Relationships between generalized Heisenberg algebras and the classical Heisenberg algebra

Marc Fabbri, and Frank Okoh. "Relationships between generalized Heisenberg algebras and the classical Heisenberg algebra." Colloquium Mathematicae 134.2 (2014): 255-265. <http://eudml.org/doc/284119>.

@article{MarcFabbri2014,
abstract = {A Lie algebra is called a generalized Heisenberg algebra of degree n if its centre coincides with its derived algebra and is n-dimensional. In this paper we define for each positive integer n a generalized Heisenberg algebra 𝓗ₙ. We show that 𝓗ₙ and 𝓗 ₁ⁿ, the Lie algebra which is the direct product of n copies of 𝓗 ₁, contain isomorphic copies of each other. We show that 𝓗ₙ is an indecomposable Lie algebra. We prove that 𝓗ₙ and 𝓗 ₁ⁿ are not quotients of each other when n ≥ 2, but 𝓗 ₁ is a quotient of 𝓗ₙ for each positive integer n. These results are used to obtain several families of 𝓗ₙ-modules from the Fock space representation of 𝓗 ₁. Analogues of Verma modules for 𝓗ₙ, n ≥ 2, are also constructed using the set of rational primes.},
author = {Marc Fabbri, Frank Okoh},
journal = {Colloquium Mathematicae},
keywords = {Heisenberg algebra; generalized Heisenberg algebra; Fock space},
language = {eng},
number = {2},
pages = {255-265},
title = {Relationships between generalized Heisenberg algebras and the classical Heisenberg algebra},
url = {http://eudml.org/doc/284119},
volume = {134},
year = {2014},
}

TY - JOUR
AU - Marc Fabbri
AU - Frank Okoh
TI - Relationships between generalized Heisenberg algebras and the classical Heisenberg algebra
JO - Colloquium Mathematicae
PY - 2014
VL - 134
IS - 2
SP - 255
EP - 265
AB - A Lie algebra is called a generalized Heisenberg algebra of degree n if its centre coincides with its derived algebra and is n-dimensional. In this paper we define for each positive integer n a generalized Heisenberg algebra 𝓗ₙ. We show that 𝓗ₙ and 𝓗 ₁ⁿ, the Lie algebra which is the direct product of n copies of 𝓗 ₁, contain isomorphic copies of each other. We show that 𝓗ₙ is an indecomposable Lie algebra. We prove that 𝓗ₙ and 𝓗 ₁ⁿ are not quotients of each other when n ≥ 2, but 𝓗 ₁ is a quotient of 𝓗ₙ for each positive integer n. These results are used to obtain several families of 𝓗ₙ-modules from the Fock space representation of 𝓗 ₁. Analogues of Verma modules for 𝓗ₙ, n ≥ 2, are also constructed using the set of rational primes.
LA - eng
KW - Heisenberg algebra; generalized Heisenberg algebra; Fock space
UR - http://eudml.org/doc/284119
ER -