Two-Sided Ideals

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.

In this paper, we present results concerning the structure of the ideals in the Leavitt path algebra of a (countable) directed graph with coefficients in an integral domain, such as, describing the set of generators for an ideal; the necessary and sufficient conditions for an ideal to be prime; the necessary and sufficient conditions for a Leavitt path algebra to be simple. Besides, some other interesting properties of ideal structure in a Leavitt path algebra are also mentioned.

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.

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).

The existence of a well-behaved dimension of a finite von Neumann algebra (see Luck, J Reine Angew Math 495:135–162, 1998) has lead to the study of such a dimension of finite Baer *-rings (see Vas, J Algebra 289(2):614–639, 2005) that satisfy certain *-ring axioms (used in Berberian, 1972). This dimension is closely related to the equivalence relation $ {\sim^{\raisebox{-.1ex}[0pc][0pc]{\scriptsize{*}}}}$ on projections defined by $p{\sim^{\raisebox{-.1ex}[0pc][0pc]{\scriptsize{*}}}} q$ iff p = xx * and q = x * x for some x. However, the equivalence ${\sim^{{\raisebox{.3ex}[0pc][0pc]{\scriptsize\em{a}}}}}$ on projections (or, in general, idempotents) defined by $p{\sim^{{\raisebox{.3ex}[0pc][0pc]{\scriptsize\em{a}}}}} q$ iff p = xy and q = yx for some x and y, can also be relevant. There were attempts to unify the two approaches (see Berberian, preprint, 1988)). In this work, our agenda is three-fold: (1) We study assumptions on a ring with involution that guarantee the existence of a well-behaved dimension defined for any general equivalence relation on projections ~. (2) By interpreting ~ as ${\sim^{{\raisebox{.3ex}[0pc][0pc]{\scriptsize\em{a}}}}},$ we prove the existence of a well-behaved dimension of strongly semihereditary *-rings with positive definite involution. This class is wider than the class of finite Baer *-rings with dimension considered in the past: it includes some non Rickart *-rings. Moreover, none of the *-ring axioms from Berberian (1972) and Vas (J Algebra 289(2):614–639, 2005) are assumed. (3) As the first corollary of (2), we obtain dimension of noetherian Leavitt path algebras over positive definite fields. Secondly, we obtain dimension of a Baer *-ring R satisfying the first seven axioms from Vas (J Algebra 289(2):614–639, 2005) (in particular, dimension of finite AW *-algebras). Assuming the eight axiom as well, R has dimension for ${\sim^{\raisebox{-.1ex}[0pc][0pc]{\scriptsize{*}}}}$ also and the two dimensions coincide. While establishing (2), we obtain some additional results for a right strongly semihereditary ring R: we prove that every finitely generated R-module M splits as a direct sum of a finitely generated projective module and a singular module; we describe right strongly semihereditary rings in terms of relations between their maximal and total rings of quotients; and we characterize extending Leavitt path algebras over finite graphs.

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.

In this paper we study some classes of rings which have a finite lattice of preradicals. We characterize commutative rings with this condition as finite representation type rings, i.e., artinian principal ideal rings. In general, it is easy to see that the lattice of preradicals of a left pure semisimple ring is a set, but it may be infinite. In fact, for a finite dimensional path algebra Λ over an algebraically closed field we prove that Λ-pr is finite if and only if its quiver is a disjoint union of finite quivers of type An; hence there are path algebras of finite representation type such that its lattice of preradicals is an infinite set. As an example, we describe the lattice of preradicals over Λ = kQ when Q is of type An and it has the canonical orientation

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. >

In this work, we realize Leavitt path algebras as partial skew groupoid rings. This yields a free path groupoid grading on Leavitt path algebras. Using this grading, we characterize free path groupoid graded isomorphisms of Leavitt path algebras that preserves generators.

The multiplicative ideal theory of Leavitt path algebras

In the past, it has been shown that the Leavitt path algebra L(E) = L K (E) of a graph E over a field K is left and right noetherian if and only if the graph E is finite and no cycle of E has an exit. If Q(E) = Q K (E) denotes the regular algebra over L(E), we prove that these two conditions are further equivalent with any of the following: L(E) contains no infinite set of orthogonal idempotents, L(E) has finite uniform dimension, L(E) is directly finite, Q(E) is directly finite, Q(E) is unit-regular, Q(E) is left (right) self-injective and a few more equivalences. In addition, if the involution on the field K is positive definite, these conditions are equivalent with the following: the involution * extends from L(E) to Q(E), Q(E) is *-regular, Q(E) is finite, Q(E) is the maximal (total or classical) symmetric ring of quotients of L(E), the maximal right ring of quotients of L(E) is the same as the total (or classical) left ring of quotients of L(E), every finitely generated nonsingular L(E)-module is projective, and the matrix ring M n (L(E)) is strongly Baer for every n. It may not be surprising that a noetherian Leavitt path algebra has these properties, but a more interesting fact is that these properties hold only if a Leavitt path algebra is noetherian (i.e. E is a finite no-exit graph). Using some of these equivalences, we give a specific description of the inverse of the isomorphism V (L(E)) → V (Q(E)) of monoids of equivalence classes of finitely generated projective modules of L(E) and Q(E) for noetherian Leavitt path algebras. We also prove that two noetherian Leavitt path algebras are isomorphic as rings if and only if they are isomorphic as *-algebras. This answers in affirmative the Isomorphism Conjecture for the class of noetherian Leavitt path algebras: if $${L_{\mathbb{C}}(E)}$$ and $${L_{\mathbb{C}}(F)}$$ are noetherian Leavitt path algebras, then $${L_{\mathbb{C}}(E) \cong L_{\mathbb{C}}(F)}$$ as rings implies $${C^{*} (E) \cong C^{*} (F)}$$ as *-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.

Let FΛ be a finite dimensional path algebra of a quiver Λ over a field F. Let L and R denote the varieties of all left and right FΛ-modules respectively. We prove the equivalence of the following statements. The subvariety lattice of L is a sublattice of the subquasivariety lattice of L. The subquasivariety lattice of R is distributive. Λ is an ordered forest.

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.

S. Alesker has shown that if G is a compact subgroup of O(n) acting transitively on the unit sphere Sn-1, then the vector space ValG of continuous, translation-invariant, G-invariant convex valuations on Rn has the structure of a finite dimensional graded algebra over R satisfying Poincaré duality. We show that the kinematic formulas for G are determined by the product pairing. Using this result we then show that the algebra ValU(n) is isomorphic to R[s, t]/(fn+1, fn+2), where s, t have degrees 2 and 1 respectively, and the polynomial fi is the degree i term of the power series log(1 + s + t).