non-degenerate, even, graded symmetric, invariant bilinear form.…”

“…In the following we shall be dealing exclusively with finite-dimensional complex Lie superalgebras which are simple, i.e., admit no non-trivial ideals. Such a Lie superalgebra is called classical if its even part g0 is reductive, that is, if it decomposes into the direct sum of its center and a semisimple subalgebra, or equivalently, if all representations of g0 (in particular that on g0 itself, which is the adjoint representation, and that on g1) are completely reducible [12,13,14,15]. Note that this property is not guaranteed automatically, as it would be for ordinary semisimple Lie algebras, according to Weyl's theorem.…”

“…(However, the term "classical" in this context is unfortunate because it suggests that "classical" for simple Lie superalgebras bears some relation to the standard term "classical", in the sense of "non-exceptional", for simple Lie algebras, which is not the case.) A standard argument then shows [14,15] that a classical Lie superalgebra necessarily belongs to one of the following two types:…”