Journal of Algebra最新文献

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Classification of charge-conserving loop braid representations

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-12-21 DOI: 10.1016/j.jalgebra.2024.12.011

Paul Martin , Eric C. Rowell , Fiona Torzewska

Here a loop braid representation is a monoidal functor

F

from the loop braid category

L

to a suitable target category, and is N-charge-conserving if the target is the category

{Match}^{N}

of charge-conserving matrices (specifically

{Match}^{N}

is the same rank-N charge-conserving monoidal subcategory of the monoidal category

Mat

used to classify braid representations in [27]) with

F

strict, and surjective on

N

, the object monoid. We classify and construct all such representations. In particular we prove that representations at given N fall into varieties indexed by a set in bijection with the set of pairs of plane partitions of total degree N.

引用次数: 0

Motivic Euler characteristic of nearby cycles and a generalised quadratic conductor formula

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-12-13 DOI: 10.1016/j.jalgebra.2024.12.010

Ran Azouri

We compute the motivic Euler characteristic of Ayoub's nearby cycles spectrum in terms of strata of a semi-stable reduction, for a degeneration to multiple semi-quasi-homogeneous singularities. This allows us to compare the local picture at the singularities with the global conductor formula for hypersurfaces developed by Levine, Pepin Lehalleur and Srinivas, revealing that the formula is local in nature, thus extending it to the more general setting considered in this paper. The result is a quadratic refinement to the Milnor number formula with multiple singularities.

引用次数: 0

Isolated torus invariants and automorphism groups of rigid varieties

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-12-12 DOI: 10.1016/j.jalgebra.2024.12.009

Viktoriia Borovik , Sergey Gaifullin

Perepechko and Zaidenberg conjectured that the neutral component of the automorphism group of a rigid affine variety is a torus. We prove this conjecture for toric varieties and varieties with a torus action of complexity one. We also obtain a criterion for an m-suspension over a rigid variety to be rigid (for every rigid variety and every regular function). Additionally, we study the automorphism group of m-suspensions satisfying this criterion.

引用次数: 0

A categorical interpretation of Morita equivalence for dynamical von Neumann algebras

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-12-10 DOI: 10.1016/j.jalgebra.2024.12.008

Joeri De Ro

<div><div>Let <math><mi>G</mi></math> be a locally compact quantum group and <math><mo>(</mo><mi>M</mi><mo>,</mo><mi>α</mi><mo>)</mo></math> a <math><mi>G</mi></math>-<math><msup><mrow><mi>W</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math>-algebra. The object of study of this paper is the <math><msup><mrow><mi>W</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math>-category <math><msup><mrow><mi>Rep</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></math> of normal, unital <math><mi>G</mi></math>-representations of M on Hilbert spaces endowed with a unitary <math><mi>G</mi></math>-representation. This category has a right action of the category <math><mi>Rep</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>Rep</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><mi>C</mi><mo>)</mo></math> for which it becomes a right <math><mi>Rep</mi><mo>(</mo><mi>G</mi><mo>)</mo></math>-module <math><msup><mrow><mi>W</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math>-category. Given another <math><mi>G</mi></math>-<math><msup><mrow><mi>W</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math>-algebra <math><mo>(</mo><mi>N</mi><mo>,</mo><mi>β</mi><mo>)</mo></math>, we denote the category of normal ⁎-functors <math><msup><mrow><mi>Rep</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><mi>N</mi><mo>)</mo><mo>→</mo><msup><mrow><mi>Rep</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></math> compatible with the <math><mi>Rep</mi><mo>(</mo><mi>G</mi><mo>)</mo></math>-module structure by <math><msub><mrow><mi>Fun</mi></mrow><mrow><mi>Rep</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>(</mo><msup><mrow><mi>Rep</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><mi>N</mi><mo>)</mo><mo>,</mo><msup><mrow><mi>Rep</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo><mo>)</mo></math> and we denote the category of <math><mi>G</mi></math>-M-N-correspondences, studied in [5], by <math><msup><mrow><mi>Corr</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><mi>M</mi><mo>,</mo><mi>N</mi><mo>)</mo></math>. We prove that there are canonical functors <math><mi>P</mi><mo>:</mo><msup><mrow><mi>Corr</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><mi>M</mi><mo>,</mo><mi>N</mi><mo>)</mo><mo>→</mo><msub><mrow><mi>Fun</mi></mrow><mrow><mi>Rep</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mo>(</mo><msup><mrow><mi>Rep</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><mi>N</mi><mo>)</mo><mo>,</mo><msup><mrow><mi>Rep</mi></mrow><mrow><mi>G</mi></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo><mo>)</mo></math> and <math><mi>Q</mi><mo>:</mo><msub><mrow><mi>Fun</mi></mrow><mrow><mi>Rep</mi><mo>(</mo><mi>G</mi><mo>)<

