scholarly journals UNIVERSAL ENVELOPING ALGEBRAS OF PBW TYPE

2011 ◽  
Vol 54 (1) ◽  
pp. 9-26 ◽  
Author(s):  
ALESSANDRO ARDIZZONI
Keyword(s):  
Type Theorem ◽  
Universal Enveloping Algebra ◽  
General Notion ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Pbw Basis ◽  
Braided Bialgebras ◽  
Nichols Algebras

AbstractWe continue our investigation of the general notion of universal enveloping algebra introduced in [A. Ardizzoni, A Milnor–Moore type theorem for primitively generated braided Bialgebras, J. Algebra 327(1) (2011), 337–365]. Namely, we study a universal enveloping algebra when it is of Poincaré–Birkhoff–Witt (PBW) type, meaning that a suitable PBW-type theorem holds. We discuss the problem of finding a basis for a universal enveloping algebra of PBW type: as an application, we recover the PBW basis both of an ordinary universal enveloping algebra and of a restricted enveloping algebra. We prove that a universal enveloping algebra is of PBW type if and only if it is cosymmetric. We characterise braided bialgebra liftings of Nichols algebras as universal enveloping algebras of PBW type.

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 Generic Lie colour algebras

2005 ◽  
Vol 71 (2) ◽  
pp. 327-335 ◽  
Author(s):  
Kenneth L. Price
Keyword(s):  
Gröbner Basis ◽  
Global Dimension ◽  
Universal Enveloping Algebra ◽  
Ore Extension ◽  
Grobner Basis ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Finite Global Dimension

We describe a type of Lie colour algebra, which we call generic, whose universal enveloping algebra is a domain with finite global dimension. Moreover, it is an iterated Ore extension. We provide an application and show Gröbner basis methods can be used to study universal enveloping algebras of factors of generic Lie colour algebras.

scholarly journals PBW Property for Associative Universal Enveloping Algebras Over an Operad

2020 ◽  
Author(s):  
Anton Khoroshkin
Keyword(s):  
Associative Algebra ◽  
Hilbert Series ◽  
Abelian Category ◽  
Universal Enveloping Algebra ◽  
Necessary Condition ◽  
Grobner Basis ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Category Of Modules

Abstract Given a symmetric operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $V$, the associative universal enveloping algebra ${\textsf{U}_{\mathcal{P}}}$ is an associative algebra whose category of modules is isomorphic to the abelian category of $V$-modules. We study the notion of PBW property for universal enveloping algebras over an operad. In case $\mathcal{P}$ is Koszul a criterion for the PBW property is found. A necessary condition on the Hilbert series for $\mathcal{P}$ is discovered. Moreover, given any symmetric operad $\mathcal{P}$, together with a Gröbner basis $G$, a condition is given in terms of the structure of the underlying trees associated with leading monomials of $G$, sufficient for the PBW property to hold. Examples are provided.

MULTIPARAMETER QUANTUM FORMS OF THE ENVELOPING ALGEBRA $U_{{\mathfrak g}{\mathfrak l}_N}$ RELATED TO THE FADDEEV-RESHETIKHIN-TAKHTAJAN U(R) CONSTRUCTIONS

1995 ◽  
Vol 04 (02) ◽  
pp. 263-317 ◽  
Author(s):  
JACOB TOWBER
Keyword(s):  
Commutative Rings ◽  
Generators And Relations ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Generating Sets ◽  
Pbw Basis ◽  
Suitable Generalization ◽  
Baxter Operator

Two quantum enveloping algebras UR and ÛR are associated in [RTF] to any Yang-Baxter operator R. These are constructed as subalgebras of A(R)* with specific generating sets. Also, [RTF] construct specific relations on the generators for UR, leaving open the question whether these generate all relations on these generators—let us say R is “perfect” when this is the case. Given an [Formula: see text] -tuple [Formula: see text] of nonzero elements qij,r in the groundfield, ([AST], [R], [S]) construct a multiparameter deformation [Formula: see text] of GLN associated with a Yang-Baxter operator [Formula: see text]. The method of ‘braiding maps’, introduced in [LT], is applied, in order to derive a PBW basis and a generators-and-relations presentation for a suitable generalization of [Formula: see text]. These results imply that [Formula: see text] is perfect, for generic [Formula: see text]. The construction [Formula: see text] is in some ways unsatisfactory if r is a root of 1. A construction [Formula: see text] is proposed, which is in some ways better behaved, coincides with [Formula: see text] if r is not a root of 1, and also makes sense over arbitrary commutative rings.

