Graded Clifford Algebras and Graded Skew Clifford Algebras and Their Role in the Classification of Artin–Schelter Regular Algebras

Graded Clifford Algebras and Graded Skew Clifford Algebras and Their Role in the Classification... This paper is a survey of work done on $$\mathbb {Z}$$ Z -graded Clifford algebras (GCAs) and $$\mathbb {Z}$$ Z -graded skew Clifford algebras (GSCAs) by Vancliff et al. (Commun Algebra 26(4): 1193–1208, 1998), Stephenson and Vancliff (J Algebra 312(1): 86–110, 2007), Cassidy and Vancliff (J Lond Math Soc 81:91–112, 2010), Nafari et al. (J Algebra 346(1): 152–164, 2011), Vancliff and Veerapen (Contemp Math 592: 241–250, 2013), (J Algebra 420:54–64, 2014). In particular, we discuss the hypotheses necessary for these algebras to be Artin Schelter-regular (Adv Math 66:171–216, 1987), (The Grothendieck Festschrift. Birkhäuser, Boston, 1990) and show how certain ‘points’ called, point modules, can be associated to them. We may view an AS-regular algebra as a noncommutative analog of the polynomial ring. We begin our survey with a fundamental result in Vancliff et al. (Commun Algebra 26(4): 1193–1208, 1998) that is essential to subsequent results discussed here: the connection between point modules and rank-two quadrics. Using, in part, this connection the authors in Stephenson and Vancliff (J Algebra 312(1): 86–110, 2007) provide a method to construct GCAs with finitely many distinct isomorphism classes of point modules. In Cassidy and Vancliff (J Lond Math Soc 81:91–112, 2010), Cassidy and Vancliff introduce a quantized analog of a GCA, called a graded skew Clifford algebra and Nafari et al. (J Algebra 346(1): 152–164, 2011) show that most Artin Schelter-regular algebras of global dimension three are either twists of graded skew Clifford algebras of global dimension three or Ore extensions of graded Clifford algebras of global dimension two. Vancliff and Veerapen (Contemp Math 592: 241–250, 2013), (J Algebra 420:54–64, 2014) go a step further and generalize the result of Vancliff et al. (Commun Algebra 26(4): 1193–1208, 1998), between point modules and rank-two quadrics, by showing that point modules over GSCAs are determined by (noncommutative) quadrics of $$\mu $$ μ -rank at most two. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Applied Clifford Algebras Springer Journals

Graded Clifford Algebras and Graded Skew Clifford Algebras and Their Role in the Classification of Artin–Schelter Regular Algebras

Loading next page...
 
/lp/springer_journal/graded-clifford-algebras-and-graded-skew-clifford-algebras-and-their-T0OjQtrE7n
Publisher
Springer International Publishing
Copyright
Copyright © 2017 by Springer International Publishing AG
Subject
Physics; Mathematical Methods in Physics; Theoretical, Mathematical and Computational Physics; Applications of Mathematics; Physics, general
ISSN
0188-7009
eISSN
1661-4909
D.O.I.
10.1007/s00006-017-0795-x
Publisher site
See Article on Publisher Site

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches

$49/month

Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588

$360/year

billed annually
Start Free Trial

14-day Free Trial