DAHA approach to iterated torus links

We extend the construction of the DAHA-Jones polynomials for any reduced root systems and DAHA-superpolynomials in type A from the iterated torus knots (our previous paper) to links, including arbitrary algebraic links. Such a passage essentially corresponds to the usage of the products of Macdonald polynomials and is directly connected to the so-called splice diagrams. The specialization t=q of our superpolynomials conjecturally results in the HOMFLY-PT polynomials. The relation of our construction to the stable Khovanov-Rozansky polynomials and the so-called ORS-polynomials of the corresponding plane curve singularities is expected for algebraic links in the uncolored case. These 2 connections are less certain, since the Khovanov-Rozansky theory for links is not sufficiently developed and the ORS polynomials are quite involved. However we provide some confirmations. For Hopf links, our construction produces the DAHA-vertex, similar to the refined topological vertex, which is an important part of our paper.Comment: v3: Further editing, essentially a somewhat extended version of our article in Contemporary Mathematic

Integrative concept of homeostasis: translating physiology into medicine

To truly understand living systems they must be viewed as a whole. In order to achieve this and to come to some law to which living systems obey, data obtained on cells, tissues and organs should be integrated. Because there are no such laws yet, there is usually a long path for physiological findings obtained by reductionist approaches to be translated into medical practice. The concept and accompanying equations of homeostasis presented here are aimed to develop biological laws and to bridge this gap between physiology and medicine. The concept of homeostasis takes into account energy input and output, enlisting all relevant contributors. In homeostasis, input should equal the output. What I suggest here is that if the system is out of homeostasis, the homeostasis may be regained by changing any of the input or output components in an adequate manner, not only the one that has changed first. The proposed equation should enable for new lab findings regarding any pathophysiological conditions to find a more direct use in medicine. It should also ease ‘decision making’ in medicine and make therapy development and treatment outcome more straightforward and predictable. Finally, to recognize the basic laws of living systems enables for evolutionary adaptations and processes to be understood better

Dimensions of irreducible modules over W-algebras and Goldie ranks

The main goal of this paper is to compute two related numerical invariants of a primitive ideal in the universal enveloping algebra of a semisimple Lie algebra. The first one, very classical, is the Goldie rank of an ideal. The second one is the dimension of an irreducible module corresponding to this ideal over an appropriate finite W-algebra. We concentrate on the integral central character case. We prove, modulo a conjecture, that in this case the two are equal. Also, modulo the same conjecture, we compute certain scale factors introduced by Joseph. Our conjecture asserts that there is a one-dimensional module over the W-algebra with certain additional properties. The conjecture is proved for the classical types. This completes a program of computing Goldie ranks proposed by Joseph in the 80's. We also provide an essentially Kazhdan-Lusztig type formula for computing the characters of the irreducibles in the Brundan-Goodwin-Kleshchev category O for a W-algebra again under the assumption that the central character is integral. The formula is based on a certain functor (a generalized Soegel functor) from an appropriate parabolic category O to the W-algebra category O. We prove a number of properties of this functor including the quotient property and the double centralizer property. We develop several topics related to our generalized Soergel functor. For example, we discuss its analog for the category of Harish-Chandra bimodules. We also discuss generalizations to the case of categories O over Dixmier algebras. The most interesting example of this situation comes from the theory of quantum groups: we prove that an algebra that is basically Luszitg's form of a quantum group at a root of unity is a Dixmier algebra. For this we check that the quantum Frobenius epimorphism splits.Comment: 51 pages, preliminary version, comments welcome; v2, 52 pages, minor changes; v3, 58 pages, improved exposition; v4 final versio