scholarly journals PBW-Basis for Universal Enveloping Algebras of Differential Graded Poisson Algebras

2018 ◽  
Vol 42 (6) ◽  
pp. 3343-3377
Author(s):  
Xianguo Hu ◽  
Jiafeng Lü ◽  
Xingting Wang
Keyword(s):  
Universal Enveloping Algebras ◽  
Poisson Algebras ◽  
Enveloping Algebras ◽  
Pbw Basis ◽  
Differential Graded Poisson Algebras

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.

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).

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