Krichever-Novikov type algebras. A general review and the genus zero case. (English) Zbl 07619843
Hervik, Sigbjørn (ed.) et al., Geometry, Lie theory and applications. Proceedings of the Abel symposium 2019, Ålesund, Norway June 24–28, 2019. Cham: Springer. Abel Symp. 16, 279-330 (2022).
The survey is a nice summary of the author’s earlier results on Krichever-Novikov type algebras. These algebras are generalizations of the classical algebras in conformal field theory, like the Witt and Virasoro algebra, affine Lie algebras, superalgebra, where the geometric setting is the genus zero Riemann surface, with two points with poles \((\{0\}\) and \(\{\infty\})\). These algebras are graded. The generalized KN type algebras are those where the set of points \(A\) where poles are allowed is arbitrary finite number. These algebras are not graded. Krichever-Novikov introduced a weaker concept of grading in higher genus case. This weaker concept, called almost grading, also works in the genus zero case with more than two points where poles are allowed. The almost grading is introduced by a splitting of the set \(A\) of allowed poles into two nonempty disjoint sets \(I \cup Q\). With a given almost grading one gets a triangular decomposition of these algebras, highest weight representations and central extensions.
In the second part of the survey the author gives the construction of generalized KN algebras for genus zero multipoint case. Such algebras include vector field algebras, function algebras, differential operator algebra, current algebras and Lie superalgebras. With the help of a given almost grading induced by the splitting \(A=I \cup Q\), the author describes the space of local cohomology with trivial coefficients, which describes local central extensions. In every case he computes the cohomology and describes central extensions. An interesting special case is the three-point genus 0 case, which has more applications. The poles are \(A=\{0, 1, \infty\}\). As an example, he gives the universal central extension of the three-point \(sl(2, \mathbb C)\) current algebra. Consider the vector space of meromorphic forms of arbitrary weights. It is associative, and commutative graded algebra. One can also define a Lie algebra structure and superalgebra on it. At the end central extensions of the considered KN-type algebras are calculated.
For the entire collection see [Zbl 1479.53006].
In the second part of the survey the author gives the construction of generalized KN algebras for genus zero multipoint case. Such algebras include vector field algebras, function algebras, differential operator algebra, current algebras and Lie superalgebras. With the help of a given almost grading induced by the splitting \(A=I \cup Q\), the author describes the space of local cohomology with trivial coefficients, which describes local central extensions. In every case he computes the cohomology and describes central extensions. An interesting special case is the three-point genus 0 case, which has more applications. The poles are \(A=\{0, 1, \infty\}\). As an example, he gives the universal central extension of the three-point \(sl(2, \mathbb C)\) current algebra. Consider the vector space of meromorphic forms of arbitrary weights. It is associative, and commutative graded algebra. One can also define a Lie algebra structure and superalgebra on it. At the end central extensions of the considered KN-type algebras are calculated.
For the entire collection see [Zbl 1479.53006].
Reviewer: Alice Fialowski (Budapest)
MSC:
| 17B66 | Lie algebras of vector fields and related (super) algebras |
| 17B56 | Cohomology of Lie (super)algebras |
| 17B65 | Infinite-dimensional Lie (super)algebras |
| 17B68 | Virasoro and related algebras |
| 30F30 | Differentials on Riemann surfaces |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
Keywords:
Krichever-Novikov type algebra; Riemann surface; conformal field theory; central extension, Virasoro algebra, current algebraReferences:
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