An Algebraic Version of the Monadic System C1

2009 ◽  
pp. 341-349 ◽  
Author(s):  
Jair Minoro Abe ◽  
Kazumi Nakamatsu ◽  
Seiki Akama
Keyword(s):  
Algebraic Version

scholarly journals A semi-algebraic version of Zarankiewicz's problem

2017 ◽  
Vol 19 (6) ◽  
pp. 1785-1810 ◽  
Author(s):  
Jacob Fox ◽  
János Pach ◽  
Adam Sheffer ◽  
Andrew Suk ◽  
Joshua Zahl
Keyword(s):  
Algebraic Version

Double centralizing theorem with respect to q-commutativity relation

2019 ◽  
Vol 18 (01) ◽  
pp. 1950003 ◽  
Author(s):  
Gregor Dolinar ◽  
Alexander Guterman ◽  
Bojan Kuzma ◽  
Olga Markova
Keyword(s):  
Classical Result ◽  
Quantum Physics ◽  
Algebraic Version ◽  
Matrix Formula

Linear algebraic version of celebrated Double Centralizing Theorem states that the set of matrices commuting with all matrices from a centralizer of a given matrix [Formula: see text] coincides with the set of polynomials in [Formula: see text]. We examine the existence of an analogue of this classical result once commutativity is substituted by commutativity up to a factor, which is an important relation in quantum physics.

scholarly journals Irregular perverse sheaves

2021 ◽  
Vol 157 (3) ◽  
pp. 573-624
Author(s):  
Tatsuki Kuwagaki
Keyword(s):  
Abelian Category ◽  
Derived Category ◽  
Perverse Sheaves ◽  
Algebraic Version ◽  
Constructible Sheaves ◽  
Straightforward Generalization ◽  
Novikov Ring

We introduce irregular constructible sheaves, which are ${\mathbb {C}}$-constructible with coefficients in a finite version of the Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic ${\mathcal {D}}$-modules by a modification of D’Agnolo and Kashiwara's irregular Riemann–Hilbert correspondence. The bounded derived category of cohomologically irregular constructible complexes is equipped with the irregular perverse $t$-structure, which is a straightforward generalization of usual perverse $t$-structure, and we prove that its heart is equivalent to the abelian category of holonomic ${\mathcal {D}}$-modules. We also develop the algebraic version of the theory.

Symbols in Berezin quantization for representation operators

2021 ◽  
pp. 296-304
Author(s):  
Vladimir F. Molchanov ◽  
Svetlana V. Tsykina
Keyword(s):  
Homogeneous Space ◽  
Lie Algebra ◽  
Grassmann Manifold ◽  
Homogeneous Spaces ◽  
Finite Multiplicity ◽  
Basic Notion ◽  
Berezin Quantization ◽  
Algebraic Version ◽  
Enveloping Algebra ◽  
Algebra Of Operators

The basic notion of the Berezin quantization on a manifold M is a correspondence which to an operator A from a class assigns the pair of functions F and F^♮ defined on M. These functions are called covariant and contravariant symbols of A. We are interested in homogeneous space M=G/H and classes of operators related to the representation theory. The most algebraic version of quantization — we call it the polynomial quantization — is obtained when operators belong to the algebra of operators corresponding in a representation T of G to elements X of the universal enveloping algebra Env g of the Lie algebra g of G. In this case symbols turn out to be polynomials on the Lie algebra g. In this paper we offer a new theme in the Berezin quantization on G/H: as an initial class of operators we take operators corresponding to elements of the group G itself in a representation T of this group. In the paper we consider two examples, here homogeneous spaces are para-Hermitian spaces of rank 1 and 2: a) G=SL(2;R), H — the subgroup of diagonal matrices, G/H — a hyperboloid of one sheet in R^3; b) G — the pseudoorthogonal group SO_0 (p; q), the subgroup H covers with finite multiplicity the group SO_0 (p-1,q -1)×SO_0 (1;1); the space G/H (a pseudo-Grassmann manifold) is an orbit in the Lie algebra g of the group G.

THE GENETIC VERSUS THE AXIOMATIC METHOD: RESPONDING TO FEFERMAN 1977

2012 ◽  
Vol 6 (1) ◽  
pp. 24-51 ◽  
Author(s):  
ELAINE LANDRY
Keyword(s):  
Category Theory ◽  
Axiomatic Method ◽  
Genetic Method ◽  
Algebraic Version ◽  
Background Theory ◽  
The Way

AbstractFeferman (1977) argues that category theory cannot stand on its own as a structuralist foundation for mathematics: he claims that, because the notions of operation and collection are both epistemically and logically prior, we require a background theory of operations and collections. Recently [2011], I have argued that in rationally reconstructing Hilbert’s organizational use of the axiomatic method, we can construct an algebraic version of category-theoretic structuralism. That is, in reply to Shapiro (2005), we can be structuralists all the way down; we do not have to appeal to some background theory to guarantee the truth of our axioms. In this paper, I again turn to Hilbert; I borrow his (Hilbert, 1900a) distinction between the genetic method and the axiomatic method to argue that even if the genetic method requires the notions of operation and collection, the axiomatic method does not. Even if the genetic method is in some sense epistemically or logically prior, the axiomatic method stands alone. Thus, if the claim that category theory can act as a structuralist foundation for mathematics arises from the organizational use of the axiomatic method, then it does not depend on the prior notions of operation or collection, and so we can be structuralists all the way up.

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