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In mathematics, a Lie algebra is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebras. We write , and for all . If the lower central series eventually arrives at the zero subalgebra, then the Lie algebra is called nilpotent. The lower central series for Lie algebras is ...
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve ...
nilpotent matrix is always less than or equal to. n {\displaystyle n} For example, every. 2 × 2 {\displaystyle 2\times 2} nilpotent matrix squares to zero. The determinant and trace of a nilpotent matrix are always zero. Consequently, a nilpotent matrix cannot be invertible. The only nilpotent diagonalizable matrix is the zero matrix.
For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algebras), it means that in some basis the ring consists entirely of upper triangular matrices with constant diagonal. This article uses the language of group theory; analogous terms are used for Lie algebras.
An example of a solvable Lie algebra is the space of upper-triangular matrices in (); this is not nilpotent when . An example of a nilpotent Lie algebra is the space u n {\displaystyle {\mathfrak {u}}_{n}} of strictly upper-triangular matrices in g l ( n ) {\displaystyle {\mathfrak {gl}}(n)} ; this is not abelian when n ≥ 3 {\displaystyle n ...
It is straightforward to check that the group U n is nilpotent. As a result, every unipotent group scheme is nilpotent. A linear algebraic group G over a field k is unipotent if and only if every element of (¯) is unipotent. [13] The group B n of upper-triangular matrices in GL(n) is a semidirect product
If a, b, c, are real numbers (in the ring R) then one has the continuous Heisenberg group H 3 (R).. It is a nilpotent real Lie group of dimension 3.. In addition to the representation as real 3×3 matrices, the continuous Heisenberg group also has several different representations in terms of function spaces.
Any nilpotent Lie algebra is a fortiori solvable but the converse is not true. ... called the Lie algebra of strictly upper triangular matrices. In addition, ...