Theory Of Associative Algebras Research Articles

Coordinate changes for network dynamics

ABSTRACTConsider networks in which all arrows are distinct and all cells are distinct. In this context, we obtain complete descriptions of the groups of diffeomorphisms that preserve network dynamics in the following sense: changing coordinates via the diffeomorphism transforms the space of admissible maps to itself. Five distinct actions are considered: left, right, contact, conjugacy, and vector field. Key features are the left core, right core, and core of the network. The core is a subnetwork with special combinatorial features, and it represents a partition of the cells into all-to-all connected subnetworks that couple to each other in a feed-forward manner. For the left/right actions, the group consists of all diffeomorphisms that are admissible for the left/right core, respectively. The contact action is a pair (B, Φ) where B is determined by the left core and Φ by the right core. For the conjugacy and vector field actions, the group is generated by diffeomorphisms that are admissible for the core together with graph automorphisms of the network; that is, permutations of the cells that map arrows to arrows but need not preserve arrow type. The proofs are combinatorial for the left and right actions, but require a mixture of Lie theory and the structure theory of associative algebras in the other cases, together with a nonlinear-to-linear reduction theorem.

Arc spaces and the vertex algebra commutant problem

Given a vertex algebra V and a subalgebra A⊂V, the commutant Com(A,V) is the subalgebra of V which commutes with all elements of A. This construction is analogous to the ordinary commutant in the theory of associative algebras, and is important in physics in the construction of coset conformal field theories. When A is an affine vertex algebra, Com(A,V) is closely related to rings of invariant functions on arc spaces. We find strong finite generating sets for a family of examples where A is affine and V is a βγ-system, bc-system, or bcβγ-system.

Valued Graphs and the Representation Theory of Lie Algebras

Quivers (directed graphs), species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K -species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K -species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.

A commutant of βγ-system associated to the highest weight module V4 of sl(2,C)

Analogous to the commutant in the theory of associative algebras, one can construct a new subalgebra of a vertex algebra, known as the vertex algebra commutant. In this paper, for the highest weight module V4 of Lie algebra sl(2,C), we describe a commutant of βγ-system S(V4) by giving a finite set of generators and operator product expansion relations among generators.

Central Simple Nonassociative Algebras with Operators

The classical central simple theory of associative algebras generalizes, in this article, to a central simple theory of nonassociative algebras with operators and a related central irreducible theory of modules. These theories are motivated by, and apply to, problems of constructing and classifying simple Jordan Lie algebras, irreducible modules, and birings.

On the Block Structure of Supersolvable Restricted Lie Algebras

Block theory is an important tool in the modular representation theory Ž w x. Ž w x. of finite groups cf. 18 . Apart from a few papers e.g. 17, 13, 16 dealing with restricted simple Lie algebras there apparently has been no effort to do the same for other classes of restricted Lie algebras despite a good Ž w x . knowledge of the simple modules cf. e.g. 21 for the solvable case . The aim of this paper is to develop the block theory for reduced universal enveloping algebras of a finite dimensional supersolvable restricted Lie algebra as far as possible in close analogy to modular group algebras. In order to organize the paper in a concise way, we include some open questions on tensor products of simple modules. Under certain conditions either on the Lie algebra or on the character they have affirmative answers which are decisively used in the proofs of our main results. But unfortunately this is not true in general as we show by an example. In the following, we are going to describe the contents of the paper in more detail. The second section provides the necessary background from the block theory of associative algebras which perhaps is not as well-known to Lie theorists. In particular, we stress that the block decomposition induces an Ž . Ž . equivalence relation ‘‘linkage relation’’ on the finite set of isomorphism classes of simple modules and indicate the proof of a cohomological characterization of the linkage relation which is fundamental for some of the following results. The latter can be used to give a combinatorial description of the linkage relation via the Gabriel quiver. We include some results on the number of blocks, resp. the principal block, that hold for any restricted Lie algebra.

Inner ideals in quadratic Jordan algebras

The inner ideals play a role in the theory of quadratic Jordan algebras analogous to that played by the one-sided ideals in the theory of associative algebras. In particular, the Jordan algebras with descending chain condition on inner ideals are intimately related to the Artinian associative algebras. In this paper we will completely characterize all inner ideals in the semisimple Jordan algebras with descending chain condition. It is well known that any left or right ideal B \mathfrak {B} in a semisimple Artinian A \mathfrak {A} is determined by an idempotent, B = A f \mathfrak {B} = \mathfrak {A}f or B = e A \mathfrak {B} = e\mathfrak {A} . We show that any inner ideal in the quadratic Jordan algebra A + {\mathfrak {A}^ + } has the form B = e A f \mathfrak {B} = e\mathfrak {A}f , and if A \mathfrak {A} has involution ∗ ^\ast the inner ideals of the Jordan algebra H ( A , ∗ ) \mathfrak {H}(\mathfrak {A}, ^ \ast ) of ∗ ^ \ast -symmetric elements are “usually” of the form B = e ∗ H e \mathfrak {B} = {e^ \ast }\mathfrak {H}e . We also characterize the inner ideals in the Jordan algebras J ( Q , c ) \mathfrak {J}(Q,c) or J ( N , c ) \mathfrak {J}(N,c) determined by a quadratic or cubic form.