dorsal/arxiv
View SchemaGeneralization and Exact Deformations of Quantum Groups
| Authors | Christian Fronsdal |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9606020 |
| URL | https://arxiv.org/abs/q-alg/9606020 |
| Journal | Pub. Res. Inst. Math. Sci. 33 (1997) 91-149 |
Abstract
A large family of "standard" coboundary Hopf algebras is investigated. The existence of a universal R-matrix is demonstrated for the case when the parameters are in general position. Algebraic surfaces in parameter space are characterized by the appearance of certain ideals; in this case the universal R-matrix exists on the associated algebraic quotient. In special cases the quotient is a "standard" quantum group; all familiar quantum groups including twisted ones are obtained in this way. In other special cases one finds new types of coboundary bi-algebras. The "standard" universal R-matrix is shown to be the unique solution of a very simple, linear recursion relation. The classical limit is obtained in the case of quantized Kac-Moody algebras of finite and affine type. Returning to the general case, we study deformations of the standard R-matrix and the associated Hopf algebras. A preliminary investigation of the first order deformations uncovers a class of deformations that incompasses the quantization of all Kac-Moody algebras of finite and affine type. The corresponding exact deformations are described as generalized twists, $ R_\epsilon = (F^t)^{-1}RF$, where $R$ is the standard R-matrix and the cocycle $F$ (a power series in the deformation parameter $\epsilon$) is the solution of a linear recursion relation of the same type as that which determines $R$. Included here is the universal R-matrix for the elliptic quantum groups associated with $sl(n)$, a big surprise! Specializing again, to the case of quantized Kac-Moody algebras, and taking the classical limit of these esoteric quantum groups, one re-discovers all the trigonometric and elliptic r-matrices of Belavin and Drinfeld. The formulas obtained here are easier to use than the original ones, and the structure of the space of classical r-matrices is more transparent. The r-matrices obtained here are more general in that they are defined on the full Kac-Moody algebras, the central extensions of the loop groups.
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"abstract": "A large family of \"standard\" coboundary Hopf algebras is investigated. The\nexistence of a universal R-matrix is demonstrated for the case when the\nparameters are in general position. Algebraic surfaces in parameter space are\ncharacterized by the appearance of certain ideals; in this case the universal\nR-matrix exists on the associated algebraic quotient. In special cases the\nquotient is a \"standard\" quantum group; all familiar quantum groups including\ntwisted ones are obtained in this way. In other special cases one finds new\ntypes of coboundary bi-algebras. The \"standard\" universal R-matrix is shown to\nbe the unique solution of a very simple, linear recursion relation. The\nclassical limit is obtained in the case of quantized Kac-Moody algebras of\nfinite and affine type. Returning to the general case, we study deformations of\nthe standard R-matrix and the associated Hopf algebras. A preliminary\ninvestigation of the first order deformations uncovers a class of deformations\nthat incompasses the quantization of all Kac-Moody algebras of finite and\naffine type. The corresponding exact deformations are described as generalized\ntwists, $ R_\\epsilon = (F^t)^{-1}RF$, where $R$ is the standard R-matrix and\nthe cocycle $F$ (a power series in the deformation parameter $\\epsilon$) is the\nsolution of a linear recursion relation of the same type as that which\ndetermines $R$. Included here is the universal R-matrix for the elliptic\nquantum groups associated with $sl(n)$, a big surprise! Specializing again, to\nthe case of quantized Kac-Moody algebras, and taking the classical limit of\nthese esoteric quantum groups, one re-discovers all the trigonometric and\nelliptic r-matrices of Belavin and Drinfeld. The formulas obtained here are\neasier to use than the original ones, and the structure of the space of\nclassical r-matrices is more transparent. The r-matrices obtained here are more\ngeneral in that they are defined on the full Kac-Moody algebras, the central\nextensions of the loop groups.",
"arxiv_id": "q-alg/9606020",
"authors": [
"Christian Fronsdal"
],
"categories": [
"q-alg",
"math.QA"
],
"journal_ref": "Pub. Res. Inst. Math. Sci. 33 (1997) 91-149",
"title": "Generalization and Exact Deformations of Quantum Groups",
"url": "https://arxiv.org/abs/q-alg/9606020"
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