Hilbert series for free lie superalgebras and related topics

Shaqaqha, Shadi (2015) Hilbert series for free lie superalgebras and related topics. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

We consider Hilbert series of ordinary Lie algebras, restricted (or p-) Lie algebras, and color Lie (p-)superalgebras. We derive a dimension formula similar to a wellknown Witt’s formula for free color Lie superalgebras and a certain class of color Lie p-superalgebras. A Lie (super)algebra analogue of a well-known Schreier’s formula for the rank of a subgroup of finite index in a free group was found by V. M. Petrogradsky. In this dissertation, Petrogradsky’s formulas are extended to the case of color Lie (p-)superalgebras. We establish more Schreier-type formulas for the ranks of submodules of free modules over free associative algebras and free group algebras. As an application, we consider Hopf subalgebras of some cocommutative Hopf algebras. Also, we apply our version of Witt and Schreier formulas to study relatively free color Lie (p-)superalgebras and to prove that the free color Lie superalgebra and its enveloping algebra have the same entropy. Y. A. Bahturin and A. Y. Olshanskii proved that the relative growth rate of a finitely generated subalgebra K of a free Lie algebra L of finite rank is strictly less than the growth rate of the free Lie algebra itself. We show that this theorem cannot be extended to free color Lie superalgebras in general. However, we establish it in a special case.

Item Type: Thesis (Doctoral (PhD))
URI: http://research.library.mun.ca/id/eprint/11599
Item ID: 11599
Additional Information: Includes bibliographical references (pages 105-108).
Keywords: Color Lie Superalgebra, Restricted Color Lie Superalgebra, Free Algebras, Witt's Formula, Character Formula, Relatively free color Lie (p-)superalgebra, Schreier Formula, Growth Rate, Relative Growth Rate, Hilbert Series, Special Universal Enveloping Algebra, Centre-by-Metabelian Lie Algebras, Commutator Subgroup of the Free Group, Actions by Free Group Algebras
Department(s): Science, Faculty of > Mathematics and Statistics
Date: August 2015
Date Type: Submission
Library of Congress Subject Heading: Lie superalgebras; Mathematics—Formulae; Hopf algebras

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