ABSTRACT
The problem of the classification of the extensions of the Virasoro algebra is discussed. It is shown that all H -reduced G ˜ r https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187674/f8337d7b-6b74-4da5-ae84-c225ed353b0a/content/in417_1.tif"/> -current algebras belong to one of the following basic algebraic structures: local quadratic W-algebras, rational W-algebras, nonlocal V-algebras, nonlocal quadratic WV-algebras and rational nonlocal tfV-algebras. The main new features of the quantum V-algebras and their heighest weight representations are demonstrated on the example of the quantum V 3 ( 1 , 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187674/f8337d7b-6b74-4da5-ae84-c225ed353b0a/content/in417_2.tif"/> -algebra.
