Gradings on associative algebras with involution and real forms of classical simple Lie algebras

We study gradings by abelian groups on associative algebras with involution over an arbitrary field. Of particular importance are the fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism (although not in a unique way). We classify up to equivalence the fine gradings on simple associative algebras with involution over the field of real numbers (or any real closed field) and, as a consequence, on the real forms of classical simple Lie algebras.

[1]  A. Elduque,et al.  Graded modules over classical simple Lie algebras with a grading , 2013, 1308.6089.

[2]  Adrián Rodrigo-Escudero.  Classification of division gradings on finite-dimensional simple real algebras , 2015, 1506.01552.

[3]  W. Waterhouse,et al.  Introduction to Affine Group Schemes , 1979 .

[4]  Group Gradings on Simple Lie Algebras of Type "A" , 2005, 0706.1045.

[5]  A. Elduque,et al.  Gradings on the Lie algebra D4 revisited , 2014, 1412.5076.

[6]  Y. Bahturin,et al.  Group Gradings on Matrix Algebras , 2002, Canadian Mathematical Bulletin.

[7]  A. Elduque,et al.  Cross products, automorphisms, and gradings , 2020, 2006.10324.

[8]  Jean-Pierre Tignol,et al.  The Book of Involutions , 1998 .

[9]  S. Sehgal,et al.  Group Gradings on Associative Algebras , 2001 .

[10]  Gradings on the simple real Lie algebras of types G 2 and D 4 , 2018, Journal of Algebra.

[11]  Y. Bahturin,et al.  Gradings on classical central simple real Lie algebras , 2017, Journal of Algebra.

[12]  Y. Bahturin,et al.  Graded polynomial identities as identities of universal algebras , 2018, Linear Algebra and its Applications.

[13]  C. Wall.  Graded Brauer Groups. , 1964 .

[14]  A. Elduque,et al.  Gradings on Simple Lie Algebras , 2013 .

[15]  J. Patera,et al.  On Lie gradings II , 1998 .

[16]  Y. Bahturin,et al.  Simple Graded Division Algebras over the Field of Real Numbers , 2015, 1506.00281.

[17]  Y. Bahturin,et al.  Gradings on simple Jordan and Lie algebras , 2005 .

[18]  Richard Elman,et al.  The Algebraic and Geometric Theory of Quadratic Forms , 2008 .

[19]  A. Elduque.  Fine gradings on simple classical Lie algebras , 2009, 0906.0655.

[20]  Y. Bahturin,et al.  Group Gradings on Associative Algebras with Involution , 2008, Canadian Mathematical Bulletin.

[21]  Y. Bahturin,et al.  Classification of group gradings on simple Lie algebras of types A, B, C and D☆ , 2009, 0908.0906.

[22]  J. Patera,et al.  On Lie gradings III. Gradings of the real forms of classical Lie algebras , 2000 .

[23]  Fine gradings on the Lie algebra , 2010 .

[24]  Classification of involutions on graded-division simple real algebras , 2017, 1707.05526.

[25]  Simeon Ivanov,et al.  Identities of Algebras and their Representations , 1994 .