VARIETIES OF REPRESENTATIONS OF GROUPS AND VARIETIES OF ASSOCIATIVE ALGEBRAS

This paper is tightly connected with the book [Varieties of Group Representations. General Theory, Connections and Applications (Zinatne, Riga, 1983) (in Russian)]. In the paper we prove new results in the spirit of the above-mentioned book. They are related to dimension subgroups, varieties of representations of groups and varieties of associative algebras. The main emphasis is put on the varieties of representations of groups induced by the varieties of associative algebras. We provide the reader with the list of open problems. For many reasons we consciously included in the text a brief review of the basic definitions and results from the theory of varieties of representations described in the book [Varieties of Group Representations. General Theory, Connections and Applications].

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Congruences induced by transitive representations of inverse semigroups

Glasgow Mathematical Journal ◽

10.1017/s0017089500006649 ◽

1987 ◽

Vol 29 (1) ◽

pp. 21-40 ◽

Cited By ~ 2

Author(s):

Mario Petrich ◽

Stuart Rankin

Keyword(s):

General Theory ◽

Inverse Semigroup ◽

Symmetric Group ◽

Transitive Group ◽

Inverse Semigroups ◽

Group Representations ◽

Symmetric Inverse Semigroup ◽

The Right ◽

Special Case

Transitive group representations have their analogue for inverse semigroups as discovered by Schein [7]. The right cosets in the group case find their counterpart in the right ω-cosets and the symmetric inverse semigroup plays the role of the symmetric group. The general theory developed by Schein admits a special case discovered independently by Ponizovskiǐ [4] and Reilly [5]. For a discussion of this topic, see [1, §7.3] and [2, Chapter IV].

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Some questions on the general theory of representations of groups

Twelve Papers on Topology, Algebra and Number Theory - American Mathematical Society Translations: Series 2 ◽

10.1090/trans2/052/10 ◽

1966 ◽

pp. 171-200

Author(s):

B. I. Plotkin

Keyword(s):

General Theory ◽

Representations Of Groups

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Synergy in the Theories of Gröbner Bases and Path Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1993-041-8 ◽

1993 ◽

Vol 45 (4) ◽

pp. 727-739 ◽

Cited By ~ 33

Author(s):

Daniel R. Farkas ◽

C. D. Feustel ◽

Edward L. Green

Keyword(s):

General Theory ◽

Gröbner Basis ◽

Gröbner Bases ◽

Grobner Bases ◽

Polynomial Rings ◽

Grobner Basis ◽

Associative Algebras ◽

Path Algebras

AbstractA general theory for Grôbner basis in path algebras is introduced which extends the known theory for commutative polynomial rings and free associative algebras.

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Some open problems on locally finite or locally nilpotent derivations and ℰ-derivations

Communications in Contemporary Mathematics ◽

10.1142/s0219199717500560 ◽

2018 ◽

Vol 20 (04) ◽

pp. 1750056 ◽

Cited By ~ 4

Author(s):

Wenhua Zhao

Keyword(s):

Commutative Ring ◽

Characteristic Zero ◽

Associative Algebras ◽

Open Problems ◽

Integral Domains ◽

Locally Finite ◽

Locally Nilpotent ◽

Base Ring ◽

Lf Formula ◽

Algebra Endomorphism

Let [Formula: see text] be a commutative ring and [Formula: see text] an [Formula: see text]-algebra. An [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] is an [Formula: see text]-linear map of the form [Formula: see text] for some [Formula: see text]-algebra endomorphism [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the identity map of [Formula: see text]. In this paper, we discuss some open problems on whether or not the image of a locally finite (LF) [Formula: see text]-derivation or [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] is a Mathieu subspace [W. Zhao, Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra 214 (2010) 1200–1216; Mathieu subspaces of associative algebras, J. Algebra 350(2) (2012) 245–272] of [Formula: see text], and whether or not a locally nilpotent (LN) [Formula: see text]-derivation or [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] maps every ideal of [Formula: see text] to a Mathieu subspace of [Formula: see text]. We propose and discuss two conjectures which state that both questions above have positive answers if the base ring [Formula: see text] is a field of characteristic zero. We give some examples to show the necessity of the conditions of the two conjectures, and discuss some positive cases known in the literature. We also show some cases of the two conjectures. In particular, both the conjectures are proved for LF or LN algebraic derivations and [Formula: see text]-[Formula: see text]-derivations of integral domains of characteristic zero.

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Problems in the General Theory of Relativity and Theory of Group Representations

10.1007/978-1-4684-0676-4 ◽

1979 ◽

Keyword(s):

General Theory ◽

Theory Of Relativity ◽

General Theory Of Relativity ◽

Group Representations

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Extreme self-sacrifice beyond fusion: Moral expansiveness and the special case of allyship

Behavioral and Brain Sciences ◽

10.1017/s0140525x18001620 ◽

2018 ◽

Vol 41 ◽

Cited By ~ 2

Author(s):

Daniel Crimston ◽

Matthew J. Hornsey

Keyword(s):

General Theory ◽

Biological Markers ◽

Natural Environment ◽

Relevant Dimension ◽

Special Case

AbstractAs a general theory of extreme self-sacrifice, Whitehouse's article misses one relevant dimension: people's willingness to fight and die in support of entities not bound by biological markers or ancestral kinship (allyship). We discuss research on moral expansiveness, which highlights individuals’ capacity to self-sacrifice for targets that lie outside traditional in-group markers, including racial out-groups, animals, and the natural environment.

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