2,015 research outputs found
Vertex operators arising from Jacobi-Trudi identities
We give an interpretation of the boson-fermion correspondence as a direct
consequence of Jacobi-Trudi identity. This viewpoint enables us to construct
from a generalized version of the Jacobi-Trudi identity the action of Clifford
algebra on polynomial algebras that arrives as analogues of the algebra of
symmetric functions. A generalized Giambelli identity is also proved to follow
from that identity. As applications, we obtain explicit formulas for vertex
operators corresponding to characters of the classical Lie algebras, shifted
Schur functions, and generalized Schur symmetric functions associated to linear
recurrence relations.Comment: 23 page
False theta functions and companions to Capparelli's identities
Capparelli conjectured two modular identities for partitions whose parts
satisfy certain gap conditions, where were motivated by the calculation of
characters for the standard modules of certain affine Lie algebras and by
vertex operator theory. These identities were subsequently proved and refined
by Andrews, who related them to Jacobi theta functions, and also by
Alladi-Andrews-Gordon, Capparelli, and Tamba-Xie. In this paper we prove two
new companions to Capparelli's identities, where the evaluations are expressed
in terms of Jacobi theta functions and false theta functions.Comment: 17 pages; references update
Acyclic Jacobi Diagrams
We propose a simple new combinatorial model to study spaces of acyclic Jacobi
diagrams, in which they are identified with algebras of words modulo
operations. This provides a starting point for a word-problem type
combinatorial investigation of such spaces, and provides fresh insights on
known results.Comment: 18 pages, 7 figures. Refernces added. Section 2 rewritten. Proof of
Theorem 1.1 rewritten. To appear in Kobe J. Mat
On the Lie enveloping algebra of a post-Lie algebra
We consider pairs of Lie algebras and , defined over a common
vector space, where the Lie brackets of and are related via a
post-Lie algebra structure. The latter can be extended to the Lie enveloping
algebra . This permits us to define another associative product on
, which gives rise to a Hopf algebra isomorphism between and
a new Hopf algebra assembled from with the new product.
For the free post-Lie algebra these constructions provide a refined
understanding of a fundamental Hopf algebra appearing in the theory of
numerical integration methods for differential equations on manifolds. In the
pre-Lie setting, the algebraic point of view developed here also provides a
concise way to develop Butcher's order theory for Runge--Kutta methods.Comment: 25 page
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