American Institute of Mathematical Sciences

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  2018, 25: 16-26. doi: 10.3934/era.2018.25.003

Signatures, sums of hermitian squares and positive cones on algebras with involution

Vincent Astier and  Thomas Unger

School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland

Received  October 23, 2017 Published  April 2018

We provide a coherent picture of our efforts thus far in extending real algebra and its links to the theory of quadratic forms over ordered fields in the noncommutative direction, using hermitian forms and "ordered" algebras with involution.

Keywords: algebra with involution, formally real field, hermitian form, signature, Witt group, positivity, sum of hermitian squares, positive cone, real algebra.
Mathematics Subject Classification: Primary: 13J30, 11E39; Secondary: 16K20, 16W10.
Citation: Vincent Astier, Thomas Unger. Signatures, sums of hermitian squares and positive cones on algebras with involution. Electronic Research Announcements, 2018, 25: 16-26. doi: 10.3934/era.2018.25.003
References:
[1]

A. A. Albert, Involutorial simple algebras and real Riemann matrices, Ann. of Math. (2), 36(1935), 886–964. doi: 10.2307/1968595.  Google Scholar

[2]

E. Artin, Über die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 100-115.  doi: 10.1007/BF02952513.  Google Scholar

[3]

E. Artin and O. Schreier, Algebraische Konstruktion reeller Körper, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 85-99.  doi: 10.1007/BF02952512.  Google Scholar

[4]

V. Astier and T. Unger, Signatures of hermitian forms and the Knebusch trace formula, Math. Ann., 358 (2014), 925-947.  doi: 10.1007/s00208-013-0977-3.  Google Scholar

[5]

V. Astier and T. Unger, Signatures of hermitian forms and "prime ideals" of Witt groups, Adv. Math., 285 (2015), 497-514.  doi: 10.1016/j.aim.2015.07.035.  Google Scholar

[6]

V. Astier and T. Unger, Positive cones on algebras with involution, preprint, 2017, arXiv: 1609.06601. Google Scholar

[7]

V. Astier and T. Unger, Signatures of hermitian forms, positivity, and an answer to a question of Procesi and Schacher, preprint, 2016, arXiv: 1511.06330. Google Scholar

[8]

V. Astier and T. Unger, Signatures, sums of hermitian squares and positive cones on algebras with involution, Séminaire de Structures Algébriques Ordonnées 2015–2016, 91 (2017). Google Scholar

[9]

V. Astier and T. Unger, Stability index of algebras with involution, Contemporary Mathematics, 697 (2017), 41-50.  doi: 10.1090/conm/697/14045.  Google Scholar

[10]

A. Auel, E. Brussel, S. Garibaldi and U. Vishne, Open problems on central simple algebras, Transform. Groups, 16 (2011), 219-264.  doi: 10.1007/s00031-011-9119-8.  Google Scholar

[11]

E. Bayer-Fluckiger and R. Parimala, Classical groups and the Hasse principle, Ann. of Math. (2), 147 (1998), 651–693. doi: 10.2307/120961.  Google Scholar

[12]

J. Bochnak, M. Coste and M. -F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03718-8.  Google Scholar

[13]

T. C. Craven, Orderings, valuations, and Hermitian forms over *-fields, in K-theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (Santa Barbara, CA, 1992), Proc. Sympos. Pure Math., 58, Part 2, American Mathematical Society, Providence, RI, (1995), 149–160.  Google Scholar

[14]

T. C. Craven, Valuations and Hermitian forms on skew fields, in Valuation Theory and its Applications, Vol. I (Saskatoon, SK, 1999), Fields Inst. Commun., 32, American Mathematical Society, Providence, RI, (2002), 103–115.  Google Scholar

[15]

D. Hilbert, Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Congress zu Paris 1900, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., (1900), 253-297.   Google Scholar

[16]

M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 142 (1969), 43-60.  doi: 10.1090/S0002-9947-1969-0251026-X.  Google Scholar

[17]

I. Klep and T. Unger, The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution, J. Algebra, 324 (2010), 256-268.  doi: 10.1016/j.jalgebra.2010.03.022.  Google Scholar

[18]

M. -A. Knus, Quadratic and Hermitian Forms Over Rings, Grundlehren der Mathematischen Wissenschaften, vol. 294, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-75401-2.  Google Scholar

[19]

M. -A. Knus, A. Merkurjev, M. Rost and J. -P. Tignol, The Book of Involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/coll/044.  Google Scholar

[20]

T. Y. Lam, An introduction to real algebra, in Ordered Fields and real Algebraic Geometry (Boulder, Colo., 1983), Rocky Mountain J. Math., 14 (1984), 767–814. doi: 10.1216/RMJ-1984-14-4-767.  Google Scholar

