Related papers: Spiders for rank 2 Lie algebras

In this paper we consider some Lie groups in complexified Clifford algebras. Using relations between operations of conjugation in Clifford algebras and matrix operations we prove isomorphisms between these groups and classical matrix groups…

The sl_3 spider is a diagrammatic category used to study the representation theory of the quantum group U_q(sl_3). The morphisms in this category are generated by a basis of non-elliptic webs. Khovanov- Kuperberg observed that non-elliptic…

We start with definitions of the general notions of the theory of $\Bbb Z_{2}$-graded algebras. Then we consider theory of inductive families of $\Bbb Z_{2}$-graded semisimple finite-dimensional algebras and its representations in the…

A method to construct in explicit form the generators of the simple roots of an arbitrary finite-dimensional representation of a quantum or standard semisimple algebra is found. The method is based on general results from the global theory…

A reducible representation of the Temperley-Lieb algebra is constructed on the tensor product of n-dimensional spaces. One obtains as a centraliser of this action a quantum algebra (a quasi-triangular Hopf algebra) U_q with a representation…

Uniform Lie algebras are combinatorially defined two-step nilpotent Lie algebras which can be used to define Einstein solvmanifolds. These Einstein spaces often have nontrivial isotropy groups. We derive basic properties of uniform Lie…

In this paper we present new formulas, which represent commutators and anticommutators of Clifford algebra elements as sums of elements of different ranks. Using these formulas we consider subalgebras of Lie algebras of pseudounitary…

An idea to present a classical Lie group of positive dimension by generators and relations sounds dubious, but happens to be fruitful. The isometry groups of classical geometries admit elegant and useful presentations by generators and…

We give a diagrammatic presentation of the A_2-Temperley-Lieb algebra. Generalizing Jones' notion of a planar algebra, we formulate an A_2-planar algebra motivated by Kuperberg's A_2-spider. This A_2-planar algebra contains a subfamily of…

We categorify the theory of Lie algebras beginning with a new notion of categorified vector space, or `2-vector space', which we define as an internal category in Vect, the category of vector spaces. We then define a `semistrict Lie…

Let g be a semisimple Lie algebra over the complex numbers. Fix a positive integer l (called the level). Let R(l,g) be the fusion algebra at level l. Then, there is an algebra homomorphism from the representation ring R(g) of g to R(l,g).…

Let $L$ be a restricted Lie algebra over a field of characteristic $p>2$ and denote by $u(L)$ its restricted enveloping algebra. We establish when the symmetric or skew elements of $u(L)$ under the principal involution are Lie metabelian.

We associate an square to any two dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behaviour of the algebra. We determine the identities of degrees…

We compare two natural bases for the invariant space of a tensor product of irreducible representations of A_2, or sl(3). One basis is the web basis, defined from a skein theory called the combinatorial A_2 spider. The other basis is the…

This paper presents some new Lie groups preserving fixed subspaces of geometric algebras (or Clifford algebras) under the twisted adjoint representation. We consider the cases of subspaces of fixed grades and subspaces determined by the…

Webs are a kind of planar, directed, edge-labeled graph that encode invariant vectors for quantum representations of $\mathfrak{sl}_n$. The theory of webs developed organically for $\mathfrak{sl}_2$, where they are also known as noncrossing…

We develop a theory of confluence of graphs. We describe an algorithm for proving that a given system of reduction rules for abstract graphs and graphs in surfaces is locally confluent. We apply this algorithm to show that each simple Lie…

In this paper, we first introduce the concept of symmetric biderivation radicals and characteristic subalgebras of Lie algebras, and study their properties. Based on these results, we precisely determine biderivations of some Lie algebras…

This is an old paper put here for archeological purposes. We derive a general formula expressing the second homology of a Lie algebra of the form L\otimes A with coefficients in the trivial module through homology of $L$, cyclic homology of…

In this note we explain, in terms of finite dimensional representations of Lie algebras $\mathfrak{sp}_{2\ell}\subset\mathfrak{sl}_{2\ell}$, a combinatorial coincidence of difference conditions in two constructions of combinatorial bases…