scholarly journals Lie Bialgebra Structures on the Extended Schrödinger-Virasoro Lie Algebra

2011 ◽  
Vol 18 (04) ◽  
pp. 709-720 ◽  
Author(s):  
Lamei Yuan ◽  
Yongping Wu ◽  
Ying Xu
Keyword(s):  
Lie Algebra ◽  
Cohomology Group ◽  
Lie Bialgebra ◽  
First Cohomology Group

In this paper, Lie bialgebra structures on the extended Schrödinger-Virasoro Lie algebra [Formula: see text] are classified. It is obtained that all the Lie bialgebra structures on [Formula: see text] are triangular coboundary. As a by-product, it is derived that the first cohomology group [Formula: see text] is trivial.

Derivations and Automorphism Group of Completed Witt Lie Algebra

2012 ◽  
Vol 19 (03) ◽  
pp. 581-590 ◽  
Author(s):  
Yongping Wu ◽  
Ying Xu ◽  
Lamei Yuan
Keyword(s):  
Finite Element ◽  
Automorphism Group ◽  
Lie Algebra ◽  
Cohomology Group ◽  
Locally Finite ◽  
Simple Lie Algebra ◽  
Derivation Algebra ◽  
First Cohomology Group

In this paper, a simple Lie algebra, referred to as the completed Witt Lie algebra, is introduced. Its derivation algebra and automorphism group are completely described. As a by-product, it is obtained that the first cohomology group of this Lie algebra with coefficients in its adjoint module is trivial. Furthermore, we completely determine the conjugate classes of this Lie algebra under its automorphism group, and also obtain that this Lie algebra does not contain any nonzero ad -locally finite element.

COHOMOLOGY OF THE VECTOR FIELDS LIE ALGEBRAS ON ℝℙ1 ACTING ON BILINEAR DIFFERENTIAL OPERATORS

2005 ◽  
Vol 02 (01) ◽  
pp. 23-40 ◽  
Author(s):  
SOFIANE BOUARROUDJ
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Vector Fields ◽  
Differential Operators ◽  
Projective Line ◽  
Smooth Vector ◽  
Relative Cohomology ◽  
First Cohomology Group ◽  
Tensor Densities

The main topic of this paper is two-fold. First, we compute the first relative cohomology group of the Lie algebra of smooth vector fields on the projective line, Vect(ℝℙ1), with coefficients in the space of bilinear differential operators that act on tensor densities, [Formula: see text], vanishing on the Lie algebra sl(2, ℝ). Second, we compute the first cohomology group of the Lie algebra sl(2, ℝ) with coefficients in [Formula: see text].

scholarly journals On the Monodromy of Milnor Fibers of Hyperplane Arrangements

2014 ◽  
Vol 57 (4) ◽  
pp. 697-707 ◽  
Author(s):  
Pauline Bailet
Keyword(s):  
Cohomology Group ◽  
Hyperplane Arrangement ◽  
General Setting ◽  
Hyperplane Arrangements ◽  
Milnor Fiber ◽  
First Cohomology Group

AbstractWe describe a general setting where the monodromy action on the first cohomology group of the Milnor fiber of a hyperplane arrangement is the identity.

scholarly journals Group Cohomology and Lp-Cohomology of Finitely Generated Groups

2003 ◽  
Vol 46 (2) ◽  
pp. 268-276 ◽  
Author(s):  
Michael J. Puls
Keyword(s):  
Banach Space ◽  
Cohomology Group ◽  
Nilpotent Groups ◽  
Group Cohomology ◽  
Infinite Group ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Cohomology Space ◽  
First Cohomology Group

AbstractLet G be a finitely generated, infinite group, let p > 1, and let Lp(G) denote the Banach space . In this paper we will study the first cohomology group of G with coefficients in Lp(G), and the first reduced Lp-cohomology space of G. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.

THE FIRST COHOMOLOGY GROUP OF BANACH INVERSE SEMIGROUP ALGEBRAS WITH COEFFICIENTS IN -EMBEDDED BANACH BIMODULES

2019 ◽  
Vol 101 (3) ◽  
pp. 488-495
Author(s):  
HOGER GHAHRAMANI
Keyword(s):  
Banach Space ◽  
Inverse Semigroup ◽  
Cohomology Group ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Mild Conditions ◽  
First Cohomology Group

Let $S$ be a discrete inverse semigroup, $l^{1}(S)$ the Banach semigroup algebra on $S$ and $\mathbb{X}$ a Banach $l^{1}(S)$-bimodule which is an $L$-embedded Banach space. We show that under some mild conditions ${\mathcal{H}}^{1}(l^{1}(S),\mathbb{X})=0$. We also provide an application of the main result.

scholarly journals LEFT-SYMMETRIC BIALGEBRAS AND AN ANALOGUE OF THE CLASSICAL YANG–BAXTER EQUATION

2008 ◽  
Vol 10 (02) ◽  
pp. 221-260 ◽  
Author(s):  
CHENGMING BAI
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symmetric Solution ◽  
Baxter Equation ◽  
Symmetric Part ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Poisson Lie Groups ◽  
Lagrangian Subalgebras

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "S-equation", which is an analogue of the classical Yang–Baxter equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the S-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "[Formula: see text]-operator", whereas a skew-symmetric solution of the classical Yang–Baxter equation corresponds to the skew-symmetric part of an [Formula: see text]-operator. Thus a method to construct symmetric solutions of the S-equation (hence parakähler Lie algebras) from [Formula: see text]-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson–Lie groups in this sense.

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