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Let G be a connected complex semisimple algebraic group, and T a maximal torus inside a Borel subgroup B , with g, t, and b their Lie algebras. Let V be a representation in the category & for g. The t-decomposition V = E9 JlEt* V Jl of V into a direct sum of finite-dimensional weight spaces is central in the representation theory of g. In this paper, we… (More)

In this paper we study universality for quantum gates acting on qudits. Qudits are states in a Hilbert space of dimension d where d can be any integer ≥ 2. We determine which 2-qudit gates V have the properties (i) the collection of all 1-qudit gates together with V produces all n-qudit gates up to arbitrary precision, or (ii) the collection of all 1-qudit… (More)

- R Brylinski, B Kostant
- Proceedings of the National Academy of Sciences…
- 1994

We explicitly construct, in a uniform fashion, the (unique) minimal and spherical representation pi0 of the split real Lie group of exceptional type E6, E7, or E8. We obtain several algebraic and analytic results about pi0.

- Ranee Brylinski
- 1998

In this paper, we begin a quantization program for nilpotent orbits OR of a real semisimple Lie group GR. These orbits arise naturally as the coadjoint orbits of GR which are stable under scaling, and thus they have a canonical symplectic structure ω where the GR-action is Hamiltonian. These orbits and their covers generalize the oscillator phase space T R,… (More)

- Alexander Astashkevich, Ranee Brylinski
- 1998

LetO be the minimal nilpotent adjoint orbit in a classical complex semisimple Lie algebra g. O is a smooth quasi-affine variety stable under the Euler dilation action C on g. The algebra of differential operators on O is D(O) = D(Cl(O)) where the closure Cl(O) is a singular cone in g. See [J] and [B-K1] for some results on the geometry and quantization of… (More)

- RANEE BRYLINSKI
- 2000

Lecomte and Ovsienko constructed SLn+1(R)-equivariant quantization maps Qλ for symbols of differential operators on λ-densities on RP . We derive some formulas for the associated graded equivariant star products ⋆λ on the symbol algebra Pol(T ∗RP). These give some measure of the failure of locality. Our main result expresses (for n odd) the coefficients… (More)

- RANEE BRYLINSKI, BERTRAM KOSTANT
- 1992

Let M be a G-covering of a nilpotent orbit in 0 where G isa complex semisimple Lie group and g = Lie(G). We prove that under Poisson bracket the space R[2] of homogeneous functions on M of degree 2 is the unique maximal semisimple Lie subalgebra of R = R{M) containing g . The action of g' ~ R[2] exponentiates to an action of the corresponding Lie group G'… (More)

- ALEXANDER ASTASHKEVICH, RANEE BRYLINSKI
- 2000

We construct a unique G-equivariant graded star product on the algebra S(g)/I of polynomial functions on the minimal nilpotent coadjoint orbit Omin of G where G is a complex simple Lie group and g 6= sl(2,C). This strengthens the result of Arnal, Benamor and Cahen. Our main result is to compute, for G classical, the star product of a momentum function μx… (More)

- R Brylinski, B Kostant
- Proceedings of the National Academy of Sciences…
- 1994

In the framework of geometric quantization we explicitly construct, in a uniform fashion, a unitary minimal representation pio of every simply-connected real Lie group Go such that the maximal compact subgroup of Go has finite center and Go admits some minimal representation. We obtain algebraic and analytic results about pio. We give several results on the… (More)

We study the polynomial functions on tensor states in (Cn)⊗k which are invariant under SU(n). We describe the space of invariant polynomials in terms of symmetric group representations. For k even, the smallest degree for invariant polynomials is n and in degree n we find a natural generalization of the determinant. For n, d fixed, we describe the… (More)