Graded associative conformal algebras of finite type. (English) Zbl 1293.16036
In 2001, B. Bakalov, A. D’Andrea and V. G. Kac proposed a generalization of conformal algebras [Adv. Math. 162, No. 1, 1-140 (2001; Zbl 1001.16021)]: a ‘pseudo-algebra’ over a Hopf algebra \(H\) is an algebra in the pseudo-tensor category of left unital \(H\)-modules. Classical conformal algebras are then the pseudo-algebras over the affine line.
The paper under review deals with pseudo-algebras over the coordinate Hopf algebras of linear algebraic groups \(G\) such that the connected component \(G^\circ\) is the affine line. These objects are described in terms of graded conformal algebras over the affine line.
A classification of the simple and semisimple graded associative conformal algebras of finite type is obtained. The simple objects turn out to be isomorphic to current graded conformal algebras over simple finite-dimensional graded associative algebras.
The paper under review deals with pseudo-algebras over the coordinate Hopf algebras of linear algebraic groups \(G\) such that the connected component \(G^\circ\) is the affine line. These objects are described in terms of graded conformal algebras over the affine line.
A classification of the simple and semisimple graded associative conformal algebras of finite type is obtained. The simple objects turn out to be isomorphic to current graded conformal algebras over simple finite-dimensional graded associative algebras.
Reviewer: Alberto Elduque (Zaragoza)
MSC:
| 16W50 | Graded rings and modules (associative rings and algebras) |
| 17B69 | Vertex operators; vertex operator algebras and related structures |
| 16S38 | Rings arising from noncommutative algebraic geometry |
| 16T05 | Hopf algebras and their applications |
| 16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
Keywords:
graded associative conformal algebras; graded algebras; pseudo-algebras over Hopf algebras; pseudo-tensor categories; coordinate Hopf algebras; linear algebraic groups; conformal algebras of finite typeCitations:
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