Lie algebras and universal enveloping algebras

2010 ◽  
pp. 251-282
Author(s):  
Joseph Neisendorfer
Keyword(s):  
Lie Algebras ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras

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.

Non-commutative unique factorization domains

1984 ◽  
Vol 95 (1) ◽  
pp. 49-54 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Lie Algebras ◽  
Commutative Algebra ◽  
Integral Domain ◽  
Zero Element ◽  
Noetherian Rings ◽  
Unique Factorization ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Unique Factorization Domains ◽  
Prime Elements

We introduce a concept of unique factorization for elements in the context of Noetherian rings which are not necessarily commutative. We will call an element p of such a ring R prime if (i) pR = Rp, (ii) pR is a height-1 prime ideal of R, and (iii) R/pR is an integral domain. We define a Noetherian u.f.d. to be a Noetherian integral domain R such that every height-1 prime P of R is principal and R/P is a domain, or equivalently every non-zero element of R is of the form cq, where q is a product of prime elements of R and c has no prime factors. Examples include the Noetherian u.f.d.'s of commutative algebra and also the universal enveloping algebras of solvable Lie algebras. The latter class provides a rich supply of genuinely non-commutative examples.

scholarly journals Principal subspaces of higher-level standard $\widehat{\mathfrak{sl}(n)}$-modules

2015 ◽  
Vol 26 (08) ◽  
pp. 1550053 ◽  
Author(s):  
Christopher Sadowski
Keyword(s):  
Lie Algebras ◽  
Operator Algebras ◽  
Intertwining Operators ◽  
Natural Setting ◽  
Affine Lie Algebras ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Defining Relations ◽  
Standard Formula ◽  
Exact Sequences

Using completions of certain universal enveloping algebras, we provide a natural setting for families of defining relations for the principal subspaces of standard modules for untwisted affine Lie algebras. We also use the theory of vertex operator algebras and intertwining operators to construct exact sequences among principal subspaces of certain standard [Formula: see text]-modules, n ≥ 3. As a consequence, we obtain the multigraded dimensions of the principal subspaces W(k1Λ1 + k2Λ2) and W(kn-2Λn-2 + kn-1Λn-1). This generalizes earlier work by Calinescu on principal subspaces of standard [Formula: see text]-modules.

scholarly journals Endofunctors and Poincaré–Birkhoff–Witt Theorems

2020 ◽  
Author(s):  
Vladimir Dotsenko ◽  
Pedro Tamaroff
Keyword(s):  
Lie Algebras ◽  
Natural Transformation ◽  
Algebraic Structures ◽  
Type Result ◽  
Unified Approach ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Categorical Framework ◽  
Dendriform Algebras ◽  
Bare Bones

Abstract We determine what appears to be the bare-bones categorical framework for Poincaré–Birkhoff–Witt (PBW)-type theorems about universal enveloping algebras of various algebraic structures. Our language is that of endofunctors; we establish that a natural transformation of monads enjoys a PBW property only if that transformation makes its codomain a free right module over its domain. We conclude with a number of applications to show how this unified approach proves various old and new PBW-type theorems. In particular, we prove a PBW-type result for universal enveloping dendriform algebras of pre-Lie algebras, answering a question of Loday.

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