Academia.eduAcademia.edu

Outline

Title
Abstract
Introduction
Clifford Algebras 1. Background
Formerly Known Results
References
All Topics
Mathematics
Pure Mathematics

$p$-Central Subspaces of Central Simple Algebras

Profile image of adam chapmanadam chapman

Abstract

We study central simple algebras in various ways, focusing on the role of $p$-central subspaces. The first part of my thesis is dedicated to the study of Clifford algebras. The standard Clifford algebra of a given form is the generic associative algebra containing a $p$-central subspace whose exponentiation form is equal to the given form. There is an old question as for whether these algebras have representations of finite rank over the center, and jointly with Daniel Krashen and Max Lieblich we managed to provide a positive answer. Different generalizations of the structure of the Clifford algebra are presented and studied in that part too. The second part is dedicated to the study of $p$-central subspaces of given central simple algebras, mainly tensor products of cyclic algebras of degree $p$. Among the results, we prove that $5$ is the upper bound for the dimension of 4-central subspaces of cyclic algebras of degree 4 containing pairs of standard generators. The third part is d...

Related papers

Open problems on central simple algebras
Asher Auel, U. Vishne

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.

View PDFchevron_right
New Simple Lie Algebras of Prime Characteristic
Richard Block

Transactions of the American Mathematical Society, 1958

RICHARD BLOCK 1. Introduction. For a number of years, the only known simple Lie algepras of characteristic p>0 not analogues of the classical algebras of characteristic zero were those of dimension pn of H. Zassenhaus , and of dimension np" of N. Jacobson , both generalizations of the ^-dimensional algebras of E. Witt. During the last few years, a number of new classes of simple Lie algebras have been determined. These are due to Kaplansky, who in noted the existence of a class of algebras of dimension rpn, 1 gr g w, generalizing those of Zassenhaus and Jacobson; to M. S. Frank, who in [2]obtained algebras ©" of dimension (n -l)(pn -l); and to A. A. Albert and M. S. Frank, who in [l ] determined new classes of simple algebras Xn, %$m, So, and Sj, of dimensions (n -l)pn, p2m -2, pn -l and pn -2 respectively, and a generalization of the Zassenhaus algebras, of dimension pn. In this paper we obtain a new class of simple Lie algebras, which we shall designate by the symbol £(®, 5, /). Here ® is an additive group, of order pn> 1, which is a direct sum of a finite number of finite elementary ^-groups ®oi • • • , ®m-These groups are then finite dimensional vector spaces over the prime field %p. We let m(@) denote the index m, allow ®0 to be zero, and assume that either p>2 or ® ?^®i. For i = l, • • • , m, we take 5< to be a nonzero element of ®<, and write 5=5i+ • • • +Sm. We index a basis of 8(®, 5,/) over a field $ of characteristic p by the elements of ® other than 0 and -S, denoting by v(a) the basis element corresponding to a. Furthermore we assume given, for each i, a nondegenerate skew-symmetric biadditive function /,-on (®,-, ®j) to §, such that, for i = l, • ■ • , m, there are additive functions gu hi, on ®i to %, with f,-(5.) =0 and/<(a, 8) = gi(°t)hi(0) -gi(0)h{(a) for every a and 8 in ®<-The definition of the algebra 8(®, S,f) may then be completed by defining multiplication in the algebra by v(a)v(8) = JZ?-ofi(ai, 0d ■v(a+8 -8i), where a< and 8i are the components of a and 0 in ®<, and where 50 and v(0) denote zero. Then we shall prove that 8(®, 5,/) is a central simple Lie algebra. Its dimension is pn -2 except when ® = ®0, in which case the

View PDFchevron_right
Clifford algebras of binary homogeneous forms
Uzi Vishne

Journal of Algebra, 2012

We study the generalized Clifford algebras associated to homogeneous binary forms of prime degree p, focusing on exponentiation forms of p-central spaces in division algebra. For a two-dimensional p-central space, we make the simplifying assumption that one basis element is a sum of two eigenvectors with respect to conjugation by the other. If the product of the eigenvalues is 1 then the Clifford algebra is a symbol Azumaya algebra of degree p, generalizing the theory developed for p = 3. Furthermore, when p = 5 and the product is not 1, we show that any quotient division algebra of the Clifford algebra is a cyclic algebra or a tensor product of two cyclic algebras, and every product of two cyclic algebras can be obtained as a quotient. Explicit presentation is given to the Clifford algebra when the form is diagonal.

View PDFchevron_right
Non-cyclic algebras with $n$-central elements
Uzi Vishne

Proceedings of the American Mathematical Society, 2012

We construct, for any prime p, a non-cyclic central simple algebra of degree p 2 with p 2 -central elements. This construction answers a problem of Peter Roquette.