[21]

T. Y. Lam, Introduction to Quadratic Forms Over Fields, Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2005.  Google Scholar

[22]

D. W. Lewis and J.-P. Tignol, On the signature of an involution, Arch. Math. (Basel), 60 (1993), 128-135.  doi: 10.1007/BF01199098.  Google Scholar

[23]

D. W. Lewis and T. Unger, A local-global principle for algebras with involution and Hermitian forms, Math. Z., 244 (2003), 469-477.  doi: 10.1007/s00209-003-0490-6.  Google Scholar

[24]

D. W. Lewis and T. Unger, Hermitian Morita theory: A matrix approach, Irish Math. Soc. Bull., 62 (2008), 37-41.   Google Scholar

[25]

F. Lorenz and J. Leicht, Die Primideale des Wittschen Ringes, Invent. Math., 10 (1970), 82-88.  doi: 10.1007/BF01402972.  Google Scholar

[26]

A. Pfister, Quadratische Formen in beliebigen Körpern, Invent. Math., 1 (1966), 116-132.  doi: 10.1007/BF01389724.  Google Scholar

[27]

A. Prestel, Quadratische Semi-Ordnungen und quadratische Formen, Math. Z., 133 (1973), 319-342.  doi: 10.1007/BF01177872.  Google Scholar

[28]

A. Prestel, Lectures on Formally Real Fields, Lecture Notes in Mathematics, vol. 1093, Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0101548.  Google Scholar

[29]

C. Procesi and M. Schacher, A non-commutative real Nullstellensatz and Hilbert's 17th problem, Ann. of Math. (2), 104 (1976), 395–406. doi: 10.2307/1970962.  Google Scholar

[30]

A. Quéguiner, Signature des involutions de deuxiéme espéce, Arch. Math. (Basel), 65 (1995), 408-412.  doi: 10.1007/BF01198071.  Google Scholar

[31]

W. Scharlau, Induction theorems and the structure of the Witt group, Invent. Math., 11 (1970), 37-44.  doi: 10.1007/BF01389804.  Google Scholar

[32]

W. Scharlau, Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften, vol. 270, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-69971-9.  Google Scholar

[33]

J. J. Sylvester, A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares, Philosophical Magazine, 4 (1852), 138-142.  doi: 10.1080/14786445208647087.  Google Scholar

[34]

J. -P. Tignol, Algebras with involution and classical groups, in European Congress of Mathematics, Vol. II (Budapest, 1996), Progr. Math., vol. 169, Birkhäuser, Basel, (1998), 244–258.  Google Scholar

[35]

A. Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. (N.S.), 24 (1961), 589-623.   Google Scholar

show all references

References:
[1]

A. A. Albert, Involutorial simple algebras and real Riemann matrices, Ann. of Math. (2), 36(1935), 886–964. doi: 10.2307/1968595.  Google Scholar

[2]

E. Artin, Über die Zerlegung definiter Funktionen in Quadrate, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 100-115.  doi: 10.1007/BF02952513.  Google Scholar

[3]

E. Artin and O. Schreier, Algebraische Konstruktion reeller Körper, Abh. Math. Sem. Univ. Hamburg, 5 (1927), 85-99.  doi: 10.1007/BF02952512.  Google Scholar

[4]

V. Astier and T. Unger, Signatures of hermitian forms and the Knebusch trace formula, Math. Ann., 358 (2014), 925-947.  doi: 10.1007/s00208-013-0977-3.  Google Scholar

[5]

V. Astier and T. Unger, Signatures of hermitian forms and "prime ideals" of Witt groups, Adv. Math., 285 (2015), 497-514.  doi: 10.1016/j.aim.2015.07.035.  Google Scholar

[6]

V. Astier and T. Unger, Positive cones on algebras with involution, preprint, 2017, arXiv: 1609.06601. Google Scholar

[7]

V. Astier and T. Unger, Signatures of hermitian forms, positivity, and an answer to a question of Procesi and Schacher, preprint, 2016, arXiv: 1511.06330. Google Scholar

[8]

V. Astier and T. Unger, Signatures, sums of hermitian squares and positive cones on algebras with involution, Séminaire de Structures Algébriques Ordonnées 2015–2016, 91 (2017). Google Scholar

[9]

V. Astier and T. Unger, Stability index of algebras with involution, Contemporary Mathematics, 697 (2017), 41-50.  doi: 10.1090/conm/697/14045.  Google Scholar

[10]

A. Auel, E. Brussel, S. Garibaldi and U. Vishne, Open problems on central simple algebras, Transform. Groups, 16 (2011), 219-264.  doi: 10.1007/s00031-011-9119-8.  Google Scholar

