Suppose that \mathfrak{g} is an n-dimensional semisimple Lie algebra. Let
\{X_i\}_{i=1}^n
be any base of \mathfrak{g}, and
\{X^i\}_{i=1}^n
be the bifold base of \mathfrak{g} with account to a anchored invariant bilinear anatomy (e.g. the Killing form) on \mathfrak{g}. The Casimir aspect Ω is an aspect of the accepted enveloping algebra U(\mathfrak{g}) accustomed by the formula
\Omega = \sum_{i=1}^n X_i X^i.
Although the analogue of the Casimir aspect refers to a accurate best of base in the Lie algebra, it is accessible to appearance that the consistent aspect Ω is absolute of this choice. Moreover, the invariance of the bilinear anatomy acclimated in the analogue implies that the Casimir aspect commutes with all elements of the Lie algebra \mathfrak{g}, and appropriately lies in the centermost of the accepted enveloping algebra U(\mathfrak{g}).
Given any representation ρ of \mathfrak{g} on a agent amplitude V, possibly infinite-dimensional, the agnate Casimir invariant is ρ(Ω), the beeline abettor on V accustomed by the formula
\rho(\Omega) = \sum_{i=1}^n \rho(X_i)\rho(X^i).
A appropriate case of this architecture plays an important role in cogwheel geometry and all-around analysis. Suppose that a affiliated Lie accumulation G with the Lie algebra \mathfrak{g} acts on a differentiable assorted M, again elements of \mathfrak{g} are represented by aboriginal adjustment cogwheel operators on M. The representation ρ is on the amplitude of bland functions on M. In this bearings the Casimir invariant is the G-invariant additional adjustment cogwheel abettor on M authentic by the aloft formula.
More accepted Casimir invariants may additionally be defined, frequently occurring in the abstraction of pseudo-differential operators in Fredholm theory.
\{X_i\}_{i=1}^n
be any base of \mathfrak{g}, and
\{X^i\}_{i=1}^n
be the bifold base of \mathfrak{g} with account to a anchored invariant bilinear anatomy (e.g. the Killing form) on \mathfrak{g}. The Casimir aspect Ω is an aspect of the accepted enveloping algebra U(\mathfrak{g}) accustomed by the formula
\Omega = \sum_{i=1}^n X_i X^i.
Although the analogue of the Casimir aspect refers to a accurate best of base in the Lie algebra, it is accessible to appearance that the consistent aspect Ω is absolute of this choice. Moreover, the invariance of the bilinear anatomy acclimated in the analogue implies that the Casimir aspect commutes with all elements of the Lie algebra \mathfrak{g}, and appropriately lies in the centermost of the accepted enveloping algebra U(\mathfrak{g}).
Given any representation ρ of \mathfrak{g} on a agent amplitude V, possibly infinite-dimensional, the agnate Casimir invariant is ρ(Ω), the beeline abettor on V accustomed by the formula
\rho(\Omega) = \sum_{i=1}^n \rho(X_i)\rho(X^i).
A appropriate case of this architecture plays an important role in cogwheel geometry and all-around analysis. Suppose that a affiliated Lie accumulation G with the Lie algebra \mathfrak{g} acts on a differentiable assorted M, again elements of \mathfrak{g} are represented by aboriginal adjustment cogwheel operators on M. The representation ρ is on the amplitude of bland functions on M. In this bearings the Casimir invariant is the G-invariant additional adjustment cogwheel abettor on M authentic by the aloft formula.
More accepted Casimir invariants may additionally be defined, frequently occurring in the abstraction of pseudo-differential operators in Fredholm theory.
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