Structure of the unitary valuation 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.

This chapter is a brief introduction into the structure of algebras, mostly finite dimensional, over any field k. The main contents are the Wedderburn theorems for a finite dimensional algebras A over an algebraically closed field k. If A has no nilpotent ideals ≠ 0, then A is a finite product of total matrix algebras over k. In this case, the set d (A) of degrees of the total matrix algebras is a complete set of invariants of A. Thus, 13.7 two finite dimensional semiprime algebras A and B over k are isomorphic if and only if d (A) = d (B).

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.

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.

Affine Chevalley 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. >

Hom-L-R-smash products, Hom-diagonal crossed products and the Drinfeld double of a Hom-Hopf algebra

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

In representation theory, the existence of a ℤ+-grading on a related finite dimensional algebra often plays an important role. For example, such a grading arises from the Koszul structure of the finite dimensional algebra representing the principal block of the BGG category 𝒪 associated to a complex semisimple Lie algebra. But Koszul gradings in positive characteristic have proved elusive. For example, except for small values of a positive integer n, it is not known if the Schur algebra S(n, n) has such a Koszul grading, assuming the characteristic p of the base field satisfies p ≥ n, though this grading would suffice to establish Lusztig's character formula for these algebras. (And even though the character formula is known for p sufficiently large [H. Andersen, J. Jantzen and W. Soergel, Representations of Quantum Groups at a pth Root of Unity and of Semisimple Groups in Characteristic p, Astérique, Vol. 220, 1994], it is not known if the Schur algebra is Koszul for p sufficiently large.) This paper introduces ℤ/2-gradings on quasi-hereditary algebras, and shows that these gradings are almost as useful as a full ℤ+-grading, while being possibly much easier to find. We define the notion of a ℤ/2-based Kazhdan–Lusztig theory, which appears to be more flexible than, and generalizes, the notion of a Kazhdan–Lusztig theory (as first defined in [E. Cline, B. Parshall and L. Scott, Abstract Kazhdan–Lusztig theories, Tôhoku Math. J. 45 (1993), 511–534]). However, its existence suffices, as was the case with the original notion, to establish character formulas in the standard settings, determine Extn-groups, and show that homological duals behave well. Finally, we present some suggestive symmetric group examples involving Schur algebras which were an outgrowth of this work.

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.

In (2009), Towers [10] presented the notion of c-ideality of a subalgebra of a Lie algebra, and gave some characterizations of solvable and supersolvable Lie algebras. In this article, we further investigate the influence of c-ideality of some subalgebras on the structure of Lie algebras. We also obtain some equivalent conditions for supersolvability of a finite dimensional Lie algebra.

D. A. Towers introduced the notion of ideal index of a maximal subalgebra of a Lie algebra, and used it to analyze the influence of maximal subalgebras on the structure of a finite dimensional Lie algebras. In this article, we generalize the ideal index from maximal subalgebras to all subalgebras, and obtain some new characterizations of solvable and supersolvable Lie algebras by the ideal indices of some certain subalgebras.

In this chapter we investigate the ideal structure of Leavitt path algebras. We start by describing the natural \(\mathbb{Z}\)-grading on L K (E). We then present the Reduction Theorem; this result describes how elements of L K (E) may be transformed in some specified way to either a vertex or a cycle without exits. Numerous consequences are discussed, including the Uniqueness Theorems. We then establish in the Structure Theorem for Graded Ideals a precise relationship between graded ideals and explicit sets of idempotents (arising from hereditary and saturated subsets of vertices, together with breaking vertices). With this description of the graded ideals having been achieved, we focus in the remainder of the chapter on the structure of all ideals. We achieve in the Structure Theorem for Ideals an explicit description of the entire ideal structure of L K (E) (including both the graded and non-graded ideals) for an arbitrary graph E and field K. This result utilizes the Structure Theorem for Graded Ideals together with the analysis of the ideal generated by vertices which lie on cycles having no exits. A number of ring-theoretic results follow almost immediately from the Structure Theorem for Ideals, including the Simplicity Theorem. Along the way, we describe the socle of a Leavitt path algebra, and we achieve a description of the finite dimensional Leavitt path algebras.

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