View PDFchevron_right
Bimodule structure of central simple algebras
Uzi Vishne

Journal of Algebra, 2017

For a maximal separable subfield K of a central simple algebra A, we provide a semiring isomorphism between K-K-bimodules A and H-H bisets of G = Gal(L/F ), where F = Z(A), L is the Galois closure of K/F , and H = Gal(L/K). This leads to a combinatorial interpretation of the growth of dim K ((KaK) i ), for fixed a ∈ A, especially in terms of Kummer sets.

View PDFchevron_right
On semi-simple group algebras. I
Peter Trojan

Pacific Journal of Mathematics, 1975

View PDFchevron_right
Decomposition of generalized Clifford algebras
Tara Smith

Quart. J. Math. Oxford, 1991

View PDFchevron_right
The p-regular subspaces of symmetric Nakayama algebras and algebras of dihedral and semidihedral type
Intan Muchtadi-Alamsyah

JP Journal of Algebra, Number Theory and Applications

Let F be a field of characteristic p&gt;0 and let G be a finite group. Consider the class sums of conjugacy classes of the elements of order prime to p and the F-vector space Z p &#39; (FG) generated by these. This is a subset of the centre of FG and it was studied by H. Meyer [Math. Comput. 77, No. 263, 1801-1821 (2008; Zbl 1195.20002)] and by Fan-Külshammer when Z p &#39; (FG) forms a subalgebra. It is easy to characterise Z p &#39; (FG) by the images of the p-power map on the degree 0 Hochschild homology, using the orthogonal space with respect to the standard dualising form on FG. This second characterisation can be used to compute spaces like Z p &#39; (A) for any symmetric F-algebra A. The authors perform these computations for symmetric Nakayama algebras and some algebras of dihedral and of semidihedral type in the sense of Erdmann. They build on work of T. Holm and A. Zimmermann [J. Algebra 320, No. 9, 3425-3437 (2008; Zbl 1160.16009)] for these computations.

View PDFchevron_right
On powerful and p-central restricted Lie algebras
thomas weigel

BULLETIN OF THE AUSTRALIAN MATHEMATICAL …, 2007

In this note we analyse the analogy between m-potent and p-central restricted Lie algebras and p-groups. For restricted Lie algebras the notion of m-potency has stronger implications than for p-groups (Theorem A). Every finite-dimensional restricted Lie algebra L is ...

View PDFchevron_right
Central ideals and Cartan invariants of symmetric algebras
erzsebet horvath

Journal of Algebra, 2005

In this paper, we investigate certain ideals in the center of a symmetric algebra A over an algebraically closed field of characteristic p > 0. These ideals include the Higman ideal and the Reynolds ideal. They are closely related to the p-power map on A. We generalize some results concerning these ideals from group algebras to symmetric algebras, and we obtain some new results as well. In case p = 2, these ideals detect odd diagonal entries in the Cartan matrix of A. In a sequel to this paper, we will apply our results to group algebras.

View PDFchevron_right

References (43)

