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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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