Leibniz and Hochschild homology Altawallbeh, Zuhier
Communications in algebra,
01/2018, Letnik:
46, Številka:
1
Journal Article
Recenzirano
We construct and study the map from Leibniz homology HL
∗
( ) of an abelian extension of a simple real Lie algebra to the Hochschild homology HH
∗−1
(U( )) of the universal envelopping algebra U( ). ...To calculate some homology groups, we use the Hochschild-Serre spectral sequences and Pirashvili spectral sequences. The result shows what part of the non-commutative Leibniz theory is detected by classical Hochschild homology, which is of interest today in string theory.
We observe several facts and make conjectures about commutative algebras satisfying the Jacobi identity. The central question is which of those algebras admit a faithful representation (i.e., in Lie ...parlance, satisfy the Ado theorem, or, in Jordan parlance, are special).
In this paper, we study non-abelian extensions of pre-Lie algebras. First we introduce the non-abelian cohomology for pre-Lie algebras by which we classify non-abelian extensions of pre-Lie algebras. ...Then we construct the bimultipliers for pre-Lie algebras, which naturally gives rise to a strict Lie 2-algebra. We show that there is a one-to-one correspondence between isomorphic classes of non-abelian extensions of pre-Lie algebras and homotopy classes of homomorphisms from the subadjacent Lie algebra to the strict Lie 2-algebra obtained from the bimultipliers. Finally we classify non-abelian extensions of pre-Lie algebras by Maurer-Cartan elements of a differential graded Lie algebra.
Let O be a nilpotent orbit of a complex semisimple Lie algebra g and let π:X→O¯ be the finite covering associated with the universal covering of O. In the previous article 14 we have explicitly ...constructed a Q-factorial terminalization X˜ of X when g is classical. In this article we count how many non-isomorphic Q-factorial terminalizations X has. We construct the universal Poisson deformation of X˜ over H2(X˜,C) and look at the action of the Weyl group W(X) on H2(X˜,C). The main result is an explicit geometric description of W(X).
We study the structure of bounded simple weight
-,
-,
-modules, which have been recently classified by D. Grantcharov and I. Penkov. Given a splitting parabolic subalgebra
of
we introduce the ...concepts of
-aligned and pseudo
-aligned
-,
-,
-modules, and give necessary and sufficient conditions for bounded simple weight modules to be
-aligned or pseudo
-aligned. The existence of pseudo
-aligned modules is a consequence of the fact that the Lie algebras considered have infinite rank.
Bidiagonal triples Funk-Neubauer, Darren
Linear algebra and its applications,
05/2017, Letnik:
521
Journal Article
Recenzirano
Odprti dostop
We introduce a linear algebraic object called a bidiagonal triple. A bidiagonal triple consists of three diagonalizable linear transformations on a finite-dimensional vector space, each of which acts ...in a bidiagonal fashion on the eigenspaces of the other two. The concept of bidiagonal triple is a generalization of the previously studied and similarly defined concept of bidiagonal pair. We show that every bidiagonal pair extends to a bidiagonal triple, and we describe the sense in which this extension is unique. In addition we generalize a number of theorems about bidiagonal pairs to the case of bidiagonal triples. In particular we use the concept of a parameter array to classify bidiagonal triples up to isomorphism. We also describe the close relationship between bidiagonal triples and the representation theory of the Lie algebra sl2 and the quantum algebra Uq(sl2).
Let F be a field, and let q∈F. The q-deformed Heisenberg algebra is the unital associative F-algebra H(q) with generators A,B and relation AB−qBA=I, where I is the multiplicative identity in H(q). ...The set of all Lie polynomials in A,B is the Lie subalgebra L(q) of H(q) generated by A,B. If q≠1 or the characteristic of F is not 2, then the equation AB−qBA=I cannot be expressed in terms of Lie algebra operations only, yet this equation still has consequences on the Lie algebra structure of L(q), which we investigate. We show that if q is not a root of unity, then L(q) is a Lie ideal of H(q), and the resulting quotient Lie algebra is infinite-dimensional and one-step nilpotent.
We consider a category of modules that admit compatible actions of the commutative algebra of Laurent polynomials and the Lie algebra of divergence zero vector fields on a torus and have a weight ...decomposition with finite dimensional weight spaces. We classify indecomposable and irreducible modules in this category.
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