Block (or Hamiltonian) Lie Symmetry of Dispersionless D-Type Drinfeld–Sokolov Hierarchy

Abstract

Abstract. In this paper, the dispersionless D type Drinfeld-Sokolov hierarchy, i.e. a reduction of the dispersionless two-component BKP hierarchy, is studied. The additional symmetry flows of this herarchy are presented. These flows form an infinite dimensional Lie algebra of Block type as well as a Lie algebra of Hamiltonian type.

Quantum torus symmetries of the CKP and multi-component CKP hierarchies

“…On the basis of the quantum parameter q, the additional symmetry flows for the time variable t m,n , t * m,n are defined as follows 16) which are equivalent to…”

Simple non-weight modules over Lie superalgebras of Block type

“…Besides, we also notice that Lie algebras of Block type have some applications in the integrable system (cf. [9,10]). Partially due to this, the study of representations of Lie (super)algebras of this kind becomes significant and has been attracting a lot of attentions.…”

Dispersionless and multicomponent BKP hierarchies with quantum torus symmetries

“…In another paper [18], we give a novel Block type additional symmetry of the bigraded Toda hierarchy(BTH). Later we did a series of studies on integrable systems and Block algebras such as in [19,20,21]. After the quantization, the Block Lie algebra becomes the so-called quantum torus Lie algebra which can be seen in several recent references as [22,23].…”

“…the BKP hierarchy and CKP hierarchy. About the BKP hierarchy, a lot of studies on additional symmetries have been done, such as additional symmetries of the BKP hierarchy [33], dispersionless BKP hierarchy [31,34], twocomponent BKP hierarchy [24] and its reduced hierarchies [20,21], supersymmetric BKP hierarchy [25] and so on. Dispersionless integrable systems [26] are very important in the study of all kinds of nonlinear sciences in physics, particularly in the application on the topological field theory [27] and matrix models [28,29].…”

“…In particular, dispersionless integrable systems have many typical properties such as the Lax pair, infinite conservation laws, symmetries and so on. On dispersionless integrable systems, we also did several studies such as [19,21]. With the above preparation, we should pay our attention to the quantum torus type additional symmetry of the multi-component BKP hierarchy [30] and dispersionless BKP hierarchy [31,32] from the points of the multi-component generalization of Lie algebras and the importance of dispersionless integrable systems.…”