Finite-dimensional Associative Algebra Research Articles

PLAIN REPRESENTATIONS OF LIE ALGEBRAS

In this paper we study representations of finite dimensional Lie algebras. In this case representations are not necessarily completely reducible. As the general problem is known to be of enormous complexity, we restrict ourselves to representations that behave particularly well on Levi subalgebras. We call such representations plain (Definition 1.1). Informally, we show that the theory of plain representations of a given Lie algebra L is equivalent to representation theory of finitely many finite dimensional associative algebras, also non-semisimple. The sense of this is to distinguish representations of Lie algebras that are of complexity comparable with that of representations of associative algebras. Non-plain representations are intrinsically much more complex than plain ones. We view our work as a step toward understanding this complexity phenomenon.We restrict ourselves also to perfect Lie algebras L, that is, such that L = [L, L]. In our main results we assume that L is perfect and [sfr ][lfr ]2-free (which means that L has no quotient isomorphic to [sfr ][lfr ]2). The ground field [ ] is always assumed to be algebraically closed and of characteristic 0.

Poisson and Hamiltonian superpairs over polarized associative algebras

A Poisson superpair is a pair of Poisson superalgebra structures on a supercommutative associative algebra, whose any linear combination is also a Poisson superalgebra structure. In this paper, we first construct certain linear and quadratic Poisson superpairs over a semi-finitely-filtered polarized 2-graded associative algebra. Then we give a construction of certain Hamiltonian superpairs in the formal variational calculus over any finite-dimensional 2-graded associative algebra with a supersymmetric nondegenerate associative bilinear form. Our constructions are based on the Adler mapping in a general sense. Our results in this paper can be viewed as noncommutative generalizations of the Adler–Gel'fand–Dikii Hamiltonian pair.

Left-right projective bimodules and stable equivalences of Morita type

We study a connection between left-right projective bimodules and stable equivalences of Morita type for finite-dimensional associative algebras over a field. Some properties of the category of all finite-dimensional left-right projective bimodules for se

Linear deformations of discrete groups and constructions of multivalued groups

We construct deformations of discrete multivalued groups described as special deformations of their group algebras in the class of finite-dimensional associative algebras. We show that the deformations of ordinary groups producing multivalued groups are defined by cocycles with coefficients in the group algebra of the original group and obtain classification theorems on these deformations. We indicate a connection between the linear deformations of discrete groups introduced in this paper and the well-known constructions of multivalued groups. We describe the manifold of three-dimensional associative commutative algebras with identity element, fixed basis, and a constant number of values. The group algebras of -valued groups of order three (three-dimensional -group algebras) form a discrete set in this manifold.

Properly stratified algebras

Abstract A class of properly stratified finite-dimensional (associative) algebras is defined recursively by means of properly stratifying idempotents inducing recollements of triangulated categories (in the sense of Beilinson, Bernstein and Deligne). A number of characterizations, as well as some recursive construction, of properly stratified algebras is given. In particular, an algebra is properly stratified if and only if its regular representation is filtered by standard (generalized Verma) as well as by proper standard (Verma) modules. Properly stratified algebras are a natural generalization of the quasi-hereditary algebras of Cline, Parshall and Scott, and appear in a recent work of Futorny, Konig and Mazorchuk on a generalization of the category O of Bernstein, Gelfand and Gelfand.

Efficient Decomposition of Associative Algebras over Finite Fields

We present new, efficient algorithms for some fundamental computations with finite-dimensional (but not necessarily commutative) associative algebras over finite fields. For a semisimple algebra A we show how to compute a complete Wedderburn decomposition of A as a direct sum of simple algebras, an isomorphism between each simple component and a full matrix algebra, and a basis for the centre of A. If A is given by a generating set of matrices inFm×m, then our algorithm requires aboutO (m3) operations inF, in addition to the cost of factoring a polynomial inF[ x ] of degree O(m), and the cost of generating a small number of random elements from A. We also show how to compute a complete set of orthogonal primitive idempotents in any associative algebra over a finite field in this same time.

Twisted representations of vertex operator algebras

This paper gives an analogue of Ag(V) theory for a vertex operator superalgebra V and an automorphism g of finite order. The relation between the g-twisted V-modules and Ag(V)-modules is established. It is proved that if V is g-rational, then Ag(V) is finite-dimensional semi-simple associative algebra and there are only finitely many irreducible g-twisted V-modules.

Linear associative algebras with finitely many idempotents

Let A be a linear (i.e., finite-dimensional) associative algebra with unity defined over K, an algebraically closed field. Then A with respect to its multiplication is an algebraic monoid over k, denoted by AM , and with respect to the the bracket forms a Lie algebra over K, denoted by AL . The following theorem is established AM is nilpotent as an algebraic monoid (equivalentlyAL is so as a Lie algebra) if and only if the set of idempotents of A is finite if and only if all irreducible closed submonoids of codimension 1 are nilpotent.

Group schemes and rigidity of algebras in positive characteristic

Rigidity criteria for a finite dimensional associative or Lie algebra of positive characteristic are given. A geometrically rigid algebra may have deformations with nontrivial infinitesimals which may be interpreted as obstructions to integrating infinitesimal automorphisms. A group scheme theoretic nature of those obstructions is revealed. For each affine group scheme G of finite type over the ground field an invariantly defined G-module Obs( G) is introduced and formal properties of the functor GObs( G) are studied.

