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A class of algebras similar to the enveloping algebra of sl(2). (English) Zbl 0732.16019
Let \(R={\mathbb{C}}[A,B,H]\) be a ring subject to the relations \[ (1)\quad [H,A]=A,\quad [H,B]=-B,\quad AB-BA=f(H) \] with an arbitrary polynomial f. These algebras have many similarities to U(sl(2)). The brief contents of the paper are:
1. Description of R as a skew polynomial ring over the enveloping algebra of the 2-dimensional nonabelian Lie algebra \([h,a]=a.\)
2. Definition of highest weight modules, V(\(\lambda\)), and the unique simple quotient of V(\(\lambda\)), L(\(\lambda\)). Every finite-dimensional simple R-module occurs among the L(\(\lambda\)). Description of which L(\(\lambda\)) are finite-dimensional in terms of properties of f. The number of finite-dimensional simples of dimension n is \(\leq \deg (f)\). Central characters and homomorphism between the V(\(\lambda\)).
3. A finite-dimensional R-module need not be semisimple. A nonsplit extension between two finite-dimensional simple modules must occur either at the “top” of some V(\(\lambda\)) or at the “bottom” of its dual. Let \(n>0\), and set \(f(x)=(x+1)^{n+1}-x^{n+1}\). Then for each \(d>0\), R has precisely n simple modules of dimension d, and every finite-dimensional R-module is semisimple.
4. The case \(\deg (f)=2\) is analyzed in detail.
The author studies the algebras having a great value for applications in mathematical physics. Algebras similar to (1) appear as ones of separation of variables in the scheme of the \({\mathcal R}\)-matrix inverse scattering method when one deals with an XXX-type \({\mathcal R}\)-matrix. In this case f is expressed in terms of the quantum determinant of polynomial \({\mathcal L}\)-operators.

MSC:
16S30 Universal enveloping algebras of Lie algebras
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16P10 Finite rings and finite-dimensional associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
81U20 \(S\)-matrix theory, etc. in quantum theory
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