scholarly journals On finite dimensionality of mixed Tate motives

2008 ◽  
Vol 4 (1) ◽  
pp. 145-161 ◽  
Author(s):  
Shahram Biglari
Keyword(s):  
Grothendieck Group ◽  
Ring Structure ◽  
Triangulated Category ◽  
Cohomology Groups ◽  
Weight Filtration ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Tate Motives

AbstractWe prove a few results concerning the notion of finite dimensionality of mixed Tate motives in the sense of Kimura and O'Sullivan. It is shown that being oddly or evenly finite dimensional is equivalent to vanishing of certain cohomology groups defined by means of the Levine weight filtration. We then explain the relation to the Grothendieck group of the triangulated category D of mixed Tate motives. This naturally gives rise to a λ–ring structure on K0(D).

scholarly journals Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky versus Hanamura

2008 ◽  
Vol 8 (1) ◽  
pp. 39-97 ◽  
Author(s):  
M. V. Bondarko
Keyword(s):  
Spectral Sequence ◽  
Finite Fields ◽  
Maximal Length ◽  
Triangulated Category ◽  
Spectral Sequences ◽  
Motivic Cohomology ◽  
Weight Filtration ◽  
Singular Cohomology ◽  
Tate Motives

AbstractWe describe explicitly the Voevodsky's triangulated category of motives $\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}$ (and give a ‘differential graded enhancement’ of it). This enables us to able to verify that DMgm ℚ is (anti)isomorphic to Hanamura's $\mathcal{D}$(k).We obtain a description of all subcategories (including those of Tate motives) and of all localizations of $\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}$. We construct a conservative weight complex functor $t:\smash{\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}}\to\smash{K^{\mathrm{b}}(\operatorname{Chow}^{\mathrm{eff}})}$; t gives an isomorphism $K_0(\smash{\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}})\to\smash{K_0(\operatorname{Chow}^{\mathrm{eff}})}$. A motif is mixed Tate whenever its weight complex is. Over finite fields the Beilinson–Parshin conjecture holds if and only if tℚ is an equivalence.For a realization D of $\operatorname{DM}^{\mathrm{eff}}_{\mathrm{gm}}$ we construct a spectral sequence S (the spectral sequence of motivic descent) converging to the cohomology of an arbitrary motif X. S is ‘motivically functorial’; it gives a canonical functorial weight filtration on the cohomology of D(X). For the ‘standard’ realizations this filtration coincides with the usual one (up to a shift of indices). For the motivic cohomology this weight filtration is non-trivial and appears to be quite new.We define the (rational) length of a motif M; modulo certain ‘standard’ conjectures this length coincides with the maximal length of the weight filtration of the singular cohomology of M.

Grothendieck Groups of a Class of Quantum Doubles

2008 ◽  
Vol 15 (03) ◽  
pp. 431-448 ◽  
Author(s):  
Yun Zhang ◽  
Feng Wu ◽  
Ling Liu ◽  
Hui-Xiang Chen
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Grothendieck Group ◽  
Ring Structure ◽  
Quantum Double ◽  
Simple Modules ◽  
Loewy Length ◽  
Finite Dimensional ◽  
Grothendieck Groups

Let k be a field and An(ω) be the Taft n2-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(An(ω)) of An(ω) is a ribbon Hopf algebra. We have constructed an n4-dimensional Hopf algebra Hn(p,q) which is isomorphic to D(An(ω)) if p ≠ 0 and q = ω-1, and studied the irreducible and finite-dimensional representations of Hn(1,q). In this paper, we continue our study of Hn(1,q), examine the Grothendieck group G0(Hn(1,q)) ≅ G0(D(An(ω)), and describe its ring structure. We also give the Loewy length of the tensor product of two simple modules over Hn(1,q).

scholarly journals How low can vacuum energy go when your fields are finite-dimensional?

