AN ALGEBRAIC APPROACH TO SYMMETRIC PRE-MONOIDAL STATISTICS

2007 ◽  
Vol 06 (01) ◽  
pp. 49-69
Author(s):  
P. S. ISAAC ◽  
W. P. JOYCE ◽  
J. LINKS
Keyword(s):  
Symmetric Group ◽  
Lie Algebras ◽  
Degrees Of Freedom ◽  
Monoidal Category ◽  
Algebraic Approach ◽  
Symmetric Groups ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Finite Dimensional ◽  
Fermi Statistics

Recently, generalized Bose–Fermi statistics was studied in a category theoretic framework and to accommodate this endeavor the notion of a pre-monoidal category was developed. Here we describe an algebraic approach for the construction of such categories. We introduce a procedure called twining which breaks the quasi-bialgebra structure of the universal enveloping algebras of semi-simple Lie algebras and renders the category of finite-dimensional modules pre-monoidal. The category is also symmetric, meaning that each object of the category provides representations of the symmetric groups, which allows for a generalized boson-fermion statistic to be defined. Exclusion and confinement principles for systems of indistinguishable particles are formulated as an invariance with respect to the actions of the symmetric group. We apply the procedure to suggest that the symmetries which can be associated to color, spin and flavor degrees of freedom lead to confinement of states.

scholarly journals Hodge decomposition of string topology

2021 ◽  
Vol 9 ◽  
Author(s):  
Yuri Berest ◽  
Ajay C. Ramadoss ◽  
Yining Zhang
Keyword(s):  
Lie Algebras ◽  
General Theorem ◽  
Hodge Decomposition ◽  
String Topology ◽  
Free Loop Space ◽  
Oriented Manifold ◽  
Universal Enveloping Algebras ◽  
Finite Dimensional ◽  
Equivariant Homology ◽  
Simply Connected

Abstract Let X be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$ -equivariant homology $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ of the free loop space of X preserves the Hodge decomposition of $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture of [7].

scholarly journals Free Lie algebras as modules for symmetric groups

1999 ◽  
Vol 67 (2) ◽  
pp. 143-156 ◽  
Author(s):  
R. M. Bryant ◽  
L. G. Kovács ◽  
Ralph Stöhr
Keyword(s):  
Lie Algebras ◽  
Symmetric Groups ◽  
Free Lie Algebra ◽  
Prime Characteristic ◽  
Specht Modules ◽  
Generating Set ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Free Lie Algebras ◽  
Direct Decompositions

AbstractLet r be a positive integer, F a field of odd prime characteristic p, and L the free Lie algebra of rank r over F. Consider L a module for the symmetric group , of all permutations of a free generating set of L. The homogeneous components Ln of L are finite dimensional submodules, and L is their direct sum. For p ≤ r ≤ 2p, the main results of this paper identify the non-porojective indecomposable direct summands of the Ln as Specht modules or dual Specht modules corresponding to certain partitions. For the case r = p, the multiplicities of these indecomposables in the direct decompositions of the Ln are also determined, as are the multiplicities of the projective indecomposables. (Corresponding results for p = 2 have been obtained elsewhere.)

scholarly journals TWO PERMANENTS IN THE UNIVERSAL ENVELOPING ALGEBRAS OF THE SYMPLECTIC LIE ALGEBRAS

2009 ◽  
Vol 20 (03) ◽  
pp. 339-368 ◽  
Author(s):  
MINORU ITOH
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Universal Enveloping Algebra ◽  
Irreducible Representations ◽  
General Linear ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Central Elements ◽  
Orthogonal Lie Algebra

This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent and it is easy to calculate their eigenvalues on irreducible representations. We can regard these generators as the counterpart of central elements of the universal enveloping algebra of the orthogonal Lie algebra given in terms of the column-determinant by Wachi. The earliest prototype of all these central elements is the Capelli determinants in the universal enveloping algebra of the general linear Lie algebra.

scholarly journals Invariants of universal enveloping algebras of relatively free lie algebras

2000 ◽  
Vol 225 (1) ◽  
pp. 261-274
Author(s):  
Vesselin Drensky ◽  
Giulia Maria Piacentini Cattaneo
Keyword(s):  
Lie Algebras ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Free Lie Algebras

Graded Lie Algebras of Dimension Five and Some Artin–Schelter Regular Algebras

2019 ◽  
Vol 26 (02) ◽  
pp. 243-258
Author(s):  
Yuan Shen
Keyword(s):  
Lie Algebras ◽  
Global Dimension ◽  
Graded Lie Algebras ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Regular Algebras

In this paper, we show that there are only seven graded Lie algebras of dimension 5 generated in degree 1 up to isomorphism. By parameterizing the relations of the universal enveloping algebras of three of those graded Lie algebras, we construct some new Artin–Schelter regular algebras of global dimension 5. We prove that those algebras are all strongly noetherian, Auslander regular and Cohen–Macaulay, and describe their Nakayama automorphisms.

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