Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over ${\mathbb{Q}}$ .     

The center of the ring of 3×3 generic matrices

Following Procesi, the center of the division ring of generic matrices over a field F is described as the fixed field of the symmetric group acting on a purely transcendental extension of F. For 3×3 matrices, the center is shown to be purely transcendental over F. In characteristic zero this is equivalent to saying that the field of simultaneous rational invariants of 3×3 matrices over F is a purely transcendental extension field of F.     

Links between cofinite prime ideals in quantum function algebras

We describe the skew primitive elements in a multiparameter enveloping algebraU=U _q,p ⁻¹ (g) and the links between cofinite maximal ideals in the corresponding quantum function algebra ℂ _q [G]. These results are applied to determine the coradical filtration forU, and to obtain a moduli space for multiparameter Drinfeld doubles. Research partially supported by NSA grant MDA 904-93-H3016.     

Weylness of 2 × 2 operator matrices

Let and be complex separable infinite‐dimensional Hilbert spaces. Given the operators and , we define where is an unknown element. In this paper, a necessary and sufficient condition is given for to be a right Weyl (left Weyl, or Weyl) operator for some . Moreover, some relevant properties and illustrating examples are also given.     

Invertibility of 2 × 2 operator matrices

Properties of right invertible row operators, i.e., of 1 × 2 surjective operator matrices are studied. This investigation is based on a specific space decomposition. Using this decomposition, we characterize the invertibility of a 2 × 2 operator matrix. As an application, the invertibility of Hamiltonian operator matrices is investigated.     

Geometry of 2×2 Hermitian matrices II

Let Z be a field of characteristic ≠2, D be a quaternion division algebra over Z and have a nonstandard involution of the first kind. The fundamental theorem of geometry of 2× 2 Hermitian matrices over D are proved. Thus, if D is a quaternion division algebra over Z with an involution of the first kind, then the fundamental theorem of geometry of 2× 2 Hermitian matrices over D are obtained.     

Symplectic 2 × 2 matrices over free algebras

P.M. Cohn has proved the remarkable theorem, that every invertible n × n matrix over a free algebra is the product of elementary n × n matrices, see [C1], [C2]. In this note we prove the analogue for symplectic 2 × 2 matrices over free algebras relative to a homogeneous involution: every symplectic 2 × 2 matrix is the product of elementary symplectic 2 × 2 matrices.In Section 1 we define the group Sp₂(R) of symplectic 2 × 2 matrices over an involutive ring R. The group ESp₂(R) generated by elementary symplectic matrices is introduced in Section 3.In Section 2 we prove a reducibility criterion for homogeneous polynomials in a free algebra KX over a commutative field K. It leads to a special form in the factorization of symmetric homogeneous polynomials, see Corollary to Proposition 2.2.We prove in Section 4 that ESp₂(KX) = Sp₂(KX), if the involution on KX is homogeneous.In a subsequent article we will show that the main result is also true for 2g × 2g symplectic matrices over free algebras relative to homogeneous involutions, g ≥ 1. It seems that a proof of this result will be much more complicated than the case g = 1.     

Lorentzian geometry for 2 × 2 real matrices

We give an interpretation and a natural proof of the lemma of Herglotz on 2 × 2 symmetric matrices from the point of view of 3-dimensional Lorentz geometry and 2-dimensional hyperbolic geometry. This interpretation leads naturally to an analogous result on 2 × 2 matrices with trace 0.     

Unsolvability in 3 × 3 Matrices

A set of 3 × 3 matrices over the integers will be said to be mortal if the zero matrix can be expressed as a finite product of members of the set. It is shown in this paper that the problem of deciding whether a given finite set is mortal is recursively unsolvable.     

Poincaré series of semi-invariants of 2×2 matrices

The Poincaré series of the algebra of -invariants of m-tuples of 2×2 matrices is presented both as a rational function and as a series of Schur functions. We show that this algebra of invariants is generated by the determinants, the mixed discriminants and the discriminants of 2×2 matrices. Consequences on invariants of three-dimensional matrices of the shape 2×2×m are discussed. For arbitrary n2, we prove an explicit functional equation for the Poincaré series of the -invariants of m-tuples of n×n matrices.     

Regularity of prime ideals

Mathematische Zeitschrift - We answer several natural questions which arise from a recent paper of McCullough and Peeva providing counterexamples to the Eisenbud–Goto Regularity Conjecture....     

Estimates for prime ideals

A modification of an old argument due to ?ebi?ev is used to obtain uniform estimates for prime ideals in algebraic number fields.     

On prime,weakly prime ideals in ordered semigroups

Communicated by J. S. Ponizovsk?     

On rings whose prime ideals are completely prime

We investigate in this paper rings whose proper ideals are 2-primal. We concentrate on the connections between this condition and related concepts to this condition, and several kinds of π-regularities of rings which satisfy this condition. Moreover, we add counterexamples to the situations that occur naturally in the process of this note.     

Commutativity for systems of (2 × 2) selfadjoint matrices

A purely analytic criterion is presented which characterises the commutativity of a finite-collection of (2x2) selfadjoint matrices. Indeed, it is always possible to associate with such matrices their so called Weyl Calculus which is a matirx-valued distribution of finite order. It is shown that the matrices common if and only if their associated Weyl Calculus is a distribution of order zero.     

Prime elements from prime ideals

The prime ideal theorem for distributive lattices (PIT) is shown to imply that any complete distributive lattice with a compact unit has a prime element, which is then used to deduce from PIT that (1) every nontrivial ring with unit has a prime ideal, and (2) every Wallman locale is spatial.