Varietäten elementarer Lie-Algebren

Friedlander and Parshall developed a theory of support varieties and rank varieties for finite-dimensional restricted Lie algebras over a feld $k$ of characteristic $p>0$. In particular, for any restricted Lie algebra \fg, the rank variety can be identified with its nullcone V(\fg) which is a conical closed subvariety of \fg. By a result of Carlson, the projectivization of V(\fg) is connected. Recently, Carlson, Friedlander and Pevtsova introduced the elementary subalgebras of restricted Lie algebras. The varieties of elementary subalgebras are natural varieties in which to define generalized support varieties for restricted representations of \fg. Farnsteiner defined an invariant for modules of restricted Lie algebras, called $j$-degrees. The invariant, together with the constant $j$-rank property can be linked by the two-dimensional elementary Lie subalgebras.\\ The aim of this thesis is to understand the geometric structure of the variety \EE(2,\fg). We first investigate the non-empty property of \EE(2,\fg). By using the sandwich elements, we give a complete description of the restricted Lie algebras which have no two-dimensional elementary subalgebras. Moreover, we show that the variety \EE(2,\fg) is always connected whenever \fg is centerless and $p>5$. This is a generalization of Carlson's result, which claims that \EE(1,\fg) is connected.\\ We also study the degree functions $\deg_M^j$ for centerless restricted Lie algebra. We show that the function $\deg_M^j$ is constant on \EE(2,\fg) if $M$ is a restricted \fg-module of constant $j$-rank.\\ In the last, we will endow the set \EE(2,\fg) with a graph structure which plays an important role in the investigation of the equal images property. We give a sufficient condition for the connectedness of the graph \EE(2,\fg). As a consequence, the graph \EE(2,W(n)) is connected, where $W(n$) is the $n$-th Witt-Jacobson algebra

Lie algebra configuration pairing

We give an algebraic construction of the topological graph-tree configuration pairing of Sinha and Walter beginning with the classical presentation of Lie coalgebras via coefficients of words in the associative Lie polynomial. Our work moves from associative algebras to preLie algebras to graph complexes, justifying the use of graph generators for Lie coalgebras by iteratively expanding the set of generators until the set of relations collapses to two simple local expressions. Our focus is on new computational methods allowed by this framework and the efficiency of the graph presentation in proofs and calculus involving free Lie algebras and coalgebras. This outlines a new way of understanding and calculating with Lie algebras arising from the graph presentation of Lie coalgebras.Comment: 21 pages; uses xypic; ver 4. added subsection 3.4 outlining another computational algorithm arising from configuration pairing with graph

Root systems from Toric Calabi-Yau Geometry. Towards new algebraic structures and symmetries in physics?

The algebraic approach to the construction of the reflexive polyhedra that yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres reveals graphs that include and generalize the Dynkin diagrams associated with gauge symmetries. In this work we continue to study the structure of graphs obtained from $CY_3$ reflexive polyhedra. We show how some particularly defined integral matrices can be assigned to these diagrams. This family of matrices and its associated graphs may be obtained by relaxing the restrictions on the individual entries of the generalized Cartan matrices associated with the Dynkin diagrams that characterize Cartan-Lie and affine Kac-Moody algebras. These graphs keep however the affine structure, as it was in Kac-Moody Dynkin diagrams. We presented a possible root structure for some simple cases. We conjecture that these generalized graphs and associated link matrices may characterize generalizations of these algebras.Comment: 24 pages, 6 figure

Integration of Lie Algebroid Comorphisms

We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the category of integrable Lie algebroids and complete Lie algebroid comorphisms to the category of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplectization functor in Poisson geometry. We include examples to show that the integrability of comorphisms and Poisson maps may not hold in the absence of a completeness assumption.Comment: 28 pages, references adde