Spectral extension of the quantum group cotangent bundle

The structure of a cotangent bundle is investigated for quantum linear groups GLq(n) and SLq(n). Using a q-version of the Cayley-Hamilton theorem we construct an extension of the algebra of differential operators on SLq(n) (otherwise called the Heisenberg double) by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators -- the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive SLq(n) type dynamical R-matrices in a surprisingly simple way. Second, we calculate an evolution operator for the model of q-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. Relation between the two operators is given by a modular functional equation for Riemann theta function.Comment: 38 pages, no figure

Manin matrices and Talalaev's formula

We study special class of matrices with noncommutative entries and demonstrate their various applications in integrable systems theory. They appeared in Yu. Manin's works in 87-92 as linear homomorphisms between polynomial rings; more explicitly they read: 1) elements in the same column commute; 2) commutators of the cross terms are equal: $[M_{ij}, M_{kl}]=[M_{kj}, M_{il}]$ (e.g. $[M_{11}, M_{22}]=[M_{21}, M_{12}]$). We claim that such matrices behave almost as well as matrices with commutative elements. Namely theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, the Newton identities and so on and so forth) holds true for them. On the other hand, we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability. For instance, Manin matrices (and their q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and the so--called Cartier-Foata matrices. Also, they enter Talalaev's hep-th/0404153 remarkable formulas: $det(\partial_z-L_{Gaudin}(z))$, det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show that theorems of linear algebra, after being established for such matrices, have various applications to quantum integrable systems and Lie algebras, e.g in the construction of new generators in $Z(U(\hat{gl_n}))$ (and, in general, in the construction of quantum conservation laws), in the Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also discuss applications to the separation of variables problem, new Capelli identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints e.g. in Newton id-s fixed, normal ordering convention turned to standard one, refs. adde

Prophylactic thyroidectomy results among RET germline mutation bearers in families with hereditary forms of medullary thyroid cancer

Genetically caused medullary thyroid cancer (MTC) is associated with unfavorable survival prognosis, so it makes necessary to develop new diagnostic techniques to reveal pre-clinical stage of disease as well as to introduce into clinical practice the effective method of tumor prevention. The article represents first in Russia summary clinical experience of prophylactic thyroidectomy have been executed in the period 1998 – 2015 yeas among ten bearers of RET gene germlinemutation in families with hereditary disease including syndrome MEN2A and familial MTC. Aim: to evaluate the results of surgical treatment of asymptomatic carriers of germinal mutations in the RET gene. Materials and methods. In the period from 1998 to 2015, in two centers: N.N. Blokhin Russian Cancer Research Center, Moscow and A. Tsyb Medical Radiological Research Centre – branch of the National Medical Research Radiological Centre, Obninsk was conducted prophylactic surgical treatment in 10 patients – asymptomatic carriers of germinal mutations in RET. Age of patients – from 2 to 23 years old. 9 patients – from families with the syndrome of multiple endocrineneoplasia type 2A (MEN2A), one – with the family of MTC. According to genealogy in 9 families of patients there have been cases of death from MTC or pheochromocytoma (PC). In all cases, surgical treatment was performed in a volume of TE, two patients additionally performed lymph node dissection VI level. The observation period after surgery ranged from 6 months to 16 years. Results. DNA diagnostics in 8 patients identified a mutation in exon 11, in one case – in exon 10 and one patient had revealed two mutations in exons 13 and 14. The age of patients ranged from 2 to 23 years. Basal calcitonin level was elevated in 7 of 10 patients. Such prophylactic TE in 2 patients was supplemented by selective lymph node dissection. Histological examination of the removal of the thyroid gland (TG) revealed foci of medullary cancer in 6 of 10. At 2 patients revealed a C-cell hyperplasia and at 2 patients were found signs of the well expressed and weakly expressed sclerosis in thyroid tissue. The earliest age to identify MTC was a child 3 years old, mutations in codon 634, from a family where relatives observed for aggressive MTC. In the course of follow-recurrence was detected in one patient, a child of 15 years, a similar mutation carrier. Conclusion. In view of the risk of MTC developing identifying a mutation in RET gene and preventive TE should be carried out as soon as possible. The high risk of MTC developing in RET-gene positive subjects was confirmed in this study (6 cases of cancer from 10 patients)