Real algebra is a branch of algebra which studies formally real rings, i. e. commutative unital rings in which -1 is not a sum of squares. The most important invariant of a formally real ring A is its real spectrum Sper(A). The motivation for studying real algebra comes from semialgebraic geometry. Related areas of algebra are:

• ordered rings
• valuation theory
• quadratic forms

The main interest of the Slovenian real algebra group is noncommutative real algebra. An associative unital ring A is formally real if -1 is not a sum of permuted products of squares. Noncommutative formally real rings include:

• enveloping algebras of real Lie algebras,
• group rings of ordered groups,
• large classes of quantum groups (including quantum polynomials),
• large classes of rings of differential operators (including real Weyl algebras).

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