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Structure of Classical Groups 69 2.1 Semisimple Elements 69 2.1.1 Toral Groups 70 2.1.2 Maximal Torus in a Classical Group 72 2.1.3 Exercises 76 2.2 Unipotent Elements 77 2.2.1 Low-Rank Examples 77 2.2.2 Unipotent Generation of Classical Groups 2.2.3 Connected Groups 81 2.2.4 Exercises 2.3 Regular Representations of SL(2, C) 2.3.1 Irreducible Representations of .5((2, C) 2.3.2 Irreducible Regular Representations of SL(2, C) 2.3.3 Complete Reducibility of SL(2, (C) 2.3.4 Exercises 2.4 The Adjoint Representation 2.4.1 Roots with Respect to a Maximal Torus 2.4.2 Commutation Relations of Root Spaces 95 2.4.3 Structure of Classical Root Systems 99 2.4.4 Irreducibility of the Adjoint Representation 2.4.5 Exercises 2.5 Semisimple Lie Algebras 2.5.1 Solvable Lie Algebras 2.5.2 Root Space Decomposition 2.5.3 Geometry of Root Systems 2.5.4 Conjugacy of Cartan Subalgebras 2.5.5 Exercises x Contents

We define a polynomial measure of multiparticle entanglement which is scalable, i.e., which applies to any number of spin-1 2 particles. By evaluating it for three particle states, for eigenstates of the one dimensional Heisenberg antiferromagnet and on quantum error correcting code subspaces, we illustrate the extent to which it quantifies global entanglement. We also apply it to track the evolution of entanglement during a quantum computation.

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Let ∆ n−1 denote the (n − 1)-dimensional simplex. Let Y be a random k-dimensional subcomplex of ∆ n−1 obtained by starting with the full (k − 1)-dimensional skeleton of ∆ n−1 and then adding each k-simplex independently with probability p. Let H k−1 (Y ; R) denote the (k − 1)-dimensional reduced homology group of Y with coefficients in a finite abelian group R. It is shown that for any fixed R and k ≥ 1 and for any function ω(n) that tends to infinity

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The purpose of this department is to provide early announcement of significant new results, with some indications of proof. Although ordinarily a research announcement should be a brief summary of a paper to be published in full elsewhere, papers giving complete proofs of results of exceptional interest are also solicited. Manuscripts more than eight typewritten double spaced pages long will not be considei ed as acceptable. All papers to be communicated by a Council member should be sent directly to M. H, Protter,

Let M (n) be the algebra (both Lie and associative) of n × n matrices over C. Then M (n) inherits a Poisson structure from its dual using the bilinear form (x, y) = −tr xy. The Gl(n) adjoint orbits are the symplectic leaves and the algebra, P (n), of polynomial functions on M (n) is a Poisson algebra. In particular if f ∈ P (n) then there is a corresponding vector field ξ f on M (n). If m ≤ n then M (m) embeds as a Lie subalgebra of M (n) (upper left hand block) and P (m) embeds as a Poisson subalgebra of P (n). Then, as an analogue of the Gelfand-Zeitlin algebra in the enveloping algebra of M (n), let J(n) be the subalgebra of P (n) generated by P (m) Gl(m) for m = 1, . . . , n. One observes that J(n) ∼ = P (1) Gl(1) ⊗ · · · ⊗ P (n) Gl(n)