引用次数: 0

The braid group action on quantum queer superalgebra

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-12-10 DOI: 10.1016/j.jalgebra.2024.11.015

Jianmin Chen , Zhenhua Li , Hongying Zhu

In his seminal work [22], Lusztig introduced a braid group action by automorphisms on the quantum group

U_{v} ({sl}_{n})

. This result and its generalizations have since become fundamental to the study of quantum algebras. In this paper, we extend the braid group action to the queer superalgebra

U (q_{n})

and its quantization

U_{v} (q_{n})

, which promises to be crucial for investigating the structure of these algebras. In particular, we are able to define root vectors for each, together with explicit expressions in terms of standard generators. For

U (q_{n})

, we moreover obtain their super commutation formulas. As a consequence, we construct PBW-type bases for both

U (q_{n})

and

U_{v} (q_{n})

involving products of these root vectors, further strengthening our understanding of their structure.

{"title":"The braid group action on quantum queer superalgebra","authors":"Jianmin Chen , Zhenhua Li , Hongying Zhu","doi":"10.1016/j.jalgebra.2024.11.015","DOIUrl":"10.1016/j.jalgebra.2024.11.015","url":null,"abstract":"<div><div>In his seminal work [22], Lusztig introduced a braid group action by automorphisms on the quantum group <math><msub><mrow><mi>U</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math>. This result and its generalizations have since become fundamental to the study of quantum algebras. In this paper, we extend the braid group action to the queer superalgebra <math><mi>U</mi><mo>(</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math> and its quantization <math><msub><mrow><mi>U</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math>, which promises to be crucial for investigating the structure of these algebras. In particular, we are able to define root vectors for each, together with explicit expressions in terms of standard generators. For <math><mi>U</mi><mo>(</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math>, we moreover obtain their super commutation formulas. As a consequence, we construct PBW-type bases for both <math><mi>U</mi><mo>(</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math> and <math><msub><mrow><mi>U</mi></mrow><mrow><mi>v</mi></mrow></msub><mo>(</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math> involving products of these root vectors, further strengthening our understanding of their structure.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 169-212"},"PeriodicalIF":0.8,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A note on module structures of source algebras of block ideals of finite groups II

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-12-06 DOI: 10.1016/j.jalgebra.2024.10.051

Hiroki Sasaki

Let b be a block ideal of the group algebra of a finite group G over an algebraically closed field k of prime characteristic p with a defect group P. Some direct summands, as

k [P \times P]

-modules, of a source algebra of the block ideal b outside of the inertial group of a maximal b-Brauer pair will be given; their multiplicities modulo p will also be given.

We shall introduce a notion, we shall call it the icc condition, which arises from the isomorphism problem of bimodules over p-subgroups. We shall show for an element

x \in G ∖ N_{G} (P)

which satisfies the

(P, P)

-icc condition, the

k [P \times P]

-module

k [P x P]

is isomorphic to a direct summand of the source algebra of b under some further condition. One of our main tools is the Brauer homomorphisms so that the multiplicities will be described using dimensions of the Brauer constructions. Our arguments to investigate these dimensions depend on the Puig's theory, especially multiplicities of points.

