Structure theorem for five-dimensional estimation algebras

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. >

Ever since Brockett, Clark and Mitter introduced the estimation algebra method, it becomes a powerful tool to classify the finite-dimensional filtering system. In this paper, we investigate finite-dimensional estimation algebra with non-maximal rank. The structure of Wong matrix Ω will be focused on since it plays a critical role in the classification of finite-dimensional estimation algebras. In this paper, we first consider general estimation algebra with non-maximal rank and determine the linear structure of the submatrix of Ω by using rank condition and property of Euler operator. In the second part, we proceed to consider the case of linear rank n−1 and prove the linear structure of Ω. Finally, we give the structure of finite-dimensional filters which implies the drift term must be a quadratic function plus a gradient of a smooth function.

This paper considers general finite-dimensional estimation algebras associated with nonlinear filtering systems. General considerations and approaches toward the classification of finite-dimensional estimation algebras are proposed. Some structural results are obtained. The properties of Euler operator and the solutions to an underdetermined partial differential equation, which inevitably arise in an estimation algebra, are studied. These tools and techniques are applied to the study of finite-dimensional estimation algebras with state dimension $2$ to obtain a complete classification result. It is shown that a finite-dimensional estimation algebra with state dimension $2$ can only have dimension less than or equal to $6$. Moreover, the Mitter conjecture and the Levine conjecture hold for finite-dimensional estimation algebras with state dimension $2$.

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.

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.

Of central importance in nonlinear filtering theory is the classification of finite-dimensional estimation algebras, through which we can construct recursive filters. We classify all t-dimensional estimation algebras, t/spl les/4, for arbitrary state space dimension.

Lie algebraic method in nonlinear filtering with state space dimension up to 6

On a new class of finite dimensional estimation algebras

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 underdetermined partial differential equations. The structure of the Wong $\Omega$-matrix is shown to be linear. The fundamental strategy we use in this paper to prove these results is to show that if they were not true, then infinite sequences could be constructed in the finite dimensional estimation algebra.

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 algebraic and homological properties of two classes of infinite-dimensional Hopf algebras over an algebraically closed field k k of characteristic 0 0 . The first class consists of those Hopf k k -algebras that are connected graded as algebras, and the second class are those Hopf k k -algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel’fand–Kirillov dimension. It is shown that if the Hopf algebra H H is a connected graded Hopf algebra of finite Gel’fand–Kirillov dimension n n , then H H is a noetherian domain which is Cohen–Macaulay, Artin–Schelter regular, and Auslander regular of global dimension n n . It has S 2 = I d H S^2 = \mathrm {Id}_H , and it is Calabi–Yau. Detailed information is also provided about the Hilbert series of H H . Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf k k -algebras of finite Gel’fand–Kirillov dimension n n with connected coalgebra, the underlying coalgebra is shown to be Artin–Schelter regular of global dimension n n . Both these classes of Hopf algebras share many features in common with enveloping algebras of finite-dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf k k -algebra H H of Gel’fand–Kirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to U ( g ) U(\mathfrak {g}) for any Lie algebra g \mathfrak {g} .

Let \mathbb M_n be the class of all multiloop algebras of finite dimensional simple Lie algebras relative to n -tuples of commuting finite order automorphisms. It is a classical result that \mathbb M_1 is the class of all derived algebras modulo their centres of affine Kac-Moody Lie algebras. This combined with the Peterson-Kac conjugacy theorem for affine algebras results in a classification of the algebras in \mathbb M_1 . In this paper, we classify the algebras in \mathbb M_2 , and further determine the relationship between \mathbb M_2 and two other classes of Lie algebras: the class of all loop algebras of affine Lie algebras and the class of all extended affine Lie algebras of nullity 2.

Affine Chevalley algebras

Recall that a triangular Hopf algebra A is said to have the Chevalley property if the tensor product of any two simple A-modules is semisimple, or, equivalently, if the radical of A is a Hopf ideal. There are two reasons to study this class of triangular Hopf algebras: First, it contains all known examples of finite-dimensional triangular Hopf algebras; second, it can be, in a sense, completely understood. Namely, it was shown in our previous work with Andruskiewitsch that any finite-dimensional triangular Hopf algebra with the Chevalley property is obtained by twisting of a finite-dimensional triangular Hopf algebra with R-matrix of rank at most 2 which, in turn, is obtained by modifying the group algebra of a finite supergroup. This provides a classification of such Hopf algebras. The goal of this paper is to make this classification more effective and explicit, i.e. to parameterize isomorphism classes of finite-dimensional triangular Hopf algebras with the Chevalley property by group-theoretical objects, similarly to how it was done in our previous work in the semisimple case. This is achieved in Theorem 2.2, where these classes are put in bijection with certain septuples of data. In the semisimple case, the septuples reduce to the quadruples, and we recover the result of our previous paper. In the minimal triangular pointed case, we recover a classification theorem from a previous paper of the second author.

Over fields of prime characteristic the centre of the universal enveloping algebra of a finite-dimensional Lie algebra contains a finitely generated polynomial algebra over which the universal envelope is a finitely generated module.This result, which is due to Curtis, is crucial in certain investigations of finitely generated soluble Lie algebras and motivates the introduction of the class Max-cu, which will be called the class of Curtis algebras, consisting of Lie algebras whose universal envelopes have the property described above.It has been an open question whether the class Max-cu consists of finite-dimensional Lie algebras.This paper gives an affirmative answer to the question.1. Notation and preliminary results.We employ the notation and conventions of Amayo and Stewart [1].If L is a Lie algebra then U(L) denotes its universal enveloping algebra.We denote by F the class of finite-dimensional Lie algebras;L G Max if every subalgebra is finitely generated;L G Max-u if U(L) is right noetherian; L G Max-cu if there exists a finitely generated polynomial subring R of the centre of U(L) and a finite number of elements "...," in U(L) such that U(L) = Rux + + Run (equivalently U(L) is a finitely generated P-module).A denotes the class of abelian Lie algebras.By the results of Amayo and Stewart [2], we have Max-cu < Max-u < Max, and F < Max-u < Max.Over fields of characteristic zero, F n A < Max-cu and over fields of prime characteristic F < Max-cu.Let L be a Lie algebra defined over a field k, K an extension field of k, U(L) the universal envelope of L and L0 = L k K, the K-Lie algebra extension of L. Then U(L0) = U(L) k K. Thus if L G Max-cu and R = k[z" . . ., z J centre of U(L) such that U(L) = Rux + + Ru", then U(L0) = P0w, + + RQun, where P0 = K[z,,. .., zn], whence L0 G Max-cu as a Lie algebra over K.We note also that the class Max-cu is closed under homomorphic images and finite direct sums.Lie algebras in the class Max-cu will be called Curtis algebras.The proof of the following lemma may be found in Curtis [4] and Amayo and Stewart [1,.Lemma 1.1.Every irreducible module of a Curtis algebra is finite-dimensional.