scholarly journals INVARIANTS OF THE COADJOINT ACTION ON THE BASIC VARIETIES OF THE UNITRIANGULAR GROUP

2014 ◽  
Vol 20 (1) ◽  
pp. 229-246 ◽  
Author(s):  
A. N. PANOV
Keyword(s):  
Coadjoint Action ◽  
Unitriangular Group

scholarly journals On the Coadjoint Orbits of the Unitriangular Group

1996 ◽  
Vol 180 (2) ◽  
pp. 587-630 ◽  
Author(s):  
Carlos A.M. André
Keyword(s):  
Coadjoint Orbits ◽  
Unitriangular Group

scholarly journals On the Structure of Monodromy Algebras and Drinfeld Doubles

1997 ◽  
Vol 09 (03) ◽  
pp. 371-395
Author(s):  
Florian Nill
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Tensor Products ◽  
Spin Chains ◽  
Coadjoint Action ◽  
Braided Group ◽  
Gauge Transformations ◽  
Loop Algebras ◽  
Algebraic Tensor Product ◽  
Quasitriangular Hopf Algebra

We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy algebras via the coadjoint action. Applying a result of Majid, the resulting crossed product is isomorphic to the Drinfeld double [Formula: see text]. Hence, under the so-called factorizability condition given by N. Reshetikhin and M. Semenov–Tian–Shansky, both algebras are isomorphic to the algebraic tensor product H ⊗ H. It is indicated that in this way the results of Alekseev et al. on lattice current algebras are consistent with the theory of more general Hopf spin chains given by K. Szlachányi and the author. In the Appendix the multi-loop algebras ℒm of Alekseev and Schomerus [3] are identified with braided tensor products of monodromy algebras in the sense of Majid, which leads to an explanation of the "bosonization formula" of [3] representing ℒm as H ⊗…⊗ H.

scholarly journals Actions and Identities on Set Partitions

10.37236/1992 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Eric Marberg
Keyword(s):  
Abelian Group ◽  
Group Action ◽  
Type B ◽  
Set Partition ◽  
Set Partitions ◽  
Unitriangular Group

A labeled set partition is a partition of a set of integers whose arcs are labeled by nonzero elements of an abelian group $\mathbb{A}$. Inspired by the action of the linear characters of the unitriangular group on its supercharacters, we define a group action of $\mathbb{A}^n$ on the set of $\mathbb{A}$-labeled partitions of an $(n+1)$-set. By investigating the orbit decomposition of various families of set partitions under this action, we derive new combinatorial proofs of Coker's identity for the Narayana polynomial and its type B analogue, and establish a number of other related identities. In return, we also prove some enumerative results concerning André and Neto's supercharacter theories of type B and D.

Triangular Automorphisms

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Commutative Ring ◽  
Lie Algebra ◽  
Polynomial Algebra ◽  
Polynomial Rings ◽  
Differential Polynomials ◽  
Graded Algebras ◽  
Linear Maps ◽  
Unitriangular Group ◽  
Filtered Algebras ◽  
Filtered Modules

This chapter focuses on triangular automorphisms, which can be analyzed by Lie techniques. Throughout the discussion K is a commutative ring containing ℚ as a subring. A formalism is introduced to analyze triangular automorphisms of such a polynomial algebra by means of their logarithms, the triangular derivations. After presenting some definitions and simple facts about filtered modules, filtered algebras, and graded algebras, the chapter considers triangular linear maps and the Lie algebra of an algebraic unitriangular group. It then describes derivations on the ring of column-finite matrices, along with iteration matrices and Riordan matrices. It also explains derivations on polynomial rings and concludes by applying triangular automorphisms to differential polynomials.

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