{"title":"A note on module structures of source algebras of block ideals of finite groups II","authors":"Hiroki Sasaki","doi":"10.1016/j.jalgebra.2024.10.051","DOIUrl":"10.1016/j.jalgebra.2024.10.051","url":null,"abstract":"<div><div>Let b be a block ideal of the group algebra of a finite group G over an algebraically closed field k of prime characteristic p with a defect group P. Some direct summands, as <math><mi>k</mi><mo>[</mo><mi>P</mi><mo>×</mo><mi>P</mi><mo>]</mo></math>-modules, of a source algebra of the block ideal b outside of the inertial group of a maximal b-Brauer pair will be given; their multiplicities modulo p will also be given.</div><div>We shall introduce a notion, we shall call it the icc condition, which arises from the isomorphism problem of bimodules over p-subgroups. We shall show for an element <math><mi>x</mi><mo>∈</mo><mi>G</mi><mo>∖</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>P</mi><mo>)</mo></math> which satisfies the <math><mo>(</mo><mi>P</mi><mo>,</mo><mi>P</mi><mo>)</mo></math>-icc condition, the <math><mi>k</mi><mo>[</mo><mi>P</mi><mo>×</mo><mi>P</mi><mo>]</mo></math>-module <math><mi>k</mi><mo>[</mo><mi>P</mi><mi>x</mi><mi>P</mi><mo>]</mo></math> is isomorphic to a direct summand of the source algebra of b under some further condition. One of our main tools is the Brauer homomorphisms so that the multiplicities will be described using dimensions of the Brauer constructions. Our arguments to investigate these dimensions depend on the Puig's theory, especially multiplicities of points.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 777-793"},"PeriodicalIF":0.8,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

From the Albert algebra to Kac's ten-dimensional Jordan superalgebra via tensor categories in characteristic 5

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-12-06 DOI: 10.1016/j.jalgebra.2024.12.006

Alberto Elduque , Pavel Etingof , Arun S. Kannan

Kac's ten-dimensional simple Jordan superalgebra over an algebraically closed field of characteristic 5 is obtained from a process of semisimplification, via tensor categories, from the exceptional simple Jordan algebra (or Albert algebra), together with a suitable order 5 automorphism.

This explains McCrimmon's ‘bizarre result’ asserting that, in characteristic 5, Kac's superalgebra is a sort of ‘degree 3 Jordan superalgebra’.

As an outcome, the exceptional simple Lie superalgebra

el (5; 5)

, specific of characteristic 5, is obtained from the simple Lie algebra of type

E_{8}

and an order 5 automorphism.

In the process, precise recipes to obtain superalgebras from algebras in

{Rep C}_{p}

(or

Rep α_{p}

p > 2

, are given.

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引用次数: 0

On Frobenius graphs of diameter 3 for finite groups

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-12-06 DOI: 10.1016/j.jalgebra.2024.11.014

T. Breuer , L. Héthelyi , E. Horváth, B. Külshammer

For a subgroup H of a finite group G, the Frobenius graph

Γ (G, H)

records the constituents of the restrictions to H of the irreducible characters of G. We investigate when this graph has diameter 3.

引用次数: 0

The 2 × 2 upper triangular matrix algebra and its generalized polynomial identities

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-12-06 DOI: 10.1016/j.jalgebra.2024.12.005

Fabrizio Martino , Carla Rizzo

Let

U T_{2}

be the algebra of

2 \times 2

upper triangular matrices over a field F of characteristic zero. Here we study the generalized polynomial identities of

U T_{2}

, i.e., identical relations holding for

U T_{2}

regarded as

U T_{2}

-algebra. We determine the generator of the

T_{U T_{2}}

-ideal of generalized polynomial identities of

U T_{2}

and compute the exact values of the corresponding sequence of generalized codimensions. Moreover, we give a complete description of the space of multilinear generalized identities in n variables in the language of Young diagrams through the representation theory of the symmetric group

S_{n}

. Finally, we prove that, unlike the ordinary case, the generalized variety of

U T_{2}

-algebras generated by

U T_{2}

has no almost polynomial growth; nevertheless, we exhibit two distinct generalized varieties of almost polynomial growth.

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引用次数: 0

Some characterizations of fundamental graded algebras

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2024-12-06 DOI: 10.1016/j.jalgebra.2024.12.007

Elena Pascucci

One of the basic notions in the theory of varieties of algebras in characteristic zero developed by Kemer [20] was that of fundamental algebras. They are used as a main tool in the solution of Specht's Problem. The aim of this paper is to extend this concept to algebras with a G-graded structure, where G is a finite group, and to develop the corresponding theory. Furthermore, we explore the connection between fundamental G-graded algebras and generators of affine varieties of G-graded PI algebras which are minimal with respect to their G-graded exponent. In some important cases, we provide necessary and sufficient conditions so that subalgebras of these generators are fundamental. Finally, for abelian groups, we give a characterization in terms of the representation theory of the group

G ≀ S_{n}

引用次数: 0

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