  1. A. Adrian Albert, Structure of algebras, Revised printing. American Mathematical Society Colloquium Publications, Vol. XXIV, American Mathematical Society, Providence, R.I., 1961. MR 0123587 (23 #A912)
  2. Michael Artin, Fernando Rodriguez-Villegas, and John Tate, On the Jacobians of plane cubics, Adv. Math. 198 (2005), no. 1, 366-382. MR 2183258 (2006h:14043)
  3. Helmer Aslaksen, Quaternionic determinants, Math. Intelligencer 18 (1996), no. 3, 57-65. MR 1412993 (97j:16028)
  4. Yik-Hoi Au-Yeung, An explicit solution for the quaternionic equation x 2 + bx + xc + d = 0, Southeast Asian Bull. Math. 26 (2003), no. 5, 717-724. MR 2045106 (2004m:16026)
  5. Adam Chapman, Polynomial equations over division rings, 2009, Thesis (M.Sc.)-Bar-Ilan University.
  6. General polynomials over division algebras and left eigenvalues, Electron. J. Linear Algebra 23 (2012), 508-513. MR 2928573
  7. Lindsay N. Childs, Linearizing of n-ic forms and generalized Clifford algebras, Linear and Multilinear Algebra 5 (1977/78), no. 4, 267-278. MR 0472880 (57 #12567)
  8. Mirela Ciperiani and Daniel Krashen, Relative Brauer groups of genus 1 curves, Israel J. Math. 192 (2012), no. 2, 921-949. MR 3009747
  9. Emre Coskun, Rajesh S. Kulkarni, and Yusuf Mustopa, On represen- tations of Clifford algebras of ternary cubic forms, New trends in non- commutative algebra, Contemp. Math., vol. 562, Amer. Math. Soc., Providence, RI, 2012, pp. 91-99. MR 2905555
  10. Adam Chapman and Uzi Vishne, Clifford algebras of binary homoge- neous forms, J. Algebra 366 (2012), 94-111. MR 2942645 [CV13] , Square-central elements and standard generators for biquater- nion algebras, Israel J. Math. 197 (2013), no. 1, 409-423. MR 3096621
  11. L. E. Dickson, Linear associative algebras and abelian equations, Trans. Amer. Math. Soc. 15 (1914), no. 1, 31-46. MR 1500963
  12. B. Gordon and T. S. Motzkin, On the zeros of polynomials over division rings, Trans. Amer. Math. Soc. 116 (1965), 218-226. MR 0195853 (33 #4050a)
  13. Darrell E. Haile, On the Clifford algebra of a binary cubic form, Amer. J. Math. 106 (1984), no. 6, 1269-1280. MR 765580 (86c:11028)
  14. When is the Clifford algebra of a binary cubic form split?, J. Algebra 146 (1992), no. 2, 514-520. MR 1152918 (93a:11029)
  15. Nickolas Heerema, An algebra determined by a binary cubic form, Duke Math. J. 21 (1954), 423-443. MR 0064030 (16,214d)
  16. Darrell Haile and Ilseop Han, On an algebra determined by a quartic curve of genus one, J. Algebra 313 (2007), no. 2, 811-823. MR 2329571 (2008f:16044)
  17. Darrell Haile, Jung-Miao Kuo, and Jean-Pierre Tignol, On chains in division algebras of degree 3, C. R. Math. Acad. Sci. Paris 347 (2009), no. 15-16, 849-852. MR 2542882 (2010h:16039)
  18. Liping Huang and Wasin So, Quadratic formulas for quaternions, Appl. Math. Lett. 15 (2002), no. 5, 533-540. MR 1889501 (2003d:12003)
  19. Drahoslava Janovská and Gerhard Opfer, The classification and the computation of the zeros of quaternionic, two-sided polynomials, Numer. Math. 115 (2010), no. 1, 81-100. MR 2594342 (2011b:16096)
  20. A note on the computation of all zeros of simple quater- nionic polynomials, SIAM J. Numer. Anal. 48 (2010), no. 1, 244-256. MR 2608368 (2011c:11170)
  21. Cemal Koç and Yosum Kurtulmaz, Structure theory of central sim- ple Z d -graded algebras, Turkish J. Math. 36 (2012), no. 4, 560-577. MR 2993587
  22. Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Col- loquium Publications, vol. 44, American Mathematical Society, Prov- idence, RI, 1998, With a preface in French by J. Tits. MR 1632779 (2000a:16031)
  23. Jung-Miao Kuo, On an algebra associated to a ternary cubic curve, J. Algebra 330 (2011), 86-102. MR 2774619 (2012b:16040)
  24. T. Y. Lam, The algebraic theory of quadratic forms, W. A. Ben- jamin, Inc., Reading, Mass., 1973, Mathematics Lecture Note Series. MR 0396410 (53 #277)
  25. F. W. Long, A generalization of the Brauer group of graded algebras, Proc. London Math. Soc. (3) 29 (1974), 237-256. MR 0354753 (50 #7230)
  26. J. Lawrence and G. E. Simons, Equations in division rings-a sur- vey, Amer. Math. Monthly 96 (1989), no. 3, 220-232. MR 991867 (90g:16015)
  27. Eliyah Matzri, Louis H. Rowen, and Uzi Vishne, Non-cyclic algebras with n-central elements, Proc. Amer. Math. Soc. 140 (2012), no. 2, 513-518. MR 2846319 (2012i:16034)
  28. A. S. Merkur ′ ev and A. A. Suslin, K-cohomology of Severi-Brauer vari- eties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011-1046, 1135-1136. MR 675529 (84i:12007)
  29. P. Mammone, J.-P. Tignol, and A. Wadsworth, Fields of characteristic 2 with prescribed u-invariants, Math. Ann. 290 (1991), no. 1, 109-128. MR 1107665 (92g:11035)
  30. Eliyahu Matzri and Uzi Vishne, Isotropic subspaces in symmetric com- position algebras and Kummer subspaces in central simple algebras of degree 3, Manuscripta Math. 137 (2012), no. 3-4, 497-523. MR 2875290 [MV14] , Composition algebras and cyclic p-algebras in characteristic 3, Manuscripta Math. 143 (2014), no. 1-2, 1-18. MR 3147442
  31. E. Macías-Virgós and M. J. Pereira-Sáez, Left eigenvalues of 2 × 2 symplectic matrices, Electron. J. Linear Algebra 18 (2009), 274-280. MR 2519914 (2010f:15042)
  32. Christopher J. Pappacena, Matrix pencils and a generalized Clifford algebra, Linear Algebra Appl. 313 (2000), no. 1-3, 1-20. MR 1770355 (2001e:15010)
  33. Mélanie Raczek, On ternary cubic forms that determine central sim- ple algebras of degree 3, J. Algebra 322 (2009), no. 5, 1803-1818. MR 2543635 (2010h:16043)
  34. Ph. Revoy, Algèbres de Clifford et algèbres extérieures, J. Algebra 46 (1977), no. 1, 268-277. MR 0472881 (57 #12568)
  35. Norbert Roby, Algèbres de Clifford des formes polynomes, C. R. Acad. Sci. Paris Sér. A-B 268 (1969), A484-A486. MR 0241454 (39 #2794)
  36. Markus Rost, The chain lemma for Kummer elements of degree 3, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 3, 185-190. MR 1674602 (2000c:12003)
  37. Louis H. Rowen, Ring theory. Vol. II, Pure and Applied Mathemat- ics, vol. 128, Academic Press Inc., Boston, MA, 1988. MR 945718 (89h:16002)
  38. Wedderburn's method and algebraic elements of simple Ar- tinian rings, Azumaya algebras, actions, and modules (Bloomington, IN, 1990), Contemp. Math., vol. 124, Amer. Math. Soc., Providence, RI, 1992, pp. 179-202. MR 1144036 (92k:16025)
  39. A. S. Sivatski, The chain lemma for biquaternion algebras, J. Algebra 350 (2012), 170-173. MR 2859881 (2012j:16037)
  40. Wasin So, Quaternionic left eigenvalue problem, Southeast Asian Bull. Math. 29 (2005), no. 3, 555-565. MR 2216293 (2006m:15030)
  41. M. Van den Bergh, Linearisations of binary and ternary forms, J. Al- gebra 109 (1987), no. 1, 172-183. MR 898344 (88j:11020)
  42. Uzi Vishne, Generators of central simple p-algebras of degree 3, Israel J. Math. 129 (2002), 175-187. MR 1910941 (2003g:16022)
  43. R. M. W. Wood, Quaternionic eigenvalues, Bull. London Math. Soc. 17 (1985), no. 2, 137-138. MR 806238 (86m:15013)