Unfoldings and APR-Tilts

Let A be a finite dimensional associative algebra with an identity over an algebraically closed field. Assume moreover that A is basic, connected, and that the ordinary quiver of A has no oriented cycles. We describe an algorithm which yields an effective criterion for deciding whether or not A is iterated tilted of Dynkin or euclidean type, or tilting-cotilting equivalent to a tubular canonical algebra.

Decomposition of algebras overF q (X 1,...,X m )

LetA be a finite dimensional associative algebra over the fieldF whereF is a finite (algebraic) extension of the function fieldF q(X 1,...,X m). Here Fq denotes the finite field ofq elements (q=pl for a primep). We address the problem of computing the Jacobson radical Rad (A) ofA and the problem of computing the minimal ideals of the radical-free part (Wedderburn decomposition). The algebraA is given by structure constants overF andF is given by structure constants overF q(X 1,...,X m). We give algorithms to find these structural components ofA. Our methods run in polynomial time ifm is constant, in particular in the casem=1. The radical algorithm is deterministic. Our method for computing the Wedderburn decomposition ofA uses randomization (for factoring univariate polynomials overF q).

A quiver quantum group

We construct quantum groups at a root of unity and we describe their monoidal module category using techniques from the representation theory of finite dimensional associative algebras.

Semilocal modules and quasi-hereditary algebras

1. Main result. Throughout this note all algebras are assumed to be finite-dimensional (associative) algebras with 1 over an algebraically closed field k and all modules are finitely generated left modules. The composition of two morphisms f: Ma--~ M2 and g: M2 ---' M3 is denoted by

A Lie algebra variation on a theorem of Wedderburn

THEOREM. Let A be a finite dimensional associative algebra over a field F. Suppose that A has a basis over F consisting of nilpotent elements. Then A itself must be nilpotent. Given a finite dimensional Lie algebra L which is a Lie subalgebra of an associative algebra A (possibly infinite dimensional), we can ask what results from supposing that L has a basis over F consisting of elements that are nilpotent in the associative algebra. We cannot conclude that L is nilpotent and finite dimensional. In fact, this is rarely the case. For example,

Canonical forms of pairs of complex matrices

Using methods of the representation theory of finite-dimensional associative algebras, the paper presents an explicit classification of the pairs of real subspaces of a finite-dimensional complex vector space in terms of complex matrices. Thus, we classify the orbits of the direct product GL(m, C) × GL(m 1, R) × GL(m 2, R) acting on the set of pairs of m × m t complex matrices A t , t = 1,2, by (A 1,A 2)(Q,P 1,P 2) = (QA 1P −1 1,QA 2P −1 2). In addition, the general case of canonical forms of pairs of complex matrices is resolved.

Computing the structure of finite algebras

In this paper we address some algorithmic problems related to computations in finite-dimensional associative algebras over finite fields. Our starting point is the structure theory of finite-dimensional associative algebras. This theory determines, mostly in a nonconstructive way, the building blocks of these algebras. Our aim is to give polynomial time algorithms to find these building blocks, the radical and the simple direct summands of the radical-free part. The radical algorithm is based on a new, tractable characterisation of the radical. The algorithm for decomposition of semisimple algebras into simple ideals involves (and generalises) factoring polynomials over the ground field. Next, we study the problem of finding zero divisors in finite algebras. We show that thisproblem is in the same complexity class as the problem of factoring polynomials over finte fields. Applications include a polynomial time Las Vegas method to find a common invariant subspace of a set of linear transformations as well as an explicit isomorphism between a given finite simple algebra and a full matrix algebra over a finite field.

A theorem of generalized Cauchy-Pompeiu type on finite-dimensional associative algebras

The author takes up the following problem: suppose given a class of regular hypercomplex functions, i.e., with domains and ranges in a finite-dimensional associative algebra (over the reals) with unity element (so that it is not, in general, a division algebra). Does there nevertheless exist for such functions an integral formula of Cauchy's type, or perhaps of Cauchy-Pompeiu tipe? The author obtains a formula of Cauchy-Pompeiu type (wich, in the commutative case, is similar to a Green identity).

On the multiplication ideal of a nonassociative algebra

In [2] it was shown that generically, finite dimensional algebras over algebraically closed fields contain no nilpotents. We show that for such an algebra the multiplication algebra and the multiplication ideal are one in the same, and that the centroid is contained in the multiplication algebra. In extending the notion of separability to general nonassociative algebras over commutative rings in [3], the author made the latter assertion for algebras that are finitely generated and projective as modules over the base ring. While this assertion does not hold in general (example 2), it does hold in the circumstances considered by that author. Let U be a finite dimensional associative algebra over the algebraically closed field K. By the Wedderburn Principal Theorem [1, p. 47] U has a subalgebra S for which S _~ U/J,

Symmetric systems: structure of the state space

It is shown that every module of every finite-dimensional associative algebra is associated with the state space of some class of symmetric systems. The general structure of symmetric systems is investigated.

On the Multiplicative Complexity of Modules over Associative Algebras

The complexity $L(A,M)$ of a finite-dimensional module M over a finite-dimensional associative algebra A is the number of nonscalar multiplications/divisions of an optimal algorithm to compute the product of an element of the algebra with an element of the module.It is known that \[ L(A,A) \geqq 2 \cdot \dim A - s, \] where s is the number of maximal two-sided ideals of A. We give a generalization of this lower bound to arbitrary A-modules M.