For I a proper, countably complete ideal on P(X) for some set X , can the quotient Boolean algebra P(X)/I be complete? This question was raised by Sikorski [Si] in 1949. By a simple projection argument as for measurable cardinals, it can be assumed that X is an uncountable cardinal κ, and that I is a κ-complete ideal on P(κ) containing all singletons. In this paper we provide consequences from ...

In this paper we present an iterative method, inspired by the inverse iteration with shift technique of finite linear algebra, designed to find the eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet boundary condition for arbitrary bounded domains X R . This method, which has a direct functional analysis approach, does not approximate the eigenvalues of the Laplacian as ...

Let g be a complex semisimple Lie algebra, and f : g → G\\g the adjoint quotient map. Springer theory of Weyl group representations can be seen as the study of the singularities of f . In this paper, we give a generalization of Springer theory to visible, polar representations. It is a class of rational representations of reductive groups over C, for which the invariant theory works by analogy ...

If a Lie group G acts on a symplectic manifold (M, ω) and preserves the symplectic form ω, then in some cases there may exist a moment map ([C]) Φ from M to the dual of the Lie algebra. When the action is (locally) free, the preimage of a point in the dual of the Lie algebra modulo the isotropy group of this point will still be a symplectic manifold (orbifold). This process is called the Marsde...

Study of the normalizer of the MAD-group corresponding to a fine grading offers the most important tool for describing symmetries in the system of non-linear equations connected with contraction of a Lie algebra. One fine grading that is always present in any Lie algebra sl(n, C) is the Pauli grading. The MAD-group corresponding to it is generated by generalized Pauli matrices. For such MAD-gro...

This paper includes two main ideas. The first one, rewriting in an algebra, was introduced in [5]. The second one, boolean rewriting, can be found in many papers but we were never able to find a clear comparison with the classic one. We prefer rewriting in an algebra to term rewriting. This is our way to give a unique theory of rewriting. If the algebra is free, then we get the term rewriting. ...

Let A denote the ring of differential operators on the affine line with its two usual generators t and d dt given degrees +1 and −1 respectively. Let X be the stack having coarse moduli space the affine line Spec k[z] and isotropy groups Z/2 at each integer point. Then the category of graded A-modules is equivalent to the category of quasicoherent sheaves on X .

This paper describes an intuitive geometric model for coupled twostate quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric algebra of a Euclidean vector space. This algebra is then lifted to Minkowski space-time and its associated geometric algebra, and the insights this ...

Let K be an infinite field and K〈X〉 = K〈X1, ..., Xn〉 the free associative algebra generated by X = {X1, ..., Xn} over K. It is proved that if I is a two-sided ideal of K〈X〉 such that the K-algebra A = K〈X〉/I is almost commutative in the sense of [3], namely, with respect to its standard N-filtration FA, the associated N-graded algebra G(A) is commutative, then I is generated by a finite Gröbner...

Fix a prime p, and let A be the polynomial part of the dual Steenrod algebra. The Frobenius map on A induces the Steenrod operation P̃0 on cohomology, and in this paper, we investigate this operation. We point out that if p = 2, then for any element in the cohomology of A, if one applies P̃0 enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and t...