Related papers

Clifford algebras of $p$-central sets
Adam Chapman

A generalization of the term &quot;generalized Clifford algebras&quot; (as appears in papers on advances in applied Clifford algebras) is introduced. This algebra is studied by means of structure theory of central simple algebras. A graph theoretical approach is proposed for studying the generating set of this algebra in case where the prime number under discussion is three. Finally, it is shown how to obtain solutions in to the equation $\alpha Y^3=\alpha X_1^3+\beta X_2^3+\alpha^2 \beta^2 X_3^3$ in $\mathbb{Z}[\rho]$ where $\rho$ is the primitive third root of unity.

View PDFchevron_right
Generators of central simplep-algebras of degree 3
U. Vishne

Israel Journal of Mathematics, 2002

We discuss standard pairs of generators of cyclic division p-algebras of degree p, and prove for p = 3 that any two Artin-Schreier elements are connected by a chain of standard pairs. This result has immediate applications to the presentations of such algebras.

View PDFchevron_right
Central simple algebras with involution viewed through centralizers
louis rowen

Journal of Algebra, 1980

View PDFchevron_right
Isotropic subspaces in symmetric composition algebras and Kummer subspaces in central simple algebras of degree 3
Uzi Vishne

Manuscripta Mathematica, 2012

The maximal isotropic subspaces in split Cayley algebras were classified by van der Blij and Springer in Nieuw Archief voor Wiskunde VIII(3): 158-169, 1960. Here we translate this classification to arbitrary composition algebras. We study intersection properties of such spaces in a symmetric composition algebra, and prove two triality results: one for two-D isotropic spaces, and another for isotropic vectors and maximal isotropic spaces. We bound the distance between isotropic spaces of various dimensions, and study the strong orthogonality relation on isotropic vectors, with its own bound on the distance. The results are used to classify maximal p-central subspaces in central simple algebras of degree p = 3. We prove various linkage properties of maximal p-central spaces and p-central elements. Analogous results are obtained for symmetric p-central elements with respect to an involution of the second kind inverting a third root of unity.

View PDFchevron_right
Locally Finite Central Simple Algebras
Uzi Vishne

Algebras and Representation Theory

We develop a comprehensive theory of algebras over a field which are locally both finite dimensional and central simple. We generalize fundamental concepts of the theory of finite dimensional central simple algebras, and introduce supernatural matrix algebras, the supernatural degree and matrix degree, and so on. We define a Brauer monoid, whose unique maximal subgroup is the classical Brauer group, and show that once infinite dimensional division algebras exist over the field, they are abundant.

View PDFchevron_right

Related topics

  • Mathematics
  • Pure Mathematics
    • 580 California St., Suite 400
    San Francisco, CA, 94104
    © 2025 Academia. All rights reserved