On theta pair for a proper subalgebra

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

We study the notion of the Killing form for Frobenius Lie algebras of dimension . The Killing form is a symmetric bilinear form on a finite dimensional Lie algebra over a field defined by where is denoted the trace and is an adjoint representation of . A Lie algebras is said to be semisimple if it has the nondegenerate Killing form. The research aims to consider the criterion for semisimplicity of Frobenius Lie algebras of dimension by using the Killing form. The results show that each Frobenius Lie algebra of dimension and is not semisimple since the the Killing form is degenerate or in other words, a determinant of a representation matrix of the Killing form is equal to zero. For the future research, it is still an open problem to consider the general formulas of the Killing form for higher dimensional Frobenius Lie algebra whether degenerate or nondegenerate such that the semisimplicity of a Lie algebra can be considered. We conjecture that each finite dimensional real Frobenius Lie algebra is not semisimple.

The authors present a new way to construct finite dimensional estimation algebra of nonmaximal rank. For a given finite dimensional estimation algebra E, one can construct a finite dimensional estimation algebra E' of nonmaximal rank which is isomorphic to E. The interesting fact is that even if E comes from a linear filtering system, there are arbitrary nonlinear filtering systems associated to E. Since every finite dimensional estimation algebra gives rise to a finite dimensional filter, we have constructed quite a large class of finite dimensional nonlinear filters.

A Decomposition Theory for Differentiable Systems

R. W. Brockett (1983) proposed to classify all finite dimensional estimation algebras. An affirmative solution to Brockett's problem will allow construction of all possible finite dimensional recursive filters from the Lie algebraic point of view. The concept of an estimation algebra with maximal rank was introduced by L. F. Tam et al. (1990). This is the most important general subclass of estimation algebras. S. S.-T. Yau and W.L. Chiou (1991) have already classified all maximal rank finite dimensional estimation algebras with state space dimension at most 2. Here, the case for state space dimension 3 is studied. >

We study Jordan-Lie inner ideals of finite dimensional associative algebras and the corresponding Lie algebras and prove that they admit Levi decompositions. Moreover, we classify Jordan-Lie inner ideals satisfying a certain minimality condition and show that they are generated by pairs of idempotents.

Some new simple Lie algebras in characteristic 2

In this paper, we study the structure of finite dimensional estimation algebras with state dimension 3 and rank 2 arising from a nonlinear filtering system by using the theories of the Euler operator and under-determined partial differential equations. It is proved that if the estimation algebra contains a degree two polynomial, then the Wong Ω-matrix must be a constant matrix. Moreover, all functions in the estimation algebra must be linear functions.

Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in Hausser and Nill ((Hausser, F., Nill, F. (1999a). Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys. 11:553–629) and (Hausser, F., Nill, F. (1999b). Doubles of quasi-quantum groups. Comm. Math. Phys. 199:547–589)). In this note, we first generalize a result of Majid (Majid, S. (1991). Doubles of quasitriangular Hopf algebras. Comm. Algebra 19:3061–3073) for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of Bulacu and Nauwelaerts (Bulacu, D., Nauwelaerts, E. (2002). Radford's biproduct for quasi-Hopf algebras and bosonization. J. Pure Appl. Algebra 179:1–42.).

Factorizable quasi-Hopf algebras—applications

Finite subgroups occuring in finite-dimensional division algebras

The structure of finite quotients of “large” subgroups of the multiplicative group of a finite dimensional division algebra is interesting and is related to the Margulis-Platonov conjecture. We develop machinery to handle such quotients and we conjecture that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. The proofs rely on Wedderburn's Factorisation Theorem.

We prove that if H is a semisimple Hopf algebra, then the action of the Drinfeld double D(H) on H and the action of the character algebra on H form a commuting pair. This result and a result of G. I. Kats imply that the dimension of every simple D(H)-submodule of H is a divisor of dim (H). Let H be a finite dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, D(H) be the Drinfeld double of H, and C(H) be the character algebra of H. C(H) is spanned by the characters of H-modules and is an associative subalgebra of H*. It is known that D(H) acts on H and that C(H) acts on H by the restriction of the action - of H* on H (these actions will be recalled below). The purpose of this note is to prove that these two actions form a commuting pair. Using this result, we prove that the dimension of every simple D(H)-submodule of H is a divisor of dim(H). It would be interesting if there exists an analog of this commuting pair in the context of Poisson Lie groups. We first recall the construction of the Drinfeld double (cf. [D], [M]) and fix necessary notations. Let H be a finite dimensional Hopf algebra over a field k (here we do not need any additional assumptions on H and k). The Drinfeld double of H, denoted by D(H), as a vector space, is the tensor space H* 0 H. The comultiplication of D(A) is given by A(f 0 a)= S (f(2) 0 a(,)) 0 (.f(i) 0 a(2)) E D(H) 0 D(H), where Af = f(l) (0 f(2), Aa =a(l) ( a(2) are comultiplictions in H and H* respectively. The multiplication in D(H) is defined as follows: for f 0 a and g 0 b in D(H), (1) (f oa)(gob) f (a(,) D 9(2)) 0 (a(2) g is the action of H on H* given by a > g = a(l)' 9 -Sla(2) and a <g is the right action of H* on H given by a< g = S-l19(l) a9(2) The notations and mean the usual left and right actions of H on H*, i.e., for a E H and g E H*, a _ gvl .2 a) LT*, .a / 2 (g_ ) Received by the editors December 6, 1995 and, in revised form, April 30, 1996. 1991 Mathematics Subject Classification. Primary 16W30. ( 1997 American Mathematical Society 2847 This content downloaded from 207.46.13.128 on Tue, 06 Sep 2016 06:06:51 UTC All use subject to http://about.jstor.org/terms

Irreducible strongly indecomposable torsion-free abelian groups of finite rank are called Reid groups, and a full subring R of a division algebra D is called Reid-realizable if it is isomorphic to the endomorphism ring of a Reid group. The main results is that a full subring R of a finite dimensional division algebra D is Reid-realizable if for infinitely many prime numbers p the quotient of the p-adic localizations D p * / div R p * > either is the product of at least two p-adic number fields, or is not a product of division algebras. In particular, a full subring of a finite dimensional central division algebra over an algebraic number field is Reid-realizable if it is locally free for infinitely many prime numbers.

The present paper is the third contribution of a series of works, where we investigate pseudo-bosonic operators and their connections with finite dimensional Lie algebras.We show that all finite dimensional nilpotent Lie algebras (over the complex field) can be realized by central extensions of Lie algebras of pseudo-bosonic operators.This result is interesting, because it provides new examples of dynamical systems for nilpotent Lie algebras of any dimension.One could ask whether these operators are intrinsic with the notion of nilpotence or not, but this is false.In fact we exibit both a simple Lie algebra and a solvable nonnilpotent Lie algebra, which can be realized in terms of pseudo-bosonic operators.