2019 ◽  
Vol 28 (14) ◽  
pp. 1944006
Author(s):  
ChunJun Cao ◽  
Aidan Chatwin-Davies ◽  
Ashmeet Singh
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Degrees Of Freedom ◽  
Vacuum Energy ◽  
Quantum Field ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Klein Gordon ◽  
Dimensional Version

According to the holographic bound, there is only a finite density of degrees of freedom in space when gravity is taken into account. Conventional quantum field theory does not conform to this bound, since in this framework, infinitely many degrees of freedom may be localized to any given region of space. In this paper, we explore the viewpoint that quantum field theory may emerge from an underlying theory that is locally finite-dimensional, and we construct a locally finite-dimensional version of a Klein–Gordon scalar field using generalized Clifford algebras. Demanding that the finite-dimensional field operators obey a suitable version of the canonical commutation relations makes this construction essentially unique. We then find that enforcing local finite dimensionality in a holographically consistent way leads to a huge suppression of the quantum contribution to vacuum energy, to the point that the theoretical prediction becomes plausibly consistent with observations.

On lambda operations on mixed motives

2013 ◽  
Vol 12 (2) ◽  
pp. 381-404 ◽  
Author(s):  
Shahram Biglari
Keyword(s):  
Ring Structure ◽  
Triangulated Category ◽  
Grothendieck Ring ◽  
Basic Properties ◽  
Natural Notion ◽  
Mixed Motives

AbstractWe study the natural λ-ring structure on the Grothendieck ring of the triangulated category of mixed motives. Basic properties of a natural notion of characteristic-like series are developed in the context of equivariant objects.

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.

scholarly journals GENERIC IRREDUCIBLE REPRESENTATIONS OF FINITE-DIMENSIONAL LIE SUPERALGEBRAS

1994 ◽  
Vol 05 (03) ◽  
pp. 389-419 ◽  
Author(s):  
IVAN PENKOV ◽  
VERA SERGANOVA
Keyword(s):  
Irreducible Module ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Irreducible Representations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Necessary And Sufficient

A theory of highest weight modules over an arbitrary finite-dimensional Lie superalgebra is constructed. A necessary and sufficient condition for the finite-dimensionality of such modules is proved. Generic finite-dimensional irreducible representations are defined and an explicit character formula for such representations is written down. It is conjectured that this formula applies to any generic finite-dimensional irreducible module over any finite-dimensional Lie superalgebra. The conjecture is proved for several classes of Lie superalgebras, in particular for all solvable ones, for all simple ones, and for certain semi-simple ones.

New exact non-linear filters by large Lie algebras

1984 ◽  
Vol 16 (1) ◽  
pp. 14-14
Author(s):  
V. E. Beneš
Keyword(s):  
Lie Algebras ◽  
Finite Dimension ◽  
Linear Filters ◽  
Finite Dimensional ◽  
Non Linear ◽  
Finite Dimensionality ◽  
Linear Differential

The ideas previously used (Beneš (1981)) to construct some finite-dimensional non-linear filters also yield related new filters of finite dimension with arbitrarily large bases; this is because the finite dimensionality is not destroyed by insertion of noiseless linear differential operations on the observations.

Relative Quotient Triangulated Categories

2014 ◽  
Vol 21 (02) ◽  
pp. 195-206 ◽  
Author(s):  
Shengyong Pan
Keyword(s):  
Triangulated Category ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Stable Category ◽  
Gorenstein Algebra ◽  
Finite Dimensional ◽  
Frobenius Category ◽  
Relative Version ◽  
Relative Derived Category ◽  
Algebra Over A Field

Let A be a finite dimensional algebra over a field k. We consider a subfunctor F of [Formula: see text], which has enough projectives and injectives such that [Formula: see text] is of finite type, where [Formula: see text] denotes the set of F-projectives. One can get the relative derived category [Formula: see text] of A-mod. For an F-self-orthogonal module TF, we discuss the relation between the relative quotient triangulated category [Formula: see text] and the relative stable category of the Frobenius category of TF-Cohen-Macaulay modules. In particular, for an F-Gorenstein algebra A and an F-tilting A-module TF, we get a triangle equivalence between [Formula: see text] and the relative stable category of TF-Cohen-Macaulay modules. This gives the relative version of a result of Chen and Zhang.

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