The Classification of Algebras by Dominant Dimension

1968 ◽  
Vol 20 ◽  
pp. 398-409 ◽  
Author(s):  
Bruno J. Mueller
Keyword(s):  
Exact Sequence ◽  
Infinite Sequence ◽  
Dominant Dimension ◽  
Finite Dimensional ◽  
Image Position ◽  
Finite Dimensional Algebras

Nakayama proposed to classify finite-dimensional algebras R over a field according to how long an exact sequenceof projective and injective R-R-bimodules Xi they allow. He conjectured that if there exists an infinite sequence of this type, then R must be quasi-Frobenius; and he proved this when R is generalized uniserial (17).

scholarly journals Auslander-Reiten triangles in subcategories

2008 ◽  
Vol 3 (3) ◽  
pp. 583-601 ◽  
Author(s):  
Peter Jørgensen
Keyword(s):  
Triangulated Category ◽  
Triangulated Categories ◽  
Approximation Properties ◽  
Finite Dimensional ◽  
Image Position ◽  
Finite Dimensional Algebras

AbstractThis paper studies Auslander-Reiten triangles in subcategories of triangulated categories. The main theorem shows that the Auslander-Reiten triangles in a subcategory are closely connected with the approximation properties of the subcategory. Namely, let C be an object in the subcategory C of the triangulated category T, and letbe an Auslander-Reiten triangle in T. Then under suitable assumptions, there is an Auslander-Reiten trianglein C if and only if there is a minimal right-C-approximation of the form.The theory is used to give a new proof of the existence of Auslander-Reiten sequences over finite dimensional algebras.

A Generalization of Commutative and Alternative Rings II

1972 ◽  
Vol 24 (4) ◽  
pp. 728-733 ◽  
Author(s):  
Erwin Kleinfeld
Keyword(s):  
Linear Function ◽  
Commutative Rings ◽  
Finite Dimensional ◽  
Image Position ◽  
Finite Dimensional Algebras

We shall call a linear function on the elements of a ring R skew-symmetric if it vanishes whenever at least two of the variables are equal. Here we shall study rings R of characteristic not 2 which satisfy the following two identities:1(2)is skew-symmetric.Both of these identities hold in alternative rings. The fact that F(w, x, y, z) is skew-symmetric in alternative rings is an important tool in the study of such rings. It is also obvious that both identities hold in commutative rings. But unlike other recent generalizations of commutative and alternative rings it turns out that there exist simple, finite dimensional algebras of degree two which are neither alternative nor commutative and satisfy (1) and (2).

On the Classification of Lie Pseudo-Algebras

1970 ◽  
Vol 22 (5) ◽  
pp. 905-915 ◽  
Author(s):  
Ngö van Quê
Keyword(s):  
Vector Bundle ◽  
Exact Sequence ◽  
Tensor Product ◽  
Vector Bundles ◽  
Canonical Morphism ◽  
Image Position

For every ( differentiable) bundle E over a manifold M, Jk(E) denotes the set of all k-jets of local (differentiable) sections of the bundle E. Jk(E) is a bundle over M such that if X is a section of E, thenis a (differentiable) section of Jk(E). If E is a vector bundle, Jk(E) is a vector bundle and we have the canonical exact sequence of vector bundleswhere Sk(T*) is the symmetric Whitney tensor product of the cotangent vector bundle T* of M. and π is the canonical morphism which associates to each k-jet of section its jet of inferior order.

On derived equivalences and homological dimensions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ming Fang ◽  
Wei Hu ◽  
Steffen Koenig
Keyword(s):  
Global Dimension ◽  
Derived Categories ◽  
Lie Theory ◽  
Restriction Theorem ◽  
Dominant Dimension ◽  
Derived Equivalences ◽  
Finite Dimensional ◽  
Cellular Algebras ◽  
Homological Dimensions ◽  
Finite Dimensional Algebras

AbstractUnlike Hochschild (co)homology and K-theory, global and dominant dimensions of algebras are far from being invariant under derived equivalences in general. We show that, however, global dimension and dominant dimension are derived invariant when restricting to a class of algebras with anti-automorphisms preserving simples. Such anti-automorphisms exist for all cellular algebras and in particular for many finite-dimensional algebras arising in algebraic Lie theory. Both dimensions then can be characterised intrinsically inside certain derived categories. On the way, a restriction theorem is proved, and used, which says that derived equivalences between algebras with positive ν-dominant dimension always restrict to derived equivalences between their associated self-injective algebras, which under this assumption do exist.

scholarly journals A derived equivalence for a degree 6 del Pezzo surface over an arbitrary field

2011 ◽  
Vol 8 (3) ◽  
pp. 481-492 ◽  
Author(s):  
M. Blunk ◽  
S.J. Sierra ◽  
S. Paul Smith
Keyword(s):  
Derived Categories ◽  
Pezzo Surface ◽  
Arbitrary Field ◽  
Derived Equivalence ◽  
Del Pezzo Surface ◽  
Finite Dimensional ◽  
Semisimple Part ◽  
Del Pezzo ◽  
Image Position

AbstractLet S be a degree six del Pezzo surface over an arbitrary field F. Motivated by the first author's classification of all such S up to isomorphism [3] in terms of a separable F-algebra B×Q×F, and by his K-theory isomorphism Kn(S) ≅ Kn(B×Q×F) for n ≥ 0, we prove an equivalence of derived categorieswhere A is an explicitly given finite dimensional F-algebra whose semisimple part is B×Q×F.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

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