Details
| Original language | English |
|---|---|
| Pages (from-to) | 850-877 |
| Number of pages | 28 |
| Journal | Journal of algebra |
| Volume | 666 |
| Early online date | 5 Dec 2024 |
| Publication status | Published - 15 Mar 2025 |
Abstract
We prove a noncommutative real Nullstellensatz for 2-step nilpotent Lie algebras that extends the classical, commutative real Nullstellensatz as follows: Instead of the real polynomial algebra R[x1,…,xd] we consider the universal enveloping ⁎-algebra of a 2-step nilpotent real Lie algebra (i.e. the universal enveloping algebra of its complexification with the canonical ⁎-involution). Evaluation at points of Rd is then generalized to evaluation through integrable ⁎-representations, which in this case are equivalent to filtered ⁎-algebra morphisms from the universal enveloping ⁎-algebra to a Weyl algebra. Our Nullstellensatz characterizes the common kernels of a set of such ⁎-algebra morphisms as the real ideals of the universal enveloping ⁎-algebra.
Keywords
- -Algebra, -Representations, Non-commutative real algebraic geometry, Nullstellensatz, Universal enveloping algebra
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Journal of algebra, Vol. 666, 15.03.2025, p. 850-877.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Real Nullstellensatz for 2-step nilpotent Lie algebras
AU - Schmitt, Philipp
AU - Schötz, Matthias
N1 - Publisher Copyright: © 2024 The Authors
PY - 2025/3/15
Y1 - 2025/3/15
N2 - We prove a noncommutative real Nullstellensatz for 2-step nilpotent Lie algebras that extends the classical, commutative real Nullstellensatz as follows: Instead of the real polynomial algebra R[x1,…,xd] we consider the universal enveloping ⁎-algebra of a 2-step nilpotent real Lie algebra (i.e. the universal enveloping algebra of its complexification with the canonical ⁎-involution). Evaluation at points of Rd is then generalized to evaluation through integrable ⁎-representations, which in this case are equivalent to filtered ⁎-algebra morphisms from the universal enveloping ⁎-algebra to a Weyl algebra. Our Nullstellensatz characterizes the common kernels of a set of such ⁎-algebra morphisms as the real ideals of the universal enveloping ⁎-algebra.
AB - We prove a noncommutative real Nullstellensatz for 2-step nilpotent Lie algebras that extends the classical, commutative real Nullstellensatz as follows: Instead of the real polynomial algebra R[x1,…,xd] we consider the universal enveloping ⁎-algebra of a 2-step nilpotent real Lie algebra (i.e. the universal enveloping algebra of its complexification with the canonical ⁎-involution). Evaluation at points of Rd is then generalized to evaluation through integrable ⁎-representations, which in this case are equivalent to filtered ⁎-algebra morphisms from the universal enveloping ⁎-algebra to a Weyl algebra. Our Nullstellensatz characterizes the common kernels of a set of such ⁎-algebra morphisms as the real ideals of the universal enveloping ⁎-algebra.
KW - -Algebra
KW - -Representations
KW - Non-commutative real algebraic geometry
KW - Nullstellensatz
KW - Universal enveloping algebra
UR - http://www.scopus.com/inward/record.url?scp=85212319470&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2403.06773
DO - 10.48550/arXiv.2403.06773
M3 - Article
AN - SCOPUS:85212319470
VL - 666
SP - 850
EP - 877
JO - Journal of algebra
JF - Journal of algebra
SN - 0021-8693
ER -