scholarly journals Primary decomposition in enveloping algebras

1981 ◽  
Vol 24 (2) ◽  
pp. 83-85 ◽  
Author(s):  
K. A. Brown ◽  
T. H. Lenagan
Keyword(s):  
Lie Algebra ◽  
Universal Enveloping Algebra ◽  
The Other ◽  
Primary Decomposition ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Factor Ring ◽  
Finite Dimensional ◽  
Other Hand

Recently, the first author and, independently, A. V. Jategaonkar have shown that every factor ring of U(g), the universal enveloping algebra of a finite dimensional complex Lie algebra, has a primary decomposition if g is solvable and almost algebraic. On the other hand, a suitable factor ring of U(SL(2, ℂ) fails to have a primary decomposition (1).

Universal Enveloping Algebras of Simple Symplectic Anti-Jordan Triple Systems

2015 ◽  
Vol 22 (02) ◽  
pp. 281-292 ◽  
Author(s):  
Marina Tvalavadze
Keyword(s):  
Associative Algebra ◽  
Triple System ◽  
Universal Enveloping Algebras ◽  
Triple Systems ◽  
Enveloping Algebras ◽  
Injective Homomorphism ◽  
Pbw Basis ◽  
Finite Dimensional ◽  
Jordan Triple Systems ◽  
Kirillov Dimension

In this work we are concerned with the universal associative envelope of a finite-dimensional simple symplectic anti-Jordan triple system (AJTS). We prove that if 𝕋 is a triple system as above, then there exists an associative algebra U(𝕋) and an injective homomorphism ε : 𝕋 → U(𝕋), where U(𝕋) is an AJTS under the triple product defined by (a,b,c) = abc - cba. Moreover, U(𝕋) is a universal object with respect to such homomorphisms. We explicitly determine the PBW-basis of U(𝕋), the center Z(U(𝕋)) and the Gelfand-Kirillov dimension of U(𝕋).

scholarly journals Filtrations on block subalgebras of reduced enveloping algebras

2021 ◽  
pp. 2350001
Author(s):  
Andrei Ionov ◽  
Dylan Pentland
Keyword(s):  
Positive Characteristic ◽  
Adjoint Representation ◽  
Universal Enveloping Algebra ◽  
Graded Algebra ◽  
Graded Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Nilpotent Cone ◽  
Algebra Of Functions ◽  
Pbw Filtration

We study the interaction between the block decompositions of reduced enveloping algebras in positive characteristic, the Poincaré-Birkhoff-Witt (PBW) filtration, and the nilpotent cone. We provide two natural versions of the PBW filtration on the block subalgebra [Formula: see text] of the restricted universal enveloping algebra [Formula: see text] and show these are dual to each other. We also consider a shifted PBW filtration for which we relate the associated graded algebra to the algebra of functions on the Frobenius neighborhood of [Formula: see text] in the nilpotent cone and the coinvariants algebra corresponding to [Formula: see text]. In the case of [Formula: see text] in characteristic [Formula: see text] we determine the associated graded algebras of these filtrations on block subalgebras of [Formula: see text]. We also apply this to determine the structure of the adjoint representation of [Formula: see text].

Some Norms on Universal Enveloping Algebras

1998 ◽  
Vol 50 (2) ◽  
pp. 356-377 ◽  
Author(s):  
Leonard Gross
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Dual Space ◽  
Universal Enveloping Algebras ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Kernel Analysis ◽  
Finite Rank Elements ◽  
Heat Kernel Analysis ◽  
Algebra Level

AbstractThe universal enveloping algebra, U(𝔤), of a Lie algebra 𝔤 supports some norms and seminorms that have arisen naturally in the context of heat kernel analysis on Lie groups. These norms and seminorms are investigated here from an algebraic viewpoint. It is shown that the norms corresponding to heat kernels on the associated Lie groups decompose as product norms under the natural isomorphism . The seminorms corresponding to Green's functions are examined at a purely Lie algebra level for sl (2, ℂ). It is also shown that the algebraic dual space U′ is spanned by its finite rank elements if and only if 𝔤 is nilpotent.

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