We study real Smirnov functions and investigate a certain ∗-closed subalgebra of the Smirnov class N+ containing them. Motivated by a result of Aleksandrov, we provide an explicit representation for the space Hp ∩ Hp. This leads to a natural analog of the Riesz projection on a certain quotient space of Lp for p ∈ (0, 1). We also study a Herglotz-like integral transform for singular measures on ...

Let (W, I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [BK] constructed a monoid structure on the set of all subsets of I using unipotent χ-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxe...

We consider an inductive scheme for quantized enveloping algebras, arising from certain inclusions of the associated root data. These inclusions determine an algebra-subalgebra pair with the subalgebra also a quantized enveloping algebra, and we want to understand the structure of the “difference” between the algebra and the subalgebra. Our point of view treats the background field and quantiza...

The resulting algebra is known as the Hall algebra HA of A. Hall algebras first appeared in the work of Steinitz [S] and Hall [H] in the case where A is the category of Abelian p-groups. They reemerged in the work of Ringel [R1]-[R3], who showed in [R1] that when A is the category of quiver representations of an A-D-E quiver ~ Q over a finite field Fq, the Hall algebra of A provides a realizati...

We will drop the compactness hypothesis on G in the results of §6, doing this in such a way that problems can be reduced to the compact case. This involves the notions of reductive Lie groups and algebras and Cartan involutions. Let © be a Lie algebra. A subalgebra S c © is called a reductive subaU gebra if the representation ad%\® of ίΐ on © is fully reducible. © is called reductive if it is a...

Let A denote the mod 2 Steenrod algebra (see Steenrod and Epstein [28]). The problem of computing its cohomology H∗,∗(A) is of great importance in algebraic topology, for this bigraded commutative algebra is the E2 term of the Adams spectral sequence (see Adams [1]) converging to the stable homotopy groups of spheres. But despite intensive investigation for nearly half a century, the structure ...

We propose a perturbation algorithm for Hamiltonian systems on Lie algebra $\mathbb{V}$, so that it can be applied to non-canonical systems. Given system preserves a subalgebra $\mathbb{B}$ of $\mathbb{V}$, when we add the subalgebra $\mathbb{B}$ will no longer preserved. show how to transform perturbed dynamical to preserve up to terms quadratic in perturbation. apply this method to ...

The vertex operator algebras associated with positive definite even lattices afford a large family of known examples of vertex operator algebras. An isometry of the lattice induces an automorphism of the lattice vertex operator algebra. The subalgebra of fixed points of an automorphism is the so-called orbifold vertex operator algebra. In this paper we deal with the case where the lattice L = √...

Let H be the quaternion algebra. Let g be a complex Lie algebra and let U(g) be the enveloping algebra of g. The quaternification g = (H ⊗ U(g), [ , ]gH ) of g is defined by the bracket [ z ⊗ X , w ⊗ Y ] gH = (z · w) ⊗ (XY ) − (w · z) ⊗ (Y X) , for z, w ∈ H and the basis vectors X and Y of U(g). Let SH be the ( non-commutative) algebra of H-valued smooth mappings over S and let Sg = SH ⊗ U(g). ...

In a series of papers [2, 3, 4, 5], B. Efimov has given sufficient conditions for a Boolean algebra to have a free subalgebra of given cardinality. For example, he has shown that if an infinite Boolean algebra A has cardinality greater than exp exp exp v, but the supremum of the cardinalities of subsets of disjoint elements in A is not greater than v, then A has a free subalgebra with (exp v) i...