Symplectic structures on quadratic Lie algebras (original) (raw)

Equivalent constructions of nilpotent quadratic Lie algebras

Linear Algebra and its Applications

The double extension and the T *-extension are classical methods for constructing finite dimensional quadratic Lie algebras. The first one gives an inductive classification in characteristic zero, while the latest produces quadratic non-associative algebras (not only Lie) out of arbitrary ones in characteristic different from 2. The classification of quadratic nilpotent Lie algebras can also be reduced to the study of free nilpotent Lie algebras and their invariant forms. In this work we will establish an equivalent characterization among these three construction methods. This equivalence reduces the classification of quadratic 2-step nilpotent to that of trivectors in a natural way. In addition, theoretical results will provide simple rules for switching among them.

Singular quadratic Lie superalgebras

In this paper, we give a generalization of results in [PU07] and [DPU] by applying the tools of graded Lie algebras to quadratic Lie superalgebras. In this way, we obtain a numerical invariant of quadratic Lie superalgebras and a classification of singular quadratic Lie superalgebras, i.e. those with a nonzero invariant. Finally, we study a class of quadratic Lie superalgebras obtained by the method of generalized double extensions.

Nilpotent Symplectic Alternating Algebras

PhD Thesis, 2015

We develop a structure theory for nilpotent symplectic alternating algebras. We then give a classification of all nilpotent symplectic alternating algebras of dimension up to 10 over any field. The study reveals a new subclasses of powerful groups that we call powerfully nilpotent groups and powerfully soluble groups.

Free nilpotent and nilpotent quadratic Lie algebras

Linear Algebra and its Applications, 2017

In this paper we introduce an equivalence between the category of the tnilpotent quadratic Lie algebras with d generators and the category of some symmetric invariant bilinear forms over the t-nilpotent free Lie algebra with d generators. Taking into account this equivalence, t-nilpotent quadratic Lie algebras with d generators are classified (up to isometric isomorphisms, and over any field of characteristic zero), in the following cases: d = 2 and t ≤ 5, d = 3 and t ≤ 3.

Symplectic groups over noncommutative algebras

2021

We introduce the symplectic group mathrmSp2(A,sigma)\mathrm{Sp}_2(A,\sigma)mathrmSp2(A,sigma) over a noncommutative algebra AAA with an anti-involution sigma\sigmasigma. We realize several classical Lie groups as mathrmSp2\mathrm{Sp}_2mathrmSp2 over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups mathrmSp2(A,sigma)\mathrm{Sp}_2(A,\sigma)mathrmSp2(A,sigma) act. We introduce the space of isotropic AAA-lines, which generalizes the projective line. We describe the action of mathrmSp2(A,sigma)\mathrm{Sp}_2(A,\sigma)mathrmSp2(A,sigma) on isotropic AAA-lines, generalize the Kashiwara-Maslov index of triples and the cross ratio of quadruples of isotropic AAA-lines as invariants of this action. When the algebra AAA is Hermitian or the complexification of a Hermitian algebra, we introduce the symmetric space XmathrmSp2(A,sigma)X_{\mathrm{Sp}_2(A,\sigma)}X_mathrmSp2(A,sigma), and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as mathrmSp2(A,sigma)\mathrm{Sp}_2(A,\sigma)mathrmSp_2(A,sigma)) and their complexifications, we ob...

Chain Equivalences for Symplectic Bases, Quadratic Forms and Tensor Products of Quaternion Algebras

We present a set of generators for the symplectic group which is different from the well-known set of transvections, from which the chain equivalence for quadratic forms in characteristic 2 is an immediate result. Based on the chain equivalences for quadratic forms, both in characteristic 2 and not 2, we provide chain equivalences for tensor products of quaternion algebras over fields with no nontrivial 3-fold Pfister forms. The chain equivalence for biquaternion algebras in characteristic 2 is also obtained in this process, without any assumption on the base-field.

Classification and Construction of Ouasisimple Lie Algebras

1986

We study a class of (possibly infinite-dimensional) Lie algebras, called the Quasisimple Lie algebras (QSLA's), and generalizing semisimple and affine Kac-Moody Lie algebras. They are characterized by the existence of a finite-dimensional Cartan subalgebra, a non-degenerate symmetric ad-invariant Killing form, and nilpotent rootspaces attached to non-isotropic roots. We are then able to derive a classification theorem for the possible quasisimple root systems; moreover, we construct explicit realizations of some of them as current algebras, generalizing the affine loop algebras.

Double constructions of quadratic and sympletic antiassociative algebras

arXiv: Rings and Algebras, 2020

This work addresses some relevant characteristics and properties of qqq-generalized associative algebras and qqq-generalized dendriform algebras such as bimodules, matched pairs. We construct for the special case of q=−1q=-1q=−1 an antiassociative algebra with a decomposition into the direct sum of the underlying vector spaces of another antiassociative algebra and its dual such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural antisymmetric bilinear form is sympletic. The former is called a double construction of quadratic antiassociative algebra and the later is a double construction of sympletic antiassociative algebra which is interpreted in terms of antidendrifom algebras. We classify the 2-dimensional antiassociative algebras and thoroughly give some double constructions of quadratic and sympletic antiassociative algebras.