[11]

E. Bayer-Fluckiger and R. Parimala, Classical groups and the Hasse principle, Ann. of Math. (2), 147 (1998), 651–693. doi: 10.2307/120961.  Google Scholar

[12]

J. Bochnak, M. Coste and M. -F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03718-8.  Google Scholar

[13]

T. C. Craven, Orderings, valuations, and Hermitian forms over *-fields, in K-theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (Santa Barbara, CA, 1992), Proc. Sympos. Pure Math., 58, Part 2, American Mathematical Society, Providence, RI, (1995), 149–160.  Google Scholar

[14]

T. C. Craven, Valuations and Hermitian forms on skew fields, in Valuation Theory and its Applications, Vol. I (Saskatoon, SK, 1999), Fields Inst. Commun., 32, American Mathematical Society, Providence, RI, (2002), 103–115.  Google Scholar

[15]

D. Hilbert, Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Congress zu Paris 1900, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., (1900), 253-297.   Google Scholar

[16]

M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 142 (1969), 43-60.  doi: 10.1090/S0002-9947-1969-0251026-X.  Google Scholar

[17]

I. Klep and T. Unger, The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution, J. Algebra, 324 (2010), 256-268.  doi: 10.1016/j.jalgebra.2010.03.022.  Google Scholar

[18]

M. -A. Knus, Quadratic and Hermitian Forms Over Rings, Grundlehren der Mathematischen Wissenschaften, vol. 294, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-75401-2.  Google Scholar

[19]

M. -A. Knus, A. Merkurjev, M. Rost and J. -P. Tignol, The Book of Involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/coll/044.  Google Scholar

[20]

T. Y. Lam, An introduction to real algebra, in Ordered Fields and real Algebraic Geometry (Boulder, Colo., 1983), Rocky Mountain J. Math., 14 (1984), 767–814. doi: 10.1216/RMJ-1984-14-4-767.  Google Scholar

[21]

T. Y. Lam, Introduction to Quadratic Forms Over Fields, Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2005.  Google Scholar

[22]

D. W. Lewis and J.-P. Tignol, On the signature of an involution, Arch. Math. (Basel), 60 (1993), 128-135.  doi: 10.1007/BF01199098.  Google Scholar

[23]

D. W. Lewis and T. Unger, A local-global principle for algebras with involution and Hermitian forms, Math. Z., 244 (2003), 469-477.  doi: 10.1007/s00209-003-0490-6.  Google Scholar

[24]

D. W. Lewis and T. Unger, Hermitian Morita theory: A matrix approach, Irish Math. Soc. Bull., 62 (2008), 37-41.   Google Scholar

[25]

F. Lorenz and J. Leicht, Die Primideale des Wittschen Ringes, Invent. Math., 10 (1970), 82-88.  doi: 10.1007/BF01402972.  Google Scholar

[26]

A. Pfister, Quadratische Formen in beliebigen Körpern, Invent. Math., 1 (1966), 116-132.  doi: 10.1007/BF01389724.  Google Scholar

[27]

A. Prestel, Quadratische Semi-Ordnungen und quadratische Formen, Math. Z., 133 (1973), 319-342.  doi: 10.1007/BF01177872.  Google Scholar

[28]

A. Prestel, Lectures on Formally Real Fields, Lecture Notes in Mathematics, vol. 1093, Springer-Verlag, Berlin, 1984. doi: 10.1007/BFb0101548.  Google Scholar

[29]

C. Procesi and M. Schacher, A non-commutative real Nullstellensatz and Hilbert's 17th problem, Ann. of Math. (2), 104 (1976), 395–406. doi: 10.2307/1970962.  Google Scholar

[30]

A. Quéguiner, Signature des involutions de deuxiéme espéce, Arch. Math. (Basel), 65 (1995), 408-412.  doi: 10.1007/BF01198071.  Google Scholar

[31]

W. Scharlau, Induction theorems and the structure of the Witt group, Invent. Math., 11 (1970), 37-44.  doi: 10.1007/BF01389804.  Google Scholar

[32]

W. Scharlau, Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften, vol. 270, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-69971-9.  Google Scholar

[33]

J. J. Sylvester, A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares, Philosophical Magazine, 4 (1852), 138-142.  doi: 10.1080/14786445208647087.  Google Scholar

[34]

J. -P. Tignol, Algebras with involution and classical groups, in European Congress of Mathematics, Vol. II (Budapest, 1996), Progr. Math., vol. 169, Birkhäuser, Basel, (1998), 244–258.  Google Scholar

[35]

A. Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. (N.S.), 24 (1961), 589-623.   